Magnetic Nanoparticles with Core Shell: Macroscopic Model and Coercive Field

Abstract

There exists nonhomogeneous structure in mechanosynthesized nanoparticles consisting of an ordered core surrounded by the shell. The shell may be structurally and magnetically disordered, or it may be ordered. These nanoparticles are found to be roughly spherical. We formulate the macroscopic model for the description of magnetic properties of nanoparticles with core-shell structure. The case of spheroids oriented in the same direction of polar axes is considered. There exits two coercive fields: 𝐻_𝑐𝛾 and 𝐻_𝑐𝜓. They are depending on quantities o and x, where 𝑥=𝑉₂/(𝑉₁−2𝑉₂), and 𝑜=𝐼_𝐹/𝐼_𝑆 , here 𝑉₁ is the volume of the nanoparticle, and 𝑉₂ is the volume of the core. 𝐼_𝐹 is the magnetization of the core, and 𝐼_𝑆 is the magnetization of the shell. There exists a transition from one type of the coercive field to another, changing x and o. We have found qualitative explanation of observed enhancement of coercive field and of its d-dependence in NiFe₂O₄ nanoparticles.

1. Introduction

Ferromagnetic and (ferri-) or antiferromagnetic nanoparticles, mechanochemically prepared, have a shell structuredifferent from the core structure. The shell has mostly disordered structural and magnetic properties, while the core is magnetically ordered. In this paper, we formulate a model describing interaction of ellipsoid with ferromagnetic/antiferro- (ferri-)magnetic/ordering with shell (ellipsoid part which is not in the volume of the core ellipsoid). We study here the description of coercive fields of magnetic nanoparticles with spheroid shape with polar axes in the same direction for both spheroids. Due to fact that nanoparticles are small (smaller than 100A^∘), we consider a single domain magnetic system. While porosity of ceramics prepared from such nanoparticles is in the range up to nearly 33 percents, and such pores do not disappear during further sintering, they undergo coalescence together with grain growth. Only subsurface pores have the possibility to go out in the surface, see in [1]. For the case of a spheroid with a pore, it can be easily shown that the pore may enhance the coercive field, or the coercive field is smaller than the bulk coercive field.

Mechanosynthesized NiFe₂O₄ nanocrystalline nanoparticles were studied in [2]. There exists nonhomogeneous structure in these nanoparticles consisting of an ordered core surrounded by a disordered grain boundary-surface region. The core has a fully inverse spinel structure with a collinear spin alignment. The surface layer (shell) is structurally and magnetically disordered due to nearly random distribution of cations and the canted spin structure. It exhibits a reduced nonsaturating magnetization, an enhanced coercivity, and a shifted hysteresis loop. The thickness of the shell of about 1 nm is estimated from size-dependent magnetization measurements. The shell thickness of crystallite radius ratio is nearly 23 percent. The nanoparticles are found to be roughly spherical with the core-shell structure. In [2], it was shown that the core part of the nanoparticle is structurally ordered and of nearly sphere shape, and the shell part is disordered structurally. The spins in the shell region are found to be canted (28^∘ and 40^∘ in magnetic sublattices), and the 𝜆 parameter (parameter which in spinels describes how cations are distributed in spinels in their sublattices) equals nearly to 0.67, in the core the 𝜆 parameter is equal to 1. It is observed that an enhanced magnetic hardness, that is, the coercive field of the material has the value 𝐻_𝐶≈0.35 T which is about 35 times larger than that for the bulk material 𝐻_𝐶≈0.01 T. The field-cooled hysteresis loop is not symmetrical and is shifted to the left, and the shift is 𝛿𝐻_𝐶≈0.03 T. Such a shift and an asymmetric hysteresis loop were explained [3, 4] in the exchange model to be present due to the exchange coupling between the collinear spins in the core and canted spins in the shell. The saturation magnetization is dependent on the diameter. The growth of NiFe₂O₄ nanoparticle is accompanied by a decrease of coercivity field value.

