A Theoretical Approach: Effects of Mn Substitution in Cobalt Ferrite

This paper reports on the effect of Mn substitution in cobalt ferrite to explore the probable correlation among the structural, magnetic, and magneto-mechanical properties by a theoretical approach. Three compositions of Mn doped Cobalt ferrites at different Mn concentration (x) = 0.125, 0.25, 0.375, 0.5 have been undertaken for their analytical study to understand the correlation among the aforesaid properties. In this approach, an empirical equation has been formulated based on idealistic cation distribution in tetrahedral and octahedral sites of cobalt ferrite at room temperature. The hopping lengths and bond lengths have also been estimated using the corresponding Stanley’s equations in idealistic condition. The estimated lattice constant is found to decrease and effective magnetic moment μferri to increase with the Mn content, substituted for Co in the octahedral site due to increased A-B interactions. This increasing effect of Mn content in cobalt ferrite may be significant to the tunability of the Curie temperature, TC and may have an influence on superparamagnetism (SPM). On the other hand, the compositions where Mn substituted for Fe may increase the porosity due to their increased bond lengths with Mn content and thus may optimize them for applications in the environmental (gas) sensors. However, the analysis of the predicted effects of Mn and correlation thereon is completely based on the theoretical approach and thereby need experimental verification to confirm and supplement them.


Introduction
Cobalt ferrites are reported as the best-known examples of the hard ferrite materials because of their excellent chemical stability, mechanical hardness, reasonable saturation magnetization and high magnetocrystalline anisotropy [1]. An extremely wide variety of total solid solution enables this cobalt ferrite to be strongly modified keeping the ferrite structure almost unchanged, which leads to a series of investigations on Mn doped cobalt ferrites. By the time, intensive researches towards the discovery and development of cobalt ferrite nanoparticles with doped and composite kind have been made them possible to be used in diversified fields of application such as electronic devices, ferrofluids, drug delivery system, magnetic resonance imaging, microwave devices and high-density information storage. These applications are mostly based on magnetic and electrical behaviors and their alteration or tuning, depending on the ionic radii of doped atoms in the host lattice and their particle size. Recently, tuning of fundamental magnetic properties of doped cobalt ferrite by adjusting dopant content, sintering temperature and their size have received renewed attention across the world to optimize them both in sensor and high frequency applications. A substantial and almost linear decrease of the Curie temperature T C but the modest decline of magnetization is observed with Mn content, substituted for Fe [2][3][4][5][6]. The dependence of structural and magnetic properties on dopant content, sintering temperature has been studied [7]. But the tunability to adjust dopant content to an optimum value of magnetic properties at which the operating temperature and external magnetic field can cause to flip the magnetic transition from one state to another is not conspicuous, which demands a theoretical study to predict at room temperature. Lattice constants of Co 1-x  other structural parameters such as bond lengths, hopping lengths, and density have also been estimated from the respective lattice constants. The obtained values of those parameters are used for analytical studies of structural and magnetic properties, and effects thereon. Keeping the above in view, the objective of this paper is to have comprehensive theoretical studies on the effects of Mn substitution in Cobalt ferrites for understanding their tenable behavior and tuning of fundamental magnetic properties by adjusting Mn content for optimizing them in advanced applications.

Structure
Bulk CoFe 2 O 4 possesses an inverse spinel structure in which ideally half of the trivalent ferric cations (Fe 3+ ) are positioned on A site and the other half Fe 3+ cations and all divalent cobalt cations (Co 2+ ) on B site as depicted in Figure 1. An inverse spinel unit cell is made up of eight face-centered cubic (fcc) cells of oxygen ions in the configuration 2×2×2. So it is a big structure consisting of 32 oxygen ions, 8 A cations and 16 B cations. Accordingly, CoFe 2 O 4 has also 8 sublattices (molecules) in its unit cell with 32 anions and 24 cations, which are ideally distributed at room temperature as shown in Table 1.   Using this equation-1, lattice constant 'a' for x =0.125, 025, 0.375 and 0.500 of each composition has been estimated. The value of a, for x = 0 found to be~8.3702 Å , which is comparable to~8.3802Å as obtained from XRD pattern [8,14] but ~8.0966 Å as estimated from Vegard's formula of the same compound. All other structural parameters like tetrahedral and octahedral bond lengths and hopping lengths have been estimated by respective Stanley's equation = , density has also been estimated for bulk CoFe 2 O 4 ferrite with a mass of 234.63 g mol -1 . The estimated density (usually known as X-ray density) is 5.315 gcm -3 and found to be almost in agreement with the literature value [4,8]. The porosity is defined as@ where, the bulk density B C depends on the dimension of the sample pellet, which can be estimated by the formula:B C F () > G ⁄ , here, m is the mass (kg), r is the radius (m) and h is the height of the sample. For a typical disk-shaped sample of 20mg each (7.6 mm diameter and 1 mm thickness), the estimated bulk density found to be~4.41g cm -3 , which leads to having the porosity of this sample~17.03%. However, the values of these structural parameters as estimated above may vary owing to redistribution of cations among the A and B sites depending on dopant (Mn) content, synthesizing technique, calcination and sintering temperature, and also on size and shape of particles when they reduce to the nanoscale.

