Nanoparticle with a ferrimagnetic interlayer coupling in the presence of single-ion anisotropis

Physica B 407 (2012) 378–383

Contents lists available at SciVerse ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Nanoparticle with a ferrimagnetic interlayer coupling in the presence of single-ion anisotropis Wei Jiang n, Hong-yu Guan, Zan Wang, An-bang Guo School of Science, Shenyang University of Technology, Shenyang 110870, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 August 2011 Received in revised form 30 October 2011 Accepted 1 November 2011 Available online 6 November 2011

The magnetic properties of a nanoparticle described by the transverse Ising model with single-ion anisotropis, which consists of a concentric spin-3/2 core and a hexagonal ring spin-5/2 shell coupled with a ferrimamagnetic interlayer coupling, are studied by the effective-field theory with self-spin correlations. Particular emphasis is given to the effects of the both the transverse field and the singleion anisotropis on the longitudinal and transverse magnetizations, phase diagrams of the nanoparticle. We have found that, for appropriate values of the system parameters, one or two compensation points may be obtained in the present systems. & 2011 Elsevier B.V. All rights reserved.

Keywords: Hexagonal nanoparticle Ferrimagnetic Phase transition Effective field theory with differential operator

1. Introduction Experimental evidence shows that the characteristic of magnetic nanoparticle is greatly deviation from bulk counterpart behavior in finite systems [1–3]. The research of magnetic nanoparticles seems to be one of the current interesting topics not only owing to its fundamental level, but also due to the potential applications. In order to investigate better behaviors of the nanoparticle, various models consisting of core/shell structure in a unit cell have been purposed based on a variety of techniques. The magnetic properties and the critical behaviors of a single spherical nanoparticle, consisting of a ferromagnetic core surrounded by a ferromagnetic shell with antiferromagnetic interface coupling, located on a simple cubic lattice has been discussed using Monte Carlo simulation [4]. They have found that, for appropriate values of the system parameters, two compensation temperatures may occur in the nanoparticle system. Vanessa and Wagner, Leite et al. [5,6] employ mean-field calculations and Monte Carlo simulations to determine the phase diagrams of a spherical uniaxial antiferromagnetic particle described by a classical Heisenberg Hamiltonian including a single-ion uniaxial anisotropy in the presence of an external magnetic field perpendicular or parallel to its easy axis. They found the explicit dependence of the zero temperature or low temperatures critical field and the Ne´el temperature on the diameter of the particle, which is in agreement with the predictions of the spin-wave theory. Oscar and Amilcar [7] presented

n

Corresponding author. Tel.: þ86 13840076827; fax: þ 86 24 25694862. E-mail address: [email protected] (W. Jiang).

0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.11.002

the results of a model of a single maghemite ferrimagnetic nanoparticle including radial surface anisotropy distinct from that in the core with the aim to clarify its role on the magnetization processes at low temperatures. The low-temperature equilibrium states are discussed and compared to those of a ferromagnetic particle with the same lattice structure. Magnetic properties of two-dimensional arrays of magnetic nanodots using an Ising model for a hexagonal lattice consisting of metallic nanodots deposited onto non-magnetic insulator substrate have shown by the differential operator technique with the approximated Van der Waerden identity. The nanodots are assumed to interact with each other via a tunneling exchange coupling. The superferromagnetic ordering temperature increases with the dot-size, magnetic anisotropy, and dot volume fraction, and is strongly dependent upon the substrate parameters [8–10]. Based on the Bethe-Peierls (pair) approximation, a method to calculate the classical magnetic properties of single-domain nanoparticles has been developed. Ising spins are treated exactly with nearest-neighbor pair correlations for various lattice symmetries. The characteristics of the model are demonstrated through nontrivial calculations of field cooling and zero-field cooling magnetization curves as well as hysteresis loops of heterogeneous nanoparticles [11]. Herna´ndez and Pinettes studied the thermal behavior of a small particles described by a Heisenberg model, including nearest-neighbor ferromagnetic interactions and radial surface anisotropy, in an applied magnetic field. The results show that the effect of the lattice structure on the magnetic behavior cannot be neglected and hysteresis cycles have different step-like characteristics [12]. A microscopic model for describing ferroelectric nanoparticles is proposed based on an Ising model in a transverse field by applying a Green’s function

