Fe3O4 magnetic nanoparticles

Physics Letters A 380 (2016) 927–936

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Physics Letters A www.elsevier.com/locate/pla

Interparticle interactions of FePt core and Fe3 O4 shell in FePt/Fe3 O4 magnetic nanoparticles Hossein Akbari a,∗ , Hossein Zeynali b , Ali Bakhshayeshi c a b c

Department of Physics, Ardabil Branch, Islamic Azad University, Ardabil, Iran Department of Physics, Kashan Branch, Islamic Azad University, Kashan, Iran Department of Physics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

a r t i c l e

i n f o

Article history: Received 25 July 2015 Received in revised form 29 December 2015 Accepted 30 December 2015 Available online 6 January 2016 Communicated by R. Wu Keywords: Magnetization reversal Interparticle interactions Exchange bias FePt/Fe3 O4 nanoparticles Magnetic recording

a b s t r a c t Monodisperse FePt nanoparticles were successfully synthesized using simple wet chemical method. Fe3 O4 was used as a magnetic shell around each FePt nanoparticles. In FePt/Fe3 O4 core/shell system, core thickness is 2 nm and shell thickness varies from zero to 2.5 nm. A theoretical model presented to calculate the shell thickness dependence of Coercivity. Presented model is compared with the results from Stoner–Wohlfarth model to interpret the shell thickness dependence of Coercivity in FePt/Fe3 O4 core/shell nanoparticles. There is a difference between the results from Stoner–Wohlfarth model and experimental data when the shell thickness increases. In the presented model, the effects of interparticle exchange and random magneto crystalline anisotropy are added to the previous models of magnetization reversal for core/shell nanostructures in order to achieve a better agreement with experimental data. For magnetic shells in FePt/Fe3 O4 core/shell, effective coupling between particles increases with increasing shell thickness which leads to Coercivity destruction for stronger couplings. According to the boundary conditions, in the harder regions with higher exchange stiffness, there is small variation in magnetization and so the magnetization modes become more localized. We discussed both localized and non-localized magnetization modes. For non-zero shell thickness, non-localized modes propagate in the soft phase which effects the quality of particle exchange interactions. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The engineering of nanostructures is an open field with enormous perspectives and much work is needed for developing the abilities of controlling the assembling of many units and obtaining new systems for new interesting applications. Nanostructured magnetic materials can be used in power electronics, sensors and advanced technological areas such as permanent magnetism and magnetic recording [1–6]. L10 -ordered FePt has attracted so much attention recently due to its large magnetization combined with high uniaxial magneto crystalline anisotropy needed to develop hysteresis in magnetic materials which is suitable for ultrahigh-density perpendicular magnetic recording technology [7,8]. To reach maximum Coercivity in FePt nano particles, thermal annealing is required for phase transition from disordered face-centered cubic (FCC) to ordered face-centered tetragonal (FCT) structure [9]. The required heat-treatment process leads to particle coalescence and loss of particle positional order which destroys

*

Corresponding author. Tel.: +98 914 455 3145; fax: +98 451 7730 218. E-mail addresses: [email protected], [email protected] (H. Akbari).

http://dx.doi.org/10.1016/j.physleta.2015.12.040 0375-9601/© 2016 Elsevier B.V. All rights reserved.

their major advantage of micro structural uniformity [10,11]. Annealing also changes interparticle spacing [12]. Magnetic shells like CoFe2 O4 [13], MnFe2 O4 [14] and Fex O y [15] are suitable to reach maximum energy products for nano materials. During annealing process, the particle size and structure, and interparticle interactions have to be well controlled. Particle size and interparticle exchange coupling in magnetic granular films usually controlled by addition of nonmagnetic shells such as MnO [16], NiO [17] and ZnO [18]. These shells provide the magnetic isolation required in magnetic recording applications. Here, we report the synthesis of FePt nanoparticles using simple wet chemical method. Fe3 O4 was used as a magnetic shell around each FePt nanoparticles. Size, structure and Coercivity of FePt nanoparticles were studied by VSM and TEM images. The characterization of as-synthesized FePt nanoparticles was carried out by using X-ray diffraction (XRD), transmission electron microscopy (TEM), High transmission electron microscopy (HRTEM), and vibrating sample magneto meter (VSM). We used FePt as core and Fe3 O4 as shell in nanoparticles. It is observed that Coercivity of FePt/Fe3 O4 core/shell nanoparticles decreases with increasing Fe3 O4 thickness. In order to interpret this observation, we presented a model in which the effects of interparticle ex-