The nonuniform core-shell structure of nanoparticles has been reported in MgFe₂O₄ [5] and in ball-milled nanocrystalline LiNbO₃ [6]. Mechanochemically prepared nanoparticles MgFe₂O₄ [5] have an unusual property: enhanced magnetization. They have core-shell structure. Inner core exhibits a partly inverse structure with a Neel-type collinear spin alignment. The shell has a nonequilibrium cation distribution and a canted spin arrangement. Although spin canting effects tend to reduce the magnetic moment, the magnetization enhancement is attributed by authors to the nearly random distribution of magnetic cations in the shell region. On heating above 623^∘K, the mechanochemical MgFe₂O₄ relaxes to the magnetic state which is similar to the bulk one [5]. The thickness 𝑡 of the shell assuming the spherical shape is estimated to be 𝑡≈1 nm and is comparable to the lattice constant.

Nanocomposite CuFe₂O₄/CeO₂ [7] has a ferrimagnetic behaviour. The nanoparticle size of CuFe₂O₄/CeO₂ decreases with increasing concentration of CeO₂. Added CeO₂ is dissolved in the CuFe₂O₄, or it is coated on the surface of CuFe₂O₄ nanoparticles [8]. There is then a surface layer-shell of CeO₂. This form of core (shell) structure is of different type, as in NiFe₂O₄. Nevertheless, also in this case there is a core and a shell, and they have magnetic properties [9]. It is possible to model such materials as materials in which there is a core-shell structure similar to that in nanoparticles prepared by mechanochemical preparation. Modelling of CeO_2−𝑥 in the LDA+U formalism has been used to account for the strong on-site Coulomb repulsion among the localized Ce 4f electrons. Results show that by choosing an appropriate U, it is possible to consistently describe structural, thermodynamic, and electronic properties oCeO_2−𝑥f, which enables modelling of redox processes involving ceria-based materials. We may expect that similar nanocomposites may be prepared using CeAl₂ as a shell. We have modelled CeAl₂ nanoparticles [10] by 1d Hubbard model. Surface valence phase transition between magnetic and nonmagnetic states was studied by us [10] in these nanoparticles. Coated CeAl₂ may form the core-shell structure. The surface properties of the shell may be described by our model [10] (surface quantum phase transition) and also by the model of magnetic nanoparticles with core-shell in which the coercive field is calculated here.

There is different surface anisotropy for the cubic and spherical nanoparticles [11]. Low temperature properties: saturation magnetization, coercivity, and spin canting are similar, while the blocking temperature 𝑇_𝐵 (temperature at which a maximum of susceptibility in nanoparticle is achieved) is different for 𝛾-Fe₂O₃ for the cubic and spherical shape. Theoretical calculations and experimental magnetic studies for nanoparticles of different shapes are [12–17] scare, see in [11]. For cobalt ferrite CoFe₂O₄, it was found that blocking temperature 𝑇_𝐵 and the saturation magnetization are independent of the shape, and the value of the coercivity field 𝐻_𝐶 is larger for spherical shape than for the cubic one. In cobalt ferrite there is a larger magnetic anisotropy. For 𝛾-Fe₂O₃, intrinsic magnetocrystalline anisotropy is small. Monte Carlo simulations are based on Heisenberg model for exchange interaction between spins in the core and in the shell experiencing different anisotropies [11]. The surface thickness is taken to be equal to one lattice constant. Experimentally it was found that the spherical nanoparticles are more ordered than cubic nanoparticles. Low temperature measurements show however that their microscopical magnetic structure properties are similar, indicating similar crystalline properties. The behaviour is ascribed to the effects of morphology on the effective surface anisotropy and the small value of intrinsic anisotropy of 𝛾-Fe₂O₃. The magnetic state in the core is ferrimagnetic and in the surface layer is of the spin-glass type. We will take into account the morphology effects in our paper by considering the ellipsoidal shape of nanoparticles. This shape enables us to describe the spherical shape of nanoparticles, the disc-like shape, or rod-like shape for extremal values of lengths axes in the ellipsoid. The case of cubic nanoparticles can be easily considered in the same way.