W. Jiang et al. / Physica B 407 (2012) 378–383

technique in real space. The coercive field, the remanent polarization, and the critical temperature of the nanoparticles differ significantly from the bulk behavior. The theoretical results are compared with a variety of different experimental data. [13]. In particular, we call attention to the recent work of Kaneyoshi, [14,15] where he investigated the magnetization and phase diagram of a nanoparticle described by the transverse Ising model (TIM) within the two theoretical frameworks, namely the standard mean-field theory and the effective-field theory. However, the study of a magnetic nanoparticle described by the TIM in the presence of different single-ion anisotropis has not been reported from theoretical due to calculation complexly as well as the experimental point of view. In this paper, a mono-domain nanoparticle consists of a two-dimensional structure, namely an atom A as a concentric core surrounded by six atoms B as a hexagonal surface, which the case of S ¼2 in the Refs. [14,15]. A schematic of a mono-domain nanoparticle is plotted in Fig. 1. Here, the model is taken into account both the transverse field and the single-ion anisotropies, which is quite different from that of Refs. [14,15]. The white circle is spin-3/2 magnetic atom A with a single-ion anisotropy DA in the core and the black circles represent spin-5/2 magnetic atom B with a single-ion anisotropy DB at the surface. The lines connecting the white (black) and black circles represent the nearestneighbor exchange interactions J (JS). To investigate the longitudinal and transverse magnetizations and phase diagrams of this model, we employ the effective field theory with self-spin correlations. The results have exhibited one or two compensation points of the total magnetization depending on the values of single-ion anisotropis, the surface exchange interaction and the transverse field, like the N-type of the bulk ferrimagnetic materials. Two compensation points of a mixed ferro–errimagnetic ternary alloy with the type ABpC1  p (ðNiIIx MnII1x Þ1:5 ½Cr III ðCN 6 Þ with different spin SNi ¼1, SMn ¼5/2 and SCr ¼3/2) by the Monte Carlo II simulations [16] or (AaBbCc)yD (ðNia MnIIb FeIIc Þ1:5 ½Cr III ðCN6 Þ with different spin SNi ¼1, SMn ¼ 5/2, SFe ¼2, SCr ¼3/2) by the Mean-field theory [17] have been also found. The paper is arranged as follows. In Section 2 we describe the Hamiltonian of a ferrimagnetic nanoparticle with single-ion anisotropies and give briefly the formulations of the effective field theory with self-spin correlations. Next, in Section 3, we present our results for the magnetizations and the phase diagrams of the system in detail. Finally, the summary is obtained in Section 4.

where J is the exchange interaction between nearest-neighbors a concentric core A and a surface shell B, Js is the surface exchange interaction between two nearest-neighbors B at the surface hexagonal ring. DA and DB are the single-ion anisotropies that come from A and B sublattices, respectively. O represents the transverse magnetic field. In the statistics of the spin system, the canonical ensemble average of any operator si at site i is given by   Tr ðiÞ si expðbHi Þ , b ¼ 1=kB T /si S ¼ ð2Þ Tr ðiÞ expðbHi Þ where T is the absolute temperature and kB is Boltzmann constant. Tr(i) denotes a partial trace at site i. The Hamiltonian Hi includes all contributions associated with the site i. Then, we give the Hamiltonian at site i in the core sublattice in the following form: 0 1 X z Sj Aszi DA ðszi Þ2 Osxi ð3Þ Hi ¼ @J j

For SA ¼3/2 the Hamiltonian Hi can rewrite as: 2 3 pffiffi 3 9 3 0 0 2 Ei þ 4 DA 2 O 7 6 pffiffi 7 6 3 1 1 O 0 6 7 2 Ei þ 4 DA 2 O 6 7 pffiffi Hi ¼ 6 7 1 1 3 O  2 Ei þ 4 DA 0 6 7 2 O 5 4 pffiffi 3 3 9 0 0 O  E þ D A i 2 4 2