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Fig. 1. (a) TEM image and (b) histogram of as-synthesized FePt nanoparticles.

change interactions and random magneto crystalline anisotropy are added to the previous models of magnetization reversal. We attribute random magneto-crystalline anisotropy to the random local magneto-crystalline anisotropy which may result from strain or disordering in the materials. In the model presented, two neighbor particles coupled to each other via effective coupling strength J eff to behave like an equivalent single particle. Presented model calculates the shell thickness dependence of Coercivity in FePt/Fe3 O4 core/shell nanoparticles. Here, we discussed localized and nonlocalized modes propagating in the core/shell system. The nonlocalized modes propagate in the soft phase for non-zero shell thickness which effect on the quality of particle exchange interactions. 2. Experimental

solution of as-synthesized FePt nanoparticles was added to the reaction flask as the cores and the content was magnetically stirred for 20 min. Then the mixture was heated to the boiling point of benzyl ether (300 ◦ C) with heating rate of 5 ◦ C/min in the presence of reflux and kept at this temperature for 15 min before cooling down to room temperature by removing the heat source. The black products were centrifuged and purified following the procedure described in the synthesis of FePt core nanoparticles. Finally, FePt/Fe3 O4 core/shell nanoparticles were obtained after a series of centrifugation. By controlling amount of Fe(acac)3 precursor from 1 mmol to 0.75 and 0.5 mmol, the Fe3 O4 shell thickness can be readily tuned from 2.5 nm to 1.5 and 0.5 nm. The FePt/Fe3 O4 nanoparticles via shell thickness varies from 0–2.5 nm were annealed under a reducing atmosphere at temperature of 750 ◦ C for 2 h using conventional quartz tube furnace.

2.1. Characterization

3. Results

The XRD patterns were collected from a diffractometer of Philips Company with X’pertpro monochromatized Cu Kα radiation (λ = 1.54 Å). The magnetic properties were studied by a model Lake Shore 7400 vibrating sample magnetometer (VSM) with the maximum field up to 20 kOe at room temperature. TEM images were obtained on EM208 Philips transmission electron microscope with an accelerating voltage of 200 kV. Image-processing software (ImageJ software) was used in the present work for the analysis of size and shape of the precipitates.

3.1. Experimental data

2.2. Synthesis of FePt core nanoparticles In a typical experimental procedure, FePt nanoparticle was prepared from a mixture of Fe(acac)3 (0.5 mmol), Pt(acac)2 (0.25 mmol), 1,2-hexadecanediol (2.5 mmol), oleic acid (5 mmol), and oleylamine (5 mmol) in the 10 ml of benzyl ether under magnetic stirring. Afterward, the above mixture was refluxed at 300 ◦ C with heating rate of 5 ◦ C/min for 15 min under the N2 atmosphere. After reflux, the system was allowed to cool to room temperature naturally, and the obtained precipitate was collected by filtration, and then washed with absolute ethanol and n-hexane for several times. Finally, the synthesized FePt cores are used as the seeds to produce FePt/Fe3 O4 core/shell.