In the next section, we formulate the core-shell model for description of magnetic nanoparticles. In the third section, the Model Free Energy is found. In the next sections we study the free energy for spheroids and spheres oriented in the same direction. Experimental behaviour of coercive field dependence on diameter of nanoparticle is compared with theoretical predictions. We have found that there exist two coercive fields 𝐻_𝑐𝛾 and 𝐻_𝑐𝜓. They are depending on quantities 𝑜 and 𝑥, defined by 𝑥=𝑉₂/(𝑉₁−2𝑉₂), and by 𝑜=𝐼_𝐹/𝐼_𝑆, here 𝑉₁ is the volume of the larger ellipsoid, and 𝑉₂ is the volume of the smaller ellipsoid. 𝐼_𝐹 is the magnetization of the volume where smaller ellipsoid occurs, and 𝐼_𝑆 is the magnetization of the shell. There exists a transition from one type of the coercive field to another changing 𝑥 and 𝑜 within range of their possible values. Critical values 𝑥_𝑐 and 𝑜_𝑐 for which values of both coercive fields are equal can be found. For spheroidal shapes of ellipsoids for 𝑥>𝑥_𝑐, the field 𝐻_𝑐𝛾 is the coercive field which may be observed in experiments. For volumes 𝑉₁ and 𝑉₂ for which 𝑥<𝑥_𝑐, the field 𝐻_𝑐𝜓 is the coercive field and should be observed. The ratio of the coercive field value for nanoparticle to the bulk coercive field value is different for 𝑥>𝑥_𝑐 and for 𝑥<𝑥_𝑐. Enhancement of the coercive field may occur in nanoparticles changing the diameter 𝑑.

2. Core-Shell Model

As it was shown above, there are many nanoparticle systems in which there exists core-shell structure. They are found in nanoparticles prepared, for example, by mechanochemical method and also by other methods. For an overview of the mechanochemical methods see in [18].

We will describe such nanoparticles by the following Core-Shell model. Due to small diameter of nanoparticles we will assume that they are magnetically monodomain nanoparticles. This is true for nanoparticles with diameter less than 100A^∘ [19]. The shape of nanoparticles described above is of the sphere type, mostly, there are deviations from this shape in the core and in the shell. To describe these observations, we assume that the nanoparticle has an ellipsoid shape in general. The ellipsoid shape is described by three parameters 𝑎, 𝑏, and 𝑐 (lengths of axes). To describe existence of the core and of the shell, we consider another ellipsoidal “nanoparticle” which is located inside the first one. The centre of that another “nanoparticle” may have almost any position inside the first ellipsoid, assuming that the second ellipsoid is as a whole inside the first ones volume.

We will assume that main axes in both ellipsoids coincide with the polar (light magnetic) crystallographic axis of the nanoparticle. In general, these two axes (light and main axes in both ellipsoids) are oriented in different directions. We will denote the angle between polar axis in both by 𝜃. The case of zero angle, 𝜃=0, corresponds to parallel polar axes. The case of angle 𝜃=𝜋/2 corresponds to perpendicular polar axes. Here in this paper we consider the case of 𝜃=0 angle which corresponds to the shell and the oriented core in the same direction.

To describe core-shell magnetic structure we will assume that the larger ellipsoid has a homogeneous magnetization 𝐈_𝐒, and that the smaller ellipsoid has a homogeneous magnetization 𝐈_𝐅−𝐈_𝐒. Thus, there is a total magnetization 𝐈_𝐅−𝐈_𝐒+𝐈_𝐒=𝐈_𝐅 in the volume where smaller ellipsoid is localized in the larger one, see Figure 1. The magnetic structure of the material in core and in shell may be ferromagnetic or ferri- (antiferro-) magnetic. We will assume that sublattice magnetizations are coupled strongly, and that then we may consider magnetizations 𝐈_𝐅 and 𝐈_𝐒 (resulting magnetization) only. We will neglect also surface magnetic phenomena. The magnetic field around an ellipsoid of the sphere shape with homogeneous magnetization 𝐈_𝐅−𝐈_𝐒 is given in our case by the dependence [19](𝐼_𝐹−𝐼_𝑆)4𝜋3(𝑑/2)³𝑟³(2𝐞_𝑟cos(𝜃)+𝐞_𝜌sin(𝜃)), (1) where 𝑟 is the distance from the centre of the sphere, and 𝑑 is a diameter of the sphere. Here, 𝐞_𝑟 is a unit vector in the direction of radius directing from the centre, and 𝐞_𝜌 is a unit vector in the direction perpendicular to the vector 𝐞_𝑟. For larger distances from the smaller ellipsoid surface, we may neglect this contribution. This approximation corresponds to smaller inner ellipsoid volume with respect to the larger ones volume.