i

j,j0

DB

X j

0

ðSzj Þ2 O@

i

1 X X x x ðsi Þ þ ðSj ÞA i

j

For SB ¼5/2 the Hamiltonian Hj can rewrite as: 25 6 6 6 6 6 Hj ¼ 6 6 6 6 4

25 2 Ej þ 4 DB pffiffi 5 2 O

pffiffi 5 2 O 3 9 2 Ej þ 4 DB

0

pffiffiffi 2O

0 pffiffiffi 2O

0

0

0

1 1 2 Ej þ 4 DB 3 2O

0

0

0

 12 Ej þ 14 DB pffiffiffi 2O

0

0

0

0

0

0

0

0

3 2O

0 pffiffiffi 2O

0

 32 Ej þ 94 DB pffiffi 5 2 O

3

0

0

pffiffi 5 2 O

 52 Ej þ 25 4 DB

7 7 7 7 7 7 7 7 7 5

ð6Þ P6

z j ¼ 1 Sj ,

Ej ¼ Js

z i J s

P2

z j0 ¼ 1 S j0

We can easily obtain the longitudinal (z-direction) magnetizations MA in the core and the longitudinal magnetization MB at the surface shell, and corresponding two transverse (x-direction) magnetizations MAx and MBx, as coupled equations [14,15,18–21],  6 MB sinhðZB JrÞ F A ðxÞ9x ¼ 0 , ð7Þ MA ¼ coshðZB J rÞ þ

The Hamiltonian of the system is expressed as follows: X X X H ¼ J szi Szj Js Szj Szj0 DA ðszi Þ2 i,j

ð4Þ

Likewise, the Hamiltonian at site j in the outer shell sublattice is correspondingly as follows: 0 1 X z z X z si Js Sj0 ASj DB ðSzj Þ2 OSxj ð5Þ Hj ¼ @J

where Ei ¼ J 2. Calculation methods

379

ð1Þ

j

ZB

 2 MB MB ¼ coshðZB J S rÞ þ sinhðZB J S rÞ

ZB

  M  coshðZA J rÞ þ A sinhðZA J rÞ F B ðxÞ9x ¼ 0 ,

ð8Þ

 6 MB MAx ¼ coshðZB J rÞ þ sinhðZB JrÞ HA ðxÞ9x ¼ 0 ,

ð9Þ

ZA

ZB

 2 MB MBx ¼ coshðZB J S rÞ þ sinhðZB J S rÞ

ZB

Fig. 1. Schematic representations of a nanoparticle. The white and black circles represent magnetic atoms A and B, respectively.

  M  coshðZA JrÞ þ A sinhðZA J rÞ HB ðxÞ9x ¼ 0 ,

ZA

ð10Þ

380

W. Jiang et al. / Physica B 407 (2012) 378–383

where r ¼ @=@x is the differential operator, and ZA and ZB are  6 M Z2A ¼ coshðZB JrÞ þ B sinhðZB JrÞ GA ðxÞ9x ¼ 0 , ð11Þ

ZB

MB

2

B   MA  coshðZA JrÞ þ sinhðZA J rÞ GB ðxÞ9x ¼ 0 ,

ð12Þ

ZA

here, the six functions FA(x), FB(x), HA(x), HB(x), GA(x) and GB(x) are defined by ( ) 4 X 1 /fi 9szi 9fi Sexpðbli Þ , ð13Þ F A ðxÞ ¼ P4 i ¼ 1 expðbli Þ i ¼ 1

F B ðxÞ ¼ P6

j¼1

HA ðxÞ ¼ P4

1

8 6
expðblj Þ :j ¼ 1 1

(

i ¼ 1 expðbli Þ

HB ðxÞ ¼ P6

j¼1

GA ðxÞ ¼ P4

i¼1

GB ðxÞ ¼ P6

1

/fj 9Szj 9fj Sexpðblj Þ

4 X

expðbli Þ 1

ð14Þ

) x i 9fi Sexpðbli Þ

/fi 9s

,

0.6

ð15Þ

i¼1

8 6
expðblj Þ :j ¼ 1 1

9 = , ;

(

4 X

9 = x /fj 9Sj 9fj Sexpðblj Þ , ;

0.5

ð17Þ

i¼1

8 6
: j¼1 j ¼ 1 expðblj Þ

/fj 9ðSzj Þ2 9fj Sexpðblj Þ

9 = , ;

−1.6

0.1 0 −1.7

−0.1

M ¼ ðM A þM B Þ=2

ð19Þ

M x ¼ ðMAx þM Ax Þ=2

ð20Þ

It is well known that there may appear a compensation point (points) in the ferrimagnetic system for some particular physical parameters. The existence of a compensation temperature in a magnetic nanoparticle has important applications in the field of thermo-magnetic recording. At a compensation point Tcomp, the total magnetization will be zero when the magnetizations of the two opposite sublattices are the same below its transition temperature TC. Moreover, the transition temperature of the system can be obtained by computer programming to solve the value of determinant being zero from the right-hand side of the Eqs. (7) and (8). This method of the numerical calculation has the enormous advantage of being mathematically simple than that of the analytical.