Fig. 1(a) shows the TEM and HRTEM image of as-synthesized core FePt nanoparticles. The lattice fringes of the particles are clearly shown in the HRTEM image which attributed to the good crystalline structure of FePt nanoparticles. According to the TEM image, the FePt nanoparticle consists of separated sphere-like nanostructures. Fig. 1(b) shows the histogram size distribution based on log-normal fitting of FePt nanoparticles which achieved by image-processing software (ImageJ software). Base on the image-processing software data (Fig. 1(b)), monodispersity with an average diameter (d) about 4.1 nm (core thickness about 2 nm) and standard deviation (σ ) of about 0.36 nm, indicate narrow size distribution (σ /d) of 0.09. The TEM and HRTEM images of FePt/Fe3 O4 core–shell nanoparticles with different shell thickness and different magnifications are shown in Figs. 2, 3 and 4 respectively. Fig. 5 shows the hysteresis loops of FePt nanoparticles and FePt/Fe3 O4 core–shell nanoparticles with different thickness (annealed at 750 ◦ C). According to the Fig. 5(a)–(c), the Coercivity ( H C ) of FePt nanoparticle is 9.7 kOe and for FePt/Fe3 O4 core–shell nanoparticles with thickness of 2.5, 1.5 and 0.5 is 6.8, 4.3 and 3.2 kOe, respectively. 3.2. Magnetization reversal

2.3. Synthesis of Fe3 O4 shell The synthesis of 2.5 nm thickness of Fe3 O4 nanoparticles as the shell, was achieved by the following procedure: Fe(acac)3 (1 mmol), 1,2 hexadecanediol (5 mmol), oleic acid (4 mmol) and oleyl amine (4 mmol) are mixed together in 10 mL benzyl ether at room temperature under a flow of N2 atmosphere. 10 mL hexane

The starting point for the models attempting to describe the magnetization reversal is based on the micro magnetic approach [19,20]. Minimizing the free (Gibbs) energy of the magnet with respect to the magnetization direction, yields states of equilibrium. The solution is the nucleation field of the material, the field in which magnetization reversal starts [20]. Determination of the lo-

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Fig. 2. TEM and HRTEM image of as-synthesized FePt/Fe3 O4 nanoparticles via shell thickness 0.5 nm.

cal magnetization configuration M (r ) starts from the well-known micro magnetic energy functional [20]:

   F=

A

∇M M

2

2   M .n 1 − K (r ) − M . H − ( M . H d )dr M

2

(1)



K eff



M eff

(2)

In equation (2) M eff = F h M h + F s M s , K eff = F h K h + F s K s , F h and F s V Shell ), V V − V Shell Fh = ( V ). M h and M s are magnetizations of hard and soft phases and K h and K s are magneto crystalline anisotropy constants

are the volume fractions of the hard and soft phases: F s = (

of hard and soft phases. The parameters used FePt/Fe3 O4 are [15, 22–25]:

erg cm3

 K s = K Fe3 O4 = 2.5 × 105



In Equation (1), M (r ) is the spontaneous magnetization, K (r ) is the first uniaxial anisotropy constant, A is the exchange stiffness and H is the applied magnetic field and H d is diamagnetization field. Equation (1) describes a generally random mixture of hard and soft phases. Equation (1) has two types of solutions-magnetization reversal modes-in the magnetic media: Coherent rotation modes and incoherent ones. SW model for non-interacting magnetic nanoparticles based on the magnetic material which is homogeneous with perfectly aligned spins. In this model, magnetization vectors of magnetic material rotate coherently [21], and the nucleation field is:

HC = 2

 K h = K FePt = 5.6 × 106

M s = M Fe3 O4 = 485

emu



 ,

 erg , 3

M h = M FePt = 1140

emu



cm3

cm



cm3

In FePt/Fe3 O4 core/shell system, core thickness is taken to be 2 nm and shell thickness varies from 0–2.5 nm. The theory result from equation (2) is compared to our experimental data in Fig. 6. Magnetically decoupled particles in SW model result in higher Coercivity than expected. The difference is due to the coherent rotation of spins suggested by SW model. The mechanism of magnetization reversal in permanent magnetic materials involves several events [19–26]. The field in which the original saturated state becomes unstable is called nucleation field [27]. To determine the nucleation ˆ = field we assume ideal alignment of magnetization M = M 0 m M 0 (m L e L + m T e T ). Here m L and m T are longitudinal and transverse components of magnetization. Assume that m L = m z e z and m T = mx e x + m y e y , we expand the free energy density with respect to the small transverse components of m T = mx e x + m y e y  1. The easy axis direction is n(r ) = (1 − a(r )2 )e z + a(r ), where a(r ) is the transverse vector component ˆ Series expansion of n(r ). A similar equation exists for M = M 0 m.