Figure 1 

Core-shell magnetic structure. The larger ellipsoid has a homogeneous magnetization 𝐈𝐒, and the smaller ellipsoid has a homogeneous magnetization 𝐈𝐅−𝐈𝐒. There is a total magnetization 𝐈𝐅−𝐈𝐒+𝐈𝐒=𝐈𝐅 in the volume where smaller ellipsoid (core) is present in the larger one. The angle of the magnetization of the shell volume with the direction of the main axis is 𝜓, and the angle of the magnetization of the core with the direction of the main axis is 𝛾.

Let us consider first the case in which both ellipsoids are spheres. Then we will consider the case of spheroids (ellipsoids with the same two main axes lengths). The angle of the magnetization of the first (larger) ellipsoid with the direction of the main axis is 𝜓, and the angle of the magnetization of the second (smaller) ellipsoid with the direction of the main axis is 𝛾, see Figure 1.

3. Model-Free Energy

Interaction of the core magnetization with the shell magnetization will be described by the effective-field method. We will assume that an effective magnetic field of the larger ellipsoid is acting on the smaller ellipsoid magnetization. On the other hand, the effective magnetic field of the smaller ellipsoid outside and inside this ellipsoid which is acting on the larger ellipsoid magnetization will be neglected.

The magnetic anisotropy of the shell structure may be described by the unit vector 𝐞_𝐒, and the contribution to the free energy is of the form 𝐾_𝑆(𝐞_𝐒⋅𝐈_𝐒)², where 𝐾_𝑆 is the anisotropy constant for the shell. The anisotropy of the core may be described by the unit vector 𝐞_𝐂, and the contribution to the free energy is of the form 𝐾_𝐶(𝐞_𝐂⋅𝐈_𝐅)², where 𝐾_𝐶 is the magnetic anisotropy constant for the core. When the structure is random then the anisotropy is random, and we can use mean anisotropy of the magnetic structure in the shell. Random anisotropy in the shell (for the constant 𝜆 unequal to one) may be described by random orientation of the vector 𝐞 and by the different values of the constant 𝐾. We will neglect magnetic anisotropy in this paper as the magnetic anisotropy constants 𝐾_𝑆 and 𝐾_𝐶 are small in NiFe₂O₄.

We will assume that the nanoparticle is monodomain, and the free energy of the magnetic system will contain contributions similar to those for the ellipsoid with an ellipsoidal hollow [19]. The inner ellipsoid is corresponding to the hollow in [19], however we consider this ellipsoid as a magnetic system with the magnetization 𝐈_𝐅−𝐈_𝐒 then the total magnetization in the volume 𝑉₂ of smaller ellipsoid is 𝐈_𝐅−𝐈_𝐒+𝐈_𝐒=𝐈_𝐅. The volume of the larger ellipsoid is 𝑉₁.