Ω/|J|=0

−1.8

Js/|J|=1.2

−0.2 −0.3

−2

0

1

2

DA/|J|=−0.5

3

4 kBT/|J|

5

6

7

8

Fig. 2. Temperature dependence of the total longitudinal magnetization M for the core spin-3/2 and the surface shell spin-5/2 when the parameters are fixed as Js/9J9 ¼1.2, DA/9J9¼  0.5 and O/9J9 ¼0. The numbers on the curves are the values of DB/9J9.

0.5 0.4

DB/|J|=−0.5

0.3 −1

−1.4

0.2

−1.5

0.1 1.58

0 −0.1 −0.2

3. Numerical results and discussions

−0.3 For the ferrimagnetic case where the spin of A in the core are aliened oppositely to the direction of the spins B at the surface shell. In the following, we will show only some typical numerical results of the transverse Ising nanoparticle in the presence of single-ion anisotropis. The longitudinal magnetization curves exhibit several different behaviors below transition temperature, depending on values

−1.66

−1.4

−1.5

0.2

ð18Þ

At this place, one should notice that the eigenvalues l and eigenfunctions f of the Hamiltonian cannot be given analytically, but can only be given numerically. The numerical method of Eqs. (13)–(18) are the similar with those of Eqs. (20)–(23) given in Ref. [22]. Furthermore, let us define the total longitudinal magnetization per site and the total transverse magnetization per site as follows:

0

−1

0.3

) /fi 9ðszi Þ2 9fi Sexpðbli Þ ,

0 DB/|J|=−0.5

0.4

ð16Þ

M

Z

sinhðZB J S rÞ

M



Z2B ¼ coshðZB JS rÞ þ

of the single-ion anisotropies DB when the parameters Js/9J9 and DA/9J9 are fixed as Js/9J9 ¼1.2 and DA/9J9¼ 0.5 in Figs. 2–4. When the transverse field O ¼ 0, there are two saturation values of the magnetization (M¼0.5, 0) at zero temperature in Fig. 2. All curves correspond to a collinear spin configuration. This result comes from the fact that the core sublattice approaches the si ¼ 73/2 state and the surface shell sublattice approaches the Sj ¼ 75/2 state for smaller DB (or the Sj ¼ 73/2 state for bigger DB), therefore the total longitudinal magnetization M ¼(975/29–9 73/29)/ 2¼0.5 (or M ¼(973/29–9 73/29)/2¼0). However, several saturation values of the longitudinal magnetization appear instead of two saturation values in Figs. 3 and 4 at zero temperature, which is due to the transverse magnetic field O/9J9 a0.The role of the transverse field is to destroy the collinear spin configurations to make more noncollinear spin configurations at zero temperature. Comparing with Figs. 3 and 4, the saturation values of the magnetization decrease with increasing the transverse field O/9J9 when other parameters are fixed. This interesting result is

0

−1.8

Ω/|J|=0.8 Js/|J|=1.2

−2

DA/|J|=−0.5

1

2

3

4

5

6

7

kBT/|J| Fig. 3. Temperature dependence of the total longitudinal magnetization M for the core spin-3/2 and the surface shell spin-5/2 when the parameters are fixed as Js/9J9 ¼1.2, DA/9J9 ¼  0.5 and O/9J9¼ 0.8. The numbers on the curves are the values of DB/9J9.