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Fig. 3. TEM and HRTEM image of as-synthesized FePt/Fe3 O4 nanoparticles via shell thickness 1.5 nm. m2

2

yields n(r ) = (1 − a2 )e z + a(r ) and M = M s (1 − 2T )e z + M s m T . Next we assume that H = H 0 e z , the differential equations are obtained by putting M and n into Equation (1) and minimizing F with respect to the small transverse magnetization component:

  ∂H δF ∂H + = −∇ =0 δm T (r ) ∂(∇ m T (r )) ∂ m T (r )

(3)

Writing m T = m we lead to differential equation:



−∇( A ∇ m) + K +

Ms H 2



m = K a(r )

(4)

In the Equation (4), we can neglect the term ∇ A ∇ m. The physical meaning of this term is that the term ∇ A (r ) in Equation (4) reflects the local character of the exchange stiffness A (r ) which can be neglected in calculations [28,29]. Except for the ∇ A ∇ m term, Equation (4) is equivalent to Schrodinger’s equation for an electron in electrostatic potential V . This allows us to apply ideas familiar from quantum mechanics to discuss micro magnetics [30–32]. In this quantum-mechanical analogy, A, K , and − M2s H are analo2

h¯ , V , and E, respectively [31]. The ground-state energy gous to m e E 0 corresponds to the nucleation field H = − H N , which determines the Coercivity of nucleation-controlled magnets [27]. In the ordered limit, Equation (4) has been solved for a number of cases [1,19]. So that the nucleation field is given by the volume-averaged anisotropy constant  K eff  = F h K h + F s K s [31]. Note that the x and

y components of m = mx e x + m y e y are decoupled and degenerate in Equation (4), In particular, the nucleation field corresponds to the quantum-mechanical ground-state energy [29], and so the small transverse magnetization or nucleation mode m(r ) has its analog in the wave function ψ(r ). Solving equation (1) for two phase magnets, a quasi-coherent purely radial angular dependence mode is defined as bulging mode [26]. The core/shell thickness required for bulging mode to occur is:



R b = 7.86

A eff 2 M eff

,

A eff = F h A h + F s A s

(5)

where A h and A s are the exchange stiffness of hard and soft phases in core/shell system. The parameters used FePt/Fe3 O4 are [21,22]:

A h = A FePt = 2.75 × 10−6



erg

 ,

cm

A s = A Fe3 O4 = 1.32 × 10−6



erg



cm

According to this model Coercivity is defined as [26]:

 HC = 2

K eff M eff



 − 2π 2

A eff M eff R b2

 (6)

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Fig. 4. TEM and HRTEM image of as-synthesized FePt/Fe3 O4 nanoparticles via shell thickness 2.5 nm.

The result from equation (6) is compared to our experimental data in Fig. 7. There is a lack of compatibility between the results from two models above and Experimental data when the shell thickness increases. 3.3. Inter particle exchange and random anisotropy In order to explain the difference between theoretical and experimental results, two mechanisms should be taken in to account: In FePt, as in other hard materials, the coercive field is much smaller than the value expected for an ideal system [33]. This might be due to the presence of random magneto crystalline anisotropy, structural defects or inter particle interactions. In some solids, nano structures and core/shells magneto crystalline anisotropy varies due to the Crystal defects or crystal plane movements or different particle size. Magneto crystalline anisotropy can be affected by mixing of materials with different phases in core/shell interface of nano particles, this intermixing leads to Coercivity reduction in magnetic materials. We can model this intermixing by using random magneto-crystalline anisotropy which attributes to the random local magneto-crystalline anisotropy results from strain or disordering in the materials. According to the SW model, in the simplest way, Coercivity field can be expressed as a function of magneto crystalline anisotropy constant K which is replaced by  K  in the presented model. When intermixing, defects or random particle size occur in the magnetic media,