Let us denote susceptibilities for larger and smaller ellipsoids as 𝜒⁰_𝑆 and 𝜒⁰_𝐹 respectively, they are defined as𝜒⁰_𝑆=1𝑁_𝑎𝑡²_a+𝑁_𝑏𝑡²_𝑏+𝑁_𝑐𝑡²_𝑐, (2) where 𝑁_𝑖 is a demagnetization factor for the direction 𝑖(𝑖=𝑎,𝑏,𝑐) for the larger ellipsoid, and where 𝑡_𝑖 is a directional cosine of the magnetization vector 𝐈_𝐒 for the direction 𝑖, and𝜒⁰_𝐹=1𝑁_𝛼𝑡²_𝛼+𝑁_𝛽𝑡²_𝛽+𝑁_𝛾𝑡²_𝛾, (3) where 𝑁_𝑗 is a demagnetization factor for the direction 𝑗(𝑗=𝛼,𝛽,𝜖) for the smaller ellipsoid, and where 𝑡_𝑗 is a directional cosine of the magnetization vector 𝐈_𝐒 for the direction 𝑗.

The free energy 𝐹_𝑆 of the larger ellipsoid is𝐹_𝑆=12𝑉₁1𝜒⁰_𝑆𝐈_𝐒². (4)

The free energy 𝐹_𝐹 of the smaller ellipsoid is𝐹_𝐹=12𝑉₂1𝜒⁰_𝐹(𝐈_𝐅−𝐈_𝐒)². (5)

The free energy 𝐹_𝑆𝐹 describing the interaction of the effective magnetic field −(1/𝜒⁰_𝑆)𝐈_𝐒 of the larger ellipsoid magnetization with the magnetization 𝐈_𝐅−𝐈_𝐒 of the smaller ellipsoid within the volume 𝑉₂ of the smaller ellipsoid is𝐹_𝑆𝐹=12𝑉₂1𝜒⁰_𝐹𝐈_𝐒⋅(𝐈_𝐅−𝐈_𝐒). (6)

The free energy describing the interaction of the magnetic field 𝐇 with the magnetization 𝐈_𝐒 of the larger ellipsoid within the volume 𝑉₁ is𝐹_𝐻1=−𝑉₁𝐈_𝐒⋅𝐇. (7)

and the free energy describing the interaction of the magnetic field 𝐇 with the magnetization 𝐈_𝐅−𝐈_𝐒 within the volume of “second nanoparticle” 𝑉₂ is𝐹_𝐻2=−𝑉₂(𝐈_𝐅−𝐈_𝐒)⋅𝐇. (8)

Let us introduce the spherical coordinates for magnetizations 𝐈_𝐒 and 𝐈_𝐅:𝐈_𝐒=𝐼_𝑆(𝑙,𝑚,𝑛), (9) where 𝐼_𝑆 is the amplitude of the magnetization 𝐈_𝐒 of the larger ellipsoid, and (𝑙,𝑚,𝑛) are coordinates of the directional unit vector of magnetization 𝐈_𝐒, and𝐈_𝐅=𝐼_𝐹(𝐿,𝑀,𝑁), (10) where 𝐼_𝐹 is the amplitude of the magnetization of the larger ellipsoid, and (𝐿,𝑀,𝑁) are coordinates of the unit vector of the direction of magnetization 𝐈_𝐅. Note that 𝑙²+𝑚²+𝑛²=1 and that 𝐿²+𝑀²+𝑁²=1.

4. Free Energy: Spheroids Oriented in the Same Direction

Let us consider the case of spheroids with polar axes in the same direction. This case corresponds to the cases of mechanochemically prepared nanoparticles in which the shell has the direction of polar axis the same as the direction of polar axis of the core spheroid and 𝜃=0. The free energy for this case has the form 𝐹=𝐹_𝑆+𝐹_𝐹+𝐹_𝑆𝐹+𝐹_𝐻1+𝐹_𝐻2.

The total free energy 𝐹 for the case of spheroids with the polar axis in the same direction has the form𝐹=𝐹₀+12𝐼²_𝑆(𝑁_𝑎−𝑁_𝑏)(𝑉₁−2𝑉₂)cos²+1(𝜓)2𝐼²_𝐹(𝑁_𝑎−𝑁_𝑏)𝑉₂cos²(𝛾)−𝑉₂𝐼_𝑆𝐼_𝐹cos(𝛾+𝜓)(𝑁_𝑎−𝑁_𝑏)(cos²(𝛾)−cos²(𝜓))−(𝑉₁−𝑉₂)𝐼_𝑆𝐻cos(𝜓)−𝑉₂𝐼_𝐹𝐻cos(𝛾), (11) where 𝐹₀ is the free energy which does not depend on angles 𝛾 and 𝜓. We assume that the deviation of magnetization vectors 𝐈_𝐒 and 𝐈_𝐅 from polar axes in core and in shell is small, thus angles 𝜓 and 𝛾 are small angles with values near 0.