Magnetic Moment
Below T C , ferric ions, Fe 3+ in A site have aligned magnetic moment (parallel spins) resulting ferromagnetism and produces net magnetic moment, while the cobalt ions Co 2+ and ferric ions Fe 3+ in B site have anti-parallel alignment that results in antiferromagnetism and balances or cancels out net magnetic moment depending upon the number of unpaired spins as reported in case of magnetite [9]. Accordingly, the situation may be hypothetically depicted in Figure 2. Hence, the exchange interaction between A and B sites lead CoFe 2 O 4 ferrite to possess ferrimagnetic behavior due to magnetic moments of bivalent cations M 2+ (Co 2+ ) in the B position as reported in the literature [9]. Accordingly, the effective magnetic moment I JKLLM for inverse spinel structure may be estimated by the following equation, modified from the formula of normal spinel structure [10] as: Where, x represents the substitution level of Mn content in the parent sample of cobalt ferrite and I ?U NO 4.898I the spin alone magnetic moment of Mn 2+ . Notable that, the measured value may be different depending upon the ionic redistribution during synthesizing and sintering temperature as reported in various literature.

Size Effect on Magnetization and SuperPara
Magnetism (SPM) Usually, bulk CoFe 2 O 4 ferrite is of multi-domains like magnetite, Fe 2 O 4 . But when the particle size of cobalt ferrites reduces to 20 nm or below, their multi-domains turn into a single domain particle. As a result, the surface-volume ratio found to increase, which in turn increases the internal pressure and reduces the external pressure on the particles. According to Laplace, this difference in pressure variables (internal and external) is inversely proportional to the radius of the particle and directly proportional to the surface tension (as mathematically expressed by an equation @ MUVKLU=W ! @ KDVKLU=W @ W=XW=YK >Z L ) owing to surface chemistry. Because of reduced external pressure, the particles get swell and thus causes the volume expansion, which ultimate to increase the lattice constant without changing the crystal structure of the unit cell as reported for Fe 2 O 4 . This change of a found to be insignificant in their structural properties but may have significance in the tuning of magnetic properties. With the decrease of particle size, the amount of exchange coupled spins, which resists the spontaneous magnetic reorientation, may decrease and tending towards paramagnetic or superparamagnetic magnetization as reported for Fe 2 O 4 [11]. This exchange coupling of spins expected to depend on the hopping lengths (L) and [L], which indicates the correlation between saturation magnetization and lattice constant' a'. So, the paramagnetic behavior in bulk CoFe 2 O 4 ferrites and superparamagnetic behavior in their nanoparticles may be made tunable with the lattice parameter and in turn dopant (Mn) content. Superparamagnetic behavior is the exclusive property of magnetic nanoparticles and found to arise when the thermal energy fluctuations or an applied field can easily move magnetic moments away from the easy axis (The preferred crystallographic axes for the magnetic moment are to point along). Then each particle behaves like paramagnetic atom but with giant magnetic moment owing to sustained magnetic order [10]. The giant magnetic moment may be thought of as the determining factor of superparamagnetism in CoFe 2 O 4 ferries, which may depend on the hopping lengths and in turn, on lattice constant 'a'.

Structural Properties
The estimated values of lattice constant a, tetrahedral hopping length (L), octahedral hopping length [ [12]. It implies that there is a correlation between lattice constant and ionic radius [13], which may be expressed by the empirical equation as: R [ ± FR, where R is the lattice constant for any value ofx (Mn content), [ the lattice constant at x =0 (for parent sample) and m is the slope to determine the rate of change of a with x. Notable here, a similar equation may be applied for hopping lengths and bond lengths as well. However, the relation as established here between the lattice constant and the concentration of dopant (Mn) is completely based on a theoretical approach. This relation may not be the same in the experimental approach due to the ionic redistribution between A and B sites depending on synthesizing technique, calcination temperature, and particle size as mentioned above. So, a series of investigations is required under different conditions.