magneto crystalline anisotropy changes, this change in magneto crystalline anisotropy leads to Coercivity field reduction. The main result is that interparticle interactions and random magneto crystalline anisotropy are the main sources of Coercivity field reduction. The effect of interparticle exchange usually controlled by the volume fraction of the shell thickness or matrix phase. When particles are well separated by matrix with weak or no exchange interaction, the individual particles rotate coherently. When the exchange within the matrix becomes stronger, the reversal mechanism has the involvement of two or more particles which is undesired in magnetic-recording media as it reduces the storage density [32]. The validity of equation (2) is restricted to coherent rotation of magnets. In fact, real systems micro magnetic exchange interactions between particles, lead to incoherent rotation of spins and decrease in Coercivity. For magnetic core/shells with fixed particle distance, increasing shell thickness will decrease the distance between neighboring inclusions. When the distance becomes too small, the soft regions can interact with each other. In this situation, the magnetization modes can tunnel through the soft region which no longer acts as an effective potential barrier [26,35,36]. In practice, exchange interactions are stronger and localized, whereas dipole–dipole interactions are weaker but long ranged. There is some exchange mediated by particle boundaries or by the shell, where the coupling is reduced [37]. The atomic modeling yields effective interparticle coupling strength that depends on particle

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Fig. 5. Hysteresis loops of annealed (a) FePt nanoparticles and FePt/Fe3 O4 nanoparticles via shell thickness (b) 0.5 nm (c) 1.5 nm (d) 2.5 nm at temperature of 750 ◦ C.

separation and size [38], interface structure [39], and matrix composition. Strong interparticle exchange between two neighbor nano particles, reduces the Coercivity of nanostructures which is undesired in magnetic recording [34]. We start from equation (1) and minimize it with respect to M (r ) to reach equation (4) [34]. The exchange term is modified by the boundaries:

  ∂ M  ∂ M  A (x) = A (x) ∂ x x0 −ε ∂ x x0 +ε

(7)

By putting calculated M (x) into equation (7) and calculating the integral of equation (1) as a function of misalignment vector, we reach to the effective interparticle exchange [6,40]:

J eff =

D 2T 1+



K eff A eff

Ds



K eff A eff 2 A eff

(8)

When the distance between two hard inclusions is small, the magnetization modes can “tunnel” through the soft region which cannot act like effective potential barrier [1]. In this situation, the

hard regions interact and Coercivity is destroyed. In fact, this micro magnetic “exchange interaction” has its quantum-mechanical analog in the formation of bonding and anti-bonding states and tends to reduce the nucleation field [32]. In equation (8), D T is the total particle size (FePt/Fe3 O4 core/shell diameter) and D s is shell surface to surface distance between particles. K eff and A eff have the same definition in equation (6). Interaction during the reversible magnetization process

can be evaluated through exchange correlation length, L exch =

A K

[20]. In a wide range of materials, the surface to volume ratio is large and exchange correlation length is larger than the particle size. Magnets with random magnetic anisotropy can be described by the random anisotropy model [41,42] which will be considered by the random-field analogy. For the local fields we have  H i  = 0 and  H i H j  = H 02 δi j [43]. The random field represents local spin disorder in magnetically correlated regions containing N atoms H 02 . When

N A the efK

and characterized by the averages  H  = 0 and  H 2  = the grain size D is smaller than the length, L exch =

fective anisotropy affecting the magnetization process results from

H. Akbari et al. / Physics Letters A 380 (2016) 927–936

Fig. 6. The theoretical results from [19] and our experimental results show a decrease in FePt/Fe3 O4 Coercivity. The two results do not match when the shell thickness increases.