The free energy constant 𝐹₀ has the form𝐹₀=12𝑉₁𝑁_𝑏𝐼²_𝑆+12𝑉₂𝑁_𝑏𝐼²_𝑆(𝐼²_𝐹+𝐼²_𝑆)−𝑉₂𝑁_𝑏𝐼²_𝑆. (12)

5. The Case of Spheres

Let us now discuss simple case of two spheres. Note that centres of these spheres are not necessary in the same point. In this case we have 𝑁_𝑎−𝑁_𝑏=0 and 𝑁_𝛼−𝑁_𝛽=0. Thus in the case of the spheres there are only last two terms present in (11). The total free energy 𝐹 for the case of spheroids with the polar axis in the same direction is𝐹=𝐹₀−(𝑉₁−𝑉₂)𝐼_𝑆𝐻cos(𝜓)−𝑉₂𝐼_𝐹𝐻cos(𝛾), (13) where 𝐹₀ is the free energy which does not depend on the angles 𝛾 and 𝜓 (see above).

Note that for 𝐻>0, the lowest free energy (13) corresponds to angles 𝜓=0 and 𝛾=0. For 𝐻<0, the lowest free energy (13) corresponds to angles 𝜓=𝜋 and 𝛾=𝜋. Thus spins in both spheres are oriented in the direction of the magnetic field. The magnetization 𝑀, its component in the direction of main axis, has in this simple case the form:𝑀=(𝑉₁−𝑉₂)𝐼_𝑆cos(𝜓)+𝑉₂𝐼_𝐹cos(𝛾). (14)

Thus the difference of magnetization for 𝐻>0 and for 𝐻<0 has the form for the angles 𝜓 and 𝛾 which minimizes free energy:Δ𝑀=2(𝑉₁−𝑉₂)𝐼_𝑆+2𝑉₂𝐼_𝐹. (15)

The coercive field has the zero value 𝐻_𝑐,±=0. The magnetic susceptibility 𝜒, has in this case value𝜒=𝛿𝑀𝛿𝐻=2Δ𝑀𝛿(𝐻). (16)

In fact the delta function above will be a peak like (a gaussian peak) with a finite and with an amplitude given by 2(𝑉₁−𝑉₂)𝐼_𝑆+2𝑉₂𝐼_𝐹.

6. The Case of Spheroids

Let us now discuss the case of spheroids with the same shape, in which 𝑁_𝑎−𝑁_𝑏 is different from 0 and 𝑁_𝛼−𝑁_𝛽 is different from 0, 𝑁_𝑎−𝑁_𝑏=𝑁_𝛼−𝑁_𝛽. Assuming small angles 𝜓 and 𝛾 we can find the following form of the free energy expansion in 𝜓 and 𝛾:𝐹=𝐹₁+𝐼²_𝑆(𝑁_𝑎−𝑁_𝑏)(𝑉₁−2𝑉₂)×[𝜓²1(−2−𝑥⋅𝑜+𝐻1+𝑥2𝐼_𝑆(𝑁_𝑎−𝑁_𝑏))+𝛾²(−𝑥1+𝑜²21+𝐻𝑥𝑜2𝐼_𝑆(𝑁_𝑎−𝑁_𝑏))], (17) where 𝐹₁ is a constant not dependent on angles 𝛾 and 𝜓. As we can see from (17) using the method from [19], there are two coercive fields, which we denote as 𝐻_𝑐𝛾 and 𝐻_𝑐𝜓, at which coefficients of 𝜓² and 𝛾² become zero:𝐻_𝑐𝛾=𝐼_𝑆(𝑁_𝑎−𝑁_𝑏)(1+𝑜²)𝑜 (18)