W. Jiang et al. / Physica B 407 (2012) 378–383

1.1

0.6 0.5 DB/|J|=−0.5

0.3

−1.49

Mx

−1.4

0

1.5

−0.1

1

−1.4

0.6

2

3

4 kBT/|J|

5

6

−1.0

0.3 7

0.2

8

Fig. 4. Temperature dependence of the total longitudinal magnetization M for the core spin-3/2 and the surface shell spin-5/2 when the parameters are fixed as Js/9J9¼ 1.2, DA/9J9¼  0.5 and O/9J9¼ 1.3. The numbers on the curves are the values of DB/9J9.

attributed to that the quantum fluctuations become stronger when the value of O/9J9 increases. Moreover, in a certain sense the transverse field has an impact similar to that of the temperature. But then the transverse field changes the spin configurations at the whole temperature range, the influence of the temperature becomes more pronounced only at intermediate/high temperatures. From Figs. 2–4, some curves of the magnetization M have N-type behavior, namely, the existence of a compensation temperature on the curves that usually can be observed only for a ferrimagnetic material depending on the direction and magnitude of the respective sublattice magnetizations. According to Neel theory only one compensation point may exist in a ferrimagnetic material. However, Kaneyoshi [23] and Ohkoshi et al. [24] reported that the molecular based ferrimagnets (such as ðNiIIa MnIIb FeIIc Þ1:5 ½Cr III ðCN 6 ÞU7:6H2 O (a þbþc ¼1)) show two compensation points. From our calculation, we also find that two compensation points could possibly occur in the nanoparticle system, for example, DB/9J9¼ 1.66 in Fig. 2 (DB/9J9 ¼  1.58 in Fig. 3 and DB/9J9 ¼ 1.49 in Fig. 4), which depending on the competition among the exchange couplings of the core and surface shell, the single-ion anisotropies, and the transverse field and also on the magnetization of the sublattices. The transverse magnetizations Mx in nanoparticle for several values of the singleion anisotropies DB/9J9 ( 1,  1.4 and 1.5) are plotted in Fig. 5. The solid line and the dashed line represent the value of the transverse field O/9J9 ¼0.8 and 1.3, respectively. One should notice that the transverse magnetizations Mx may exhibit weak temperature dependences below TC when the value of DB/9J9 is weaker as DB/9J9¼ 1, while Mx decreases with the increasing temperature below the TC when the value of DB/9J9 is stronger as DB/9J9¼ 1.4 and 1.5. By looking at the above typical longitudinal magnetization curves as a function of temperature and the single-ion anisotropy DB/9J9, we can find the transition and compensation points. The phase diagrams for nanoparticles with various system parameters are shown in Figs. 6–8. In these figures the solid and the dashed lines on the curves represent the phase transition temperature TC curves and the compensation temperature Tcomp curves, respectively. The solid lines show part of the second-order transitions separating the paramagnetic and ferrimagnetic phases. For fixed the values of Js/9J9¼1.2 and DA/9J9¼  0.5 and changing the value of O/9J9

0

1

2

3

4 kBT/|J|

5

6

7

8

Fig. 5. Temperature dependence of the total transverse magnetization Mx for the core spin-3/2 and the surface shell spin-5/2 when the parameters are fixed as Js/9J9¼1.2 and DA/9J9¼  0.5. The numbers on the curves are the values of DB/9J9. The solid and dashed line represents O/9J9 ¼0.8 and 1.3, respectively.

7 6

Tc Tcomp

5 Ω/|J|=0.3

kBT/|J|

−2

0

0.7

0.4

Ω/|J|=1.3 Js/|J|=1.2 DA/|J|=−0.5

−1.8

−0.3

−1.5

0.5

−1.6

−0.2

DB/|J|=−1,−1.4,−1.5

0.8

0.2 0.1

Ω/|J|=1.3

0.9

−1

−1.3

Ω/|J|=0.8

1

0

0.4

M

381

4 3

0.85 Js/|J|=1.2 DA/|J|=−0.5

1.3

2 1 0 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 DB/|J| Fig. 6. Phase diagram of the nanoparticle with the core spin-3/2 and the surface shell spin-5/2 plotted as a function of DB/9J9, when the parameters are fixed as DA/9J9 ¼  0.5 and Js/9J9¼ 1.2. The numbers on the curves are the values of O/9J9. The solid and dashed lines represent the phase transition temperature Tc and the compensation temperature Tcomp, respectively.