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Fig. 8. Effective coupling between particles increase with increasing magnetic shell thickness.

strength is determined by soft region. This is because the energy product is defined as the usable magneto static energy stored outside of the magnet or hard regions. So, regions such as voids or shells must be included in the equivalent particle volume. For two coupled particles behaving like single particle hard regions with exchange stiffness A h are coupled together through softer Fe3 O4 shell region with exchange stiffness A s . Here the magnetic shell determines the coupling strength of two hard cores so A s is replaced by A Coupl . The parameter A Coupl is defined by J eff D CC

Fig. 7. The theoretical results from [19,24] and our experimental data show a decrease in FePt/Fe3 O4 Coercivity. These results do not match when the shell thickness increases.

averaging over the N grains. Since nano particles have a large surL2

face to volume ratio [44], we define N as N = ( Dex2 ). For a region containing N atoms we have:

K

K  = √

N

2 K eff D2

A eff

(10)

3.4. Coercivity for core/shell systems

(11)

Equivalent single particle consists of two hard FePt particles coupling to each other through softer Fe3 O4 shell region. Coupling

K 



(12)

M eff

In equation (12),  K  is defined as:

K  =

2 K eff D 2E

A eff

,

A eff = F h A h + F s A Coupl

(13)

We use random magneto crystalline anisotropy presented in equations (12), (13) instead of coherent rotation of equation (2) to reach a better conclusion. Results from equations (12), (13) are shown in Fig. 8. The results seem to have a better agreement with experimental data when we add random magneto crystalline anisotropy and incoherent rotation of magnetization reversal in equations (12), (13), (6):



We assume two neighbor nano particles to be coupled to each other and behave like an equivalent single particle with total size D = D E which is the nearest-neighbor center-to-center distance:

D E = D s + 2( R Core + R Shell )



HC = 2

(9)

Inserting parameters L ex and N in equation (9) we have [41,42]:

K  =

where D CC is center-to-center distance between two particles. The total exchange stiffness in equation (10) is defined as A eff = F s A Coupl + F h A h . To reach an exchange dependent Coercivity, the parameters A eff and D E are inserted in equation (11). Increasing effective coupling in magnetic shells might be one of the reasons of Coercivity reduction. According to the model presented, the effective coupling J eff and exchange stiffness A Coupl increase with increasing shell thickness (Fig. 8) which is in agreement with Ref. [45]. The average of D CC is taken to be 7 nm. Replacing the Coercivity from this model in equation (2), we are able to add the effect of interparticle interactions to the Coercivity of core/shell system:

HC = 2

K  M eff





− 2π 2

A eff M eff R b2



(14)

Equation (14) is plotted in Fig. 9. The theoretical results seem to fit better with our experimental data than the previous models. According to the industrial point of view, experimental procedures try to reach an ideal single size nano particles which can be used as magnetic recording media. In the FePt case, the size of

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Fig. 9. The theoretical results from [19,24] and experimental results from [14] show a decrease in FePt/Fe3 O4 Coercivity. The results from equations (11), (13) seem to have a better agreement with experimental data when the shell thickness increases. It might be due to the increase in effective coupling between particles with increasing shell thickness.

According to the section 2 and Ref. [31], there is a formal analogy between micromagnetics and quantum mechanics. The ∂m boundary condition A h ∂ rh = A s ∂∂mr s corresponds to A h (e .∇)ψh = A s (e .∇)ψs . In the harder regions with higher exchange stiffness ∂m A h , ∂ rh = (e .∇ψh ) should be small and magnetization modes become more localized. But in the softer regions when A s is small we have larger ∂∂mr s = (e .∇ψs ) so the magnetization modes in softer regions are more non-localized. The localization and the Coercivity reduction depend on the ratio of exchange and anisotropy energies. In this situation, local variations of the magnetization cost some exchange energy. Exchange energy is not the only consideration in calculations, because local variations of the magnetization may be affected from local anisotropy inhomogeneities. Here we only considered exchange energy variations at phase boundaries of soft and hard Core/Shells which affects the coupling between two nano grains and, indirectly, the extrinsic properties of the nano structures. Reduced grain-boundary exchange leads to a quasi-discontinuity of the magnetization. Experimentally, reduced interface exchange reflects real-structure features such as impurity atoms diluting the interatomic exchange, oxide layers covering the grains, and interface amorphization. For the Equations (13), (14) in Fig. 8, when the shell thickness ∂m is equal to zero we have: A h ∂ rh = A h (e .∇ψh ) = 0, since A h = 0,