(for 𝑥≠0 ) and𝐻_𝑐𝜓=𝐼_𝑆(𝑁_𝑎−𝑁_𝑏)1+2𝑥𝑜.1+𝑥 (19)

Here the quantities 𝑜 and 𝑥 are defined by 𝑥=𝑉₂/(𝑉₁−2𝑉₂) and 𝑜=𝐼_𝐹/𝐼_𝑆. The coercive field which will be measured in experiments is that field from coercive fields 𝐻_𝑐𝛾 and 𝐻_𝑐𝜓, which is smaller and in this case both magnetizations 𝐈_𝐒 and 𝐈_𝐅 are reoriented. When this field is approached decreasing the magnetic field from above, there will be a change in the sign of the coefficient corresponding free energy terms. Note that for small values of 𝑥 for which the core is small, this field becomes 𝐻_𝑐𝜓=𝐼_𝑆(𝑁_𝑎−𝑁_𝑏)(1+𝑥(2𝑜−1)) and thus is increasing with increasing 𝑥 for 2𝑜>1, is larger than the bulk coercive field. For 2𝑜<1 it is smaller than the bulk coercive field

From (18) and (19) we can find conditions for critical values of 𝑥 and 𝑜 quantities for which values of both coercive fields 𝐻_𝑐𝛾 and 𝐻_𝑐𝜓 are equal:1+2𝑥𝑜=1+𝑥(1+𝑜²)𝑜. (20)

The coercive fields 𝐻_𝑐𝜓 and 𝐻_𝑐𝛾 change their role as the observed coercive field at those values of 𝑜 and 𝑥 for which (20) holds.

The critical value 𝑥_𝑐 at which this change takes place is given by:𝑥_𝑐=1−𝑜+𝑜²𝑜².−1 (21)

We see that at 𝑜=0 where there is zero magnetization in the core, we have the critical value of 𝑥 such that 𝑉₁=𝑉₂. At 𝑜=∞ we have the critical value of 𝑥 such that 𝑉₂=𝑉₁/3. There is zero magnetization 𝐼_𝑆=0 in the shell now. For 𝑜=1₊, we find 𝑥_𝑐=+∞ and 𝑉₂=𝑉₁/2, for 𝑜=1₋ we obtain 𝑥_𝑐=−∞ and again 𝑉₂=𝑉₁/2. There is the same magnetic moment in both core and shell for 𝑜=1₊ and 𝑜=1₋.

For the same spheroidal shapes of ellipsoids for 𝑥>𝑥_𝑐 the field 𝐻_𝑐𝛾 is the coercive field observed in experiments. For volumes 𝑉₁ and 𝑉₂ for which 𝑥<𝑥_𝑐, the field 𝐻_𝑐𝜓 is the coercive field observed, now the smaller spheroid is very small. The bulk value of the coercive field as can be easily shown is given by𝐻_𝑐𝑏𝜓=𝐼_𝑆(𝑁_𝑎−𝑁_𝑏). (22)

7. Discussion

The coercive field depends on the diameter 𝑑 in nanoparticles prepared by mechanochemical methods. The nanoparticle growth of NiFe₂O₄ is accompanied by the decrease of coercivity [2]. The surface to volume fraction in this material decreases with increasing annealing temperature. In [20], it was found that increasing the diameter 𝑑 of the nanoparticle the width of the shell decreases for BaFe₁₂O₁₉ and for Co_0.56Zn_1.44W. While this fact was found to be true for nanoparticles with the diameter 𝑑 from the interval 100A^∘ to 400A^∘ we will assume that the same qualitative dependence of thickness 𝑡 on the diameter 𝑑 is present for diameter 𝑑<100A^∘ in NiFe₂O₄ nanoparticles. From two coercive fields, which we have found above, the field 𝐻_𝑐𝜓 depends on the volume of nanoparticles: 𝐻_𝑐𝜓=𝐻_𝑐𝑏𝜓((1+2𝑥𝑜)/(1+𝑥)). Here 𝐻_𝑐𝑏𝜓 is a bulk value of the coercive field, 𝐻_𝑐𝑏𝜓=𝐼_𝑆(𝑁_𝑎−𝑁_𝑏). This coercive field changes with the diameter 𝑑 as𝑑𝐻𝑑𝑑^𝑐𝜓=𝐻_𝑐𝑏2𝑜−1(1+𝑥)²𝑑𝑥.𝑑𝑑 (23)