(O/9J9¼0.3, 0.85, 1.3), the variations of TC in the system with DB/9J9 are plotted in Fig. 6. With the decrease of DB/9J9, the TC curves labeled O/9J9¼0.3 firstly falls, then gradually becomes flat and finally takes an early horizontal line, which is approximately kBTc/9J9¼1.14. However, when O/9J9 is bigger than the critical value of Oc/9J9¼0.85, the peculiarity of the TC curve is clearly different from such a behavior. In particular, the TC curve labeled O/9J9¼ 0.85 may show a feature form, which is also different from the TC curve labeled O/9J9¼1.3. This interesting phenomenon comes from the fact that the z-direction of spins is easily disordered with the increase of T for O Z Oc. On the other hand, as shown in Fig. 6, the Tcomp curves (dashed curves) exhibit monotonical behavior, which gradually changes to a non-monotonical one with the decreasing of the O/9J9, such as the dashed curve labeled O/9J9¼0.3. It indicates that the two compensation points may occur in the nanoparticle with an appropriate value of O/9J9, for a very

382

W. Jiang et al. / Physica B 407 (2012) 378–383

12

6 Tc

10

Tcomp

5 −1.4

8

4

DB/|J|=−0.5

6

kB T/|J|

kBT/|J|

−1

3

4 2 Ω/|J|=0.8

2 0

−2

0

0.5

1

1

1.5

2

2.5

0

Js/|J|

4. Conclusions We have studied, via the effect-field theory with self-spin correlations, the temperature dependence of longitudinal and

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Ω /|J|

Fig. 7. Phase diagram of the nanoparticle with the core spin-3/2 and the surface shell spin-5/2 plotted as a function of Js/9J9, when the parameters are fixed as DA/9J9 ¼  0.5 and O/9J9¼0.8. The numbers on the curves are the values of DB/9J9. The solid and dashed lines represent the phase transition temperature Tc and the compensation temperature Tcomp, respectively.

6

5

4 kB T/|J|

narrow range of value of DB/9J9. Further, the region in which the Tcomp can be obtained is moved to the small value of DB/9J9 when O/9J9 is decreased. In Fig. 7, the TC curves of the nanoparticle are plotted as a function of Js/9J9, when the parameters are fixed as DA/9J9¼  0.5 and O/9J9¼0.8. Here, Js is the ferromagnetic surface exchange interaction between two nearest-neighbors B at the surface hexagonal ring, which is no negative. Generally, for the smaller value the DB/9J9(¼  0.5, 1.0 and  1.4), TC curves of the system almost linearly decrease with the decrease of JS/9J9. However, from our calculation we can know that when DB/9J9Z1.618, the Tc curve labeled DB/9J9¼  2 takes particularly interesting feature. Namely, the system becomes disorder at paramagnetic phase for too small value of JS/9J9. This result comes from the competition among the single-ion anisotropy DB, the surface exchange interaction JS, and the transverse field O. In Fig. 7, it is also seen that the compensation temperature Tcomp curves (dashed curves) indicates two compensation points for a very large range of value of JS/9J9 with appropriate values of DB/9J9 and O/9J9. The region of the Tcomp is moved to the large value of JS/9J9 when DB/9J9 is decreased. In Fig. 8(a) and (b), the phase transition temperature TC and the compensation temperature Tcomp curves of the nanoparticle with DB/9J9¼  1.4 are plotted as a function of O/9J9, respectively. In Fig. 8(a) the Tc curve exhibits a rapid decrease from the value at O/9J9¼0, which exist critical values Oc/9J9¼4.086 for DB/9J9¼ 1.2, Oc/9J9¼3.5279 for DB/9J9¼ 1.4 and Oc/9J9¼2.9982 for DB/9J9¼  1.6. The phenomenon comes from the fact that the z-direction of spins is easily disordered with the increase of T, because of the large value of O/9J9 and the small value of surface exchange interaction DB/9J9. Here, one should notice that for the bigger DB/9J9, the curves Tcomp labeled DB/9J9¼  1.2, and  1.4 become smaller with the increasing transverse field O/9J9. Whereas the curves Tcomp labeled DB/9J9¼ 1.5, 1.58, and  1.6 indicate that the two compensation points may occur in the nanoparticle with an appropriate value of O/9J9 in Fig. 8(b), for a very narrow range of value of DB/9J9, which are in agreement with those given of Fig. 6.