synthesized nano particles are not uniform. When we have an ideal material with particles having equal size, presented model transforms to the simple SW model for non-interacting magnetically isolated particles. In the experiments, we try to control interparticle interactions with shells to avoid sintering effect but interparticle exchange interactions and random particle size are present in the synthesized nanoparticles. The amount of randomness in particle size depends on the experimental procedure. Even if we have a single sized nano particles, there is an intermixing of core and shell phases at the core/shell interface. Nano particles have a large surface to volume ratio and surface effects have a large impact in their magnetic properties. Intermixing of two phases at the core/shell interface effects on the surface anisotropy of core or shell and since the intermixing of phases is random, we can also use random anisotropy model for nano particles with equal size which makes this model useful. However, the effect of random models can be used not only for random particle size or random intermixing of core and shell phases, but in the sol–gel process when there is a random place for each Fe or Pt particles during phase transition which leads to random interparticle interactions in the media [46]. The word “random magneto-crystalline anisotropy” may results from strain or disordering in the materials which changes local magneto-crystalline anisotropy.

4. Discussion

3.5. Mechanism of presented magnetization reversal Micro magnetic localization is important because it determines the nucleation of reverse domains and affects the Coercivity [47–50]. The hard/soft phases have different values of the parameters A, M, and K . The term ∇( A ∇ m) in Equation (4) reflects the local character of exchange stiffness A (r ). In the boundary of hard/soft phases, the exchange term reduces to the general ∂m boundary condition of equation A h ∂ rh = A s ∂∂mr s [30]. Where the respective indicates the hard and soft regions. A jump in A (r ) leaves the magnetization continuous but yields a change in the slope of the Perpendicular magnetization component of magnetization. This situation can be applied for adjacent grains in the interface region [6,40]. The nucleation mode penetrates from the soft phase into the hard phase when the exchange energy density is able to compete against the random anisotropy energy density [30].

∂m

we reach ∂ rh = (e .∇ψh ) = 0. So in this situation we have more localized nucleation modes in hard phase. Results obey the rules of partial differential Schrodinger’s equation (equation (4)). By replacing K in the equation (4) by coupling dependent anisotropy constant  K  from Equation (13) we will reach to the different answers and so different modes of magnetization reversal. Equation (14) contains bout localized and nonlocalized modes. The non-localized modes propagating in the soft phase for the non-zero shell thickness can effect on the quality of particle exchange interactions. When the shell thickness is equal to zero, the boundary conditions lead to form more localized nucleation modes in hard phase and we can only have these localized modes. In this situation, the presented model would have the same results of SW model as seen in Figs. 6, 8. For the shell thickness equal to zero these modes would be omitted and so we can see a little difference in the Coercivity predicted by model presented and SW model. As the shell thickness increases, the non-localized modes are likely to happen in the shell and since the distance between the hard inclusions is small, the magnetization modes can “tunnel” through the soft region. In this situation, the hard regions interact and Coercivity is destroyed. We can see this effect in Figs. 6, 8 as shell thickness increases.

Magnetic properties of materials can be classed into intrinsic and extrinsic properties. Intrinsic properties such as Curie temperature, saturation magnetization, magneto-crystalline anisotropy etc. are mainly defined by the chemical composition as well as by the crystal structure. Extrinsic properties such as coercivity, remanent magnetization, and energy product are strongly affected by the specific shape, size and grain boundaries of the magnetic object as well as by their microstructure. These properties are affected by spin dependence of the impurity, defect potentials and exchange energy variation at grain boundaries which play an important role for all kinds of technical applications. In magnetic materials, how a material is processed can greatly impact the extrinsic properties. Coercivity is a function of both intrinsic factors (such as magneto crystalline anisotropy) and extrinsic factors (such as microstructure). For example, the coercivity of technical iron doubles by adding 0.01 wt.% nitrogen [51]. Such small concentrations have little effect on the intrinsic properties but lead to inhomogeneous