As we can see the change of the coercive field 𝐻_𝑐𝜓 on the diameter 𝑑 is given by 𝑑𝑥/𝑑𝑑. This change is found to be𝑑𝑥=𝑉𝑑𝑑¹𝑉^′₂−𝑉₂𝑉^′₁(𝑉₁−2𝑉₂)². (24)

Here 𝑉^′₁=𝑑𝑉₁/𝑑𝑑 and 𝑉^′₂=𝑑𝑉₂/𝑑𝑑. We can find that the coercive field 𝐻_𝑐𝜓 is decreasing for 𝑜>1/2 with increasing diameter 𝑑 when the condition 𝑡(𝑑)>𝑑(𝑑𝑡(𝑑)/𝑑𝑑), where 𝑡(𝑑) is 𝑑-dependent width of the shell. 𝑑-dependent width of the shell is decreasing with increasing diameter 𝑑 as is observed in some experiments [20], in [2], it is assumed that the thickness of the shell in NiFe₂O₄ does not depend on 𝑑. In both cases the condition 𝑡(𝑑)>𝑑(𝑑𝑡(𝑑)/𝑑𝑑) is fulfilled. While the spin 𝐼_𝑆 of the shell is canted from the spin 𝐼_𝐹 of the core by angles 20^∘ and 40^∘ depending on the sublattice, the value of the spin in the shell may be expected to be smaller than the value of the spin in the core due to disorder in the shell and the condition 𝑜>1/2 may be assumed to be satisfied. Thus our theory qualitatively explains observed decrease of the coercive field with increasing diameter 𝑑 of nanoparticles for nanoparticles with the width of the shell independent of diameter or with decreasing width of the shell with increasing diameter 𝑑. In the shell of nanocrystalline complex oxides a nonequilibrium cation distribution was found accompanied by a deformation of polyhedron geometries [21]. Note for example that in some nanoparticles like BPFTO ((Bi_0.9Pb_0.1)(Fe_0.9Ti_0.1)𝑂₃), the width of the shell with increasing diameter may increase. For small increase of the shell width which authors are describing as almost constant, see [22].

The coercive field observed in experiment in NiFe₂O₄ is enhanced by the factor 35 with respect to the coercive field of the bulk [2]. The ratio of the coercive field 𝐻_𝑐𝜓 to the bulk coercive field 𝐻_𝑐𝑏𝜓 is given as (1+2𝑥𝑜)/(1+𝑥). We then obtain that the value of 𝑜 corresponding to the enhancement factor in NiFe₂O₄ is 𝑜=35+17/𝑥 and we see that for small volumes 𝑉₂ of the core the ratio o of spin values is too large. The spins in the shell do not have large value. For larger values of the volume 𝑉₂ and larger values of shell spins the enhancement factor may be expected to have the value 35. In our approximation of a small 𝑥 the enhancement of the coercive field may exist, however this enhancement does not reach the value 35; one can find (𝑑/𝑑𝑥)((1+2𝑥𝑜)/(1+𝑥))=2(2𝑜−1)/(1+𝑥)²>0 for 𝑜>1/2 and for 𝑥≈0. We have (1+2𝑥𝑜)/(1+𝑥)≈1+𝑥(2𝑜−1)>1, however this ratio does not reach the value 35 in NiFe₂O₄.

Acknowledgments

The author would like to express his thanks to Professor V. Sepelak for discussions on nanoparticles prepared mechanochemically. This paper was supported by the Slovak Research and Development Agency under the contract no. APVV-0728-07.