Magnetic Properties
Figure6 shows the variation of the estimated effective magnetic moment, I JKLLM as a function of Mn content (x). The increase of a, on the other hand, increases (L) and [L], which may reduce the super-exchange interactions between magnetic ions of A and B sites at room temperature and resulting this decrease in the effective magnetic moment I JKLLM for CoMn x Fe 2-x O 4 with Mn content(x) and thereby leading to decrease of saturation magnetization, M s as reported [12]. Conversely, the decrease of a causes shortening in (L) and [L] for Co 1-x Mn x Fe 2 O 4. , which may enhance the super-exchange interactions between magnetic ions of A and B sites at the room temperature, which results in the increase of I JKLLM with Mn content(x) and therefore may leading to increase M s and associated H c . The increasing trend in I JKLLM is the signature of ferri-to-ferro and decreasing trend of ferri-to-para magnetic phase transitions for Co substituted and Fe substituted compositions respectively. But I JKLLM is found to increase for Co 1+x Mn x Fe 2-x O 4 as shown in Figure 6 because of concurrent addition of Co along with Mn substitution for Fe, which may lead to supersede the effect of longer hopping lengths through the formation of new probable phases due to more influence of inherent ferromagnetic ordering of Co in this system. This is found in agreement with the recently published experimental result [10,14]. Thus this eventual variation ofI JKLLM leads to predict that thermally fluctuated magnetic ions require more thermal energy ( κ] ) inCo 1-x Mn x Fe 2 O 4 (where Mn substituted for Co) than inCoMn x Fe 2-x O 4 (where Mn substituted for Fe) to decrease spontaneous magnetization, which is indicative to make T C tunable by adjusting Mn content and almost in agreement with published experimental result [13]. As the particle size reduces to the nanoscale (typically <20nm) range, 'a' increases with the Laplace pressure difference [11] and eventually may decrease I JKLLM due to increase of hopping length. Hence, counterbalance of I JKLLM may be possible inCo 1-x Mn x Fe 2 O 4 by adjusting Mn content, which may lead to tune T C or T N at an optimum point for making superparamagnetism (SPM)tunable closer to the room temperature.

Magneto-Mechanical Properties
The magneto-mechanical property includes the magnetoelastic and magnetostriction properties of magnetic materials. Magnetoelastic property is also known as inverse magnetostriction, which refers to the change of magnetization with mechanical stress. The stress may develop from strain during synthesizing the material. This strain is closely related to the bond length in ferromagnetic materials. As such the increase of bond lengths both in A and B sites may decrease

A-A and B-B interactions in CoMn x Fe 2-x O 4 but increase A-A
and B-B interactions with the decrease of bond lengths inCo 1-x Mn x Fe 2 O 4 and resulting increase in the magnetization of the later. The shape or dimension of the material may have a phenomenological relationship with the bulk density, which in turn a measure of porosity of the ferromagnetic material. The longer is the bond length, the higher is the porosity and vice versa. Accordingly, the lower is the magnetization, the higher is the porosity and vice versa. As such, CoMn x Fe 2-x O 4 may exhibit higher porosity compared to Co 1-x Mn x Fe 2 O 4 as reported [15]. But in Co 1+x Mn x Fe 2-x O 4 , higher porosity might be observed due to the probable formation of new phases during synthesizing and therefore more suitable for magneto-mechanical sensor applications, which is also a subject to further experimental investigation. The imaginary part of magnetic susceptibility is responsible for converting magnetic energy into heat in a fluid when exposed to an alternating field, which is governed by the relaxation process. The relaxation time, τ determines the maximum achievable loss power as reported [16]. Apart from the field parameter, the power loss is related to the susceptibility of the materials of nanoparticles, which again depends on saturation magnetization, particle volume, and field amplitude. With required particle size, this saturation magnetization might be controlled to an optimum value of temperature by adjusting   4 . The increase in lattice constant according to Laplace pressure difference is also expected in all three systems undertaken for analytical study when the size of the particle reduces to the nanoscale. For Co 1-x Mn x Fe 2 O 4 , it might be possible to control and tune M s and T C by adjusting Mn content and the particle size depending on the synthesizing method. This is a unique and significant property for Co 1-x Mn x Fe 2 O 4 , which may make it suitable for advanced applications like magnetic hyperthermia etc. whereas in the other two compositions, reducing of particle size to nanoscale may increase the porosity due to lengthening of A-O and B-O bond lengths by affecting on bulk density and may make them suitable for environmental sensors. However, since the analysis is completely based on theoretical aspects, so further investigations with synthesized samples of these compositions are required to supplement and confirm the findings and results to assess suitability for advanced applications in the days to come.