0

3

2

1

0

0

0.5

1

1.5

2

2.5

Ω /|J| Fig. 8. Phase diagram of the nanoparticle with the core spin-3/2 and the surface shell spin-5/2 plotted as a function of O/9J9, when the parameters are fixed as DA/9J9¼  0.5 and Js/9J9¼ 1.2. The numbers on the curves are the values of DB/9J9. (a) The phase transition temperature Tc. (b) The compensation temperature Tcomp.

transverse magnetizations and phase diagram in a ferrimagnetic nanoparticle described by the transverse Ising model, which was simulated on a two-dimensional hexagonal structure, with the two types of single-ion anisotropis (DA from the core and DB from the surface). As shown in Figs. 2–8, the magnetic properties of the system may exhibit some interesting behaviors depending on the single-ion anisotropies, transverse field and surface exchange interaction. The magnetization cures have N-type behavior depending on the direction and magnitude of the respective sublattice magnetizations. The transverse field O/9J9 is to destroy the collinear spin configurations to make more noncollinear spin configurations at zero temperature. The regions of the single-ion anisotropies DB/9J9 in which compensation points may exist are calculated for different value of the transverse field O/9J9. In this paper, a new method is developed to expand directly the right-hand side of the Eqs. (7)–(12) and others of physical quantities by the computation program, which is more simple mathematically than those of Refs. [14,15,20,21]. The study of the nanoparticle described by the transverse Ising model with the

W. Jiang et al. / Physica B 407 (2012) 378–383

single-ion anisotropies for core and surface shell may open a new field in the research of the critical phenomena and the new type of ferrimagnetism.

[4] [5] [6] [7] [8] [9]

Acknowledgment [10]

This work has been supported by Natural Sciences Foundation of Liaoning province, China (Grant no 20102171), Scientific technology plan of Shenyang, China (Grant no F10-205-1-33), and Program for Liaoning Excellent Talents in University, China (Grant no. LR201031). References [1] V. Skumryev, S. Stoyanov, Y. Zhang, G. Hadjipanayis, D. Givord, J. Nogue´s, Nature 423 (2003) 850. [2] C. Frandsen, C.W. Ostenfeld, M. Xu, C.S. Jacobsen, L. Keller, K. Lefmann, S. Mørup, Phys. Rev. B 70 (2004) 134416. [3] X. He, Z.H. Wang, D.Y. Geng, Z.D. Zhang, J. Mater. Sci. Tech. 27 (2011) 503.

[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

383

Z. Ahmed, K. Mohamed, Physica A 389 (2010) 3435. S.L. Vanessa, F. Wagner, Phys. Lett. A 372 (2008) 898. V.S. Leite, B.C.S. Grandi, W. Figueiredo, Phys. Rev. B 74 (2006) 094408. I. Oscar, L. Amilcar, Physica B 343 (2004) 286. A.F. Bakuzis, P.C. Morais, J. Magn. Magn. Mater. 285 (2005) 145. M.R. Scheinfein, K.E. Schmidt, K.R. Heim, G.G. Hembree, Phys. Rev. Lett. 76 (1996) 1541. K.R. Heim, G.G. Hembree, K.E. Schmidt, M.R. Scheinfein, Appl. Phys. Lett. 67 (1995) 2878. L.G.C. Rego, W. Figueiredo, Phys. Rev. B 64 (2001) 144424. L. Herna´ndez, C. Pinettes, J. Magn. Magn. Mater. 295 (2005) 82. Th. Michael, S. Trimper, J.M. Wesselinowa, Phys. Rev. B74 (2006) 214113. T. Kaneyoshi, Phys. Stat. Sol. (b) 242 (2005) 2938. T. Kaneyoshi, J. Magn. Magn. Mater. 321 (2009) 3430. ˇ J. Dely, A. Boba´k, M. Zukovi, Phys. Lett. A 373 (2009) 3197. G.Z. Wei, Z.H. Xin, Y.Y. Liang, Q. Zhang, Phys. Lett. A 321 (2004) 56. T. Kaneyoshi, Acta Phys. Pol. A 83 (1993) 703. R. Honmura, T. Kaneyoshi, J. Phys. C 12 (1979) 3979. W. Jiang, G