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lattice strains that affect the propagation of magnetic domain walls and explain the observed coercivity decrease. In single domain particles the coercivity increases to a maximum at a critical particle-size diameter. Future increase of the diameter results in a multi-domain state particle and incoherent rotation. In this case, coercivity is determined by pinning mechanisms of the domain wall. These mechanisms are determined by the magnetically inhomogeneous regions like columnar boundaries, chemical inhomogeneities, stacking faults, etc. resulting in an incoherent switching which decreases coercivity. Even in a matrix of particles, with magnetostatic interaction the coercivity will again be influenced. Multi-domain particles and thin films switch by domain-wall motion and again the coercivity decreases [52,53]. An important consideration in measuring the magnetic properties of a material is its uniformity. Inhomogeneities may occur on the microscopic scale (point defects such as vacancies and interstitials, line and planar defects such as dislocations, or composition fluctuations in disordered alloys). Many magnetic materials of interest are multi-component alloys, amorphous, or Nano-crystalline composites. In such materials, local magnetic anisotropy may be large but is expected to be randomly oriented at different sites, possibly decreasing coercivity [54–56]. However, for films made from these materials, strain or local chemical or structural disordering can lead random alignment of the local anisotropy axes, causing smaller macroscopic anisotropy. For example, M (r ) and K (r ) reflect the existence of local chemistry disordering in the media, and the unit vector n(r ) is easy magnetization direction corresponds to the local c-axis orientation of the crystallites. Local chemistry disorder distributes randomly in the media which is not uniform and leads to the changes in K (r ). The source of random K (r ) may be quite fundamental: the growth process of a film, for example, sintering, rapid quenching, sputtering, annealing conditions, or molecular beam epitaxy (including thermal and mechanical treatments). The appearance of inhomogeneities influences both intrinsic and extrinsic properties. In most instances, magnetic measurements are extrinsic made by measuring a sample’s response to an externally applied magnetic field. These measurements can depend on the sample’s shape and grain boundaries: the inhomogeneities and discontinuities of the magnetization in the sample’s interior and on its surface are sources of a magneto static demagnetizing field which depends on the sample’s shape and affects internal field. By changing sample’s internal magnetization in a response to the applied fields, the internal field will also change due to the demagnetizing field variations. These field variations affect related extrinsic value which is coercivity in this discussion. Magneto static demagnetizing fields can also be microstructure dependent because typical polycrystalline or particulate materials exhibit effects due to the exchange energy variations at grain boundaries or grain orientation. From a structural point of view, surfaces, interfaces, and junctions have a strong impact on Nano magnetism. Variation of exchange energy at grain boundaries is a source to alter interparticle interactions. Reduced grain-boundary exchange leads to a quasi-discontinuity of the magnetization and changes internal field. One issue is that exchange at grain boundaries affects the coupling between Nano grains and, indirectly, the extrinsic properties of the structures. In addition, for Nanoparticles, grain boundary conditions and interface properties also influence the coercivity. This leads to the assumption that the coercivity (an extrinsic property) can only be determined by means of the macroscopically hysteresis loop in combination with the theory of micro magnetism [57]. Knowledge of the microstructural properties of the material cannot be omitted [58,59]. In the first place the size and shape of the grains in the material plays an important role. In nanostructures, we have some distances between two adjacent grains in the media these distances are not equal: According to the

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random particle size distribution present in the media, the distance between each two grains varies randomly, so random size in the particles generates random exchange interactions present in the media. According to the grain boundary dependence of extrinsic values such as coercivity, these random interactions affect related extrinsic value which is coercivity in this discussion. 5. Conclusion In conclusion, FePt nanoparticles were synthesized using simple wet chemical method. We have calculated the effect of interparticle interactions in core/shell systems. Interparticle interactions and random magneto crystalline anisotropy are the main source of Coercivity reduction in the presented model. Increase in shell thickness leads to increase in effective particle coupling which reduces the core/shell system Coercivity. The nucleation mode penetrates from the soft phase into the hard phase when the exchange energy density is able to compete against the random anisotropy. We discussed bout localized and non-localized modes. The nonlocalized modes propagating in the soft phase for the non-zero shell thickness can effect on the quality of particle exchange interactions. Acknowledgements This work was supported by the Islamic Azad University of Ardabil Branch and the Islamic Azad University of Kashan Branch and Islamic Azad University of Mashhad Branch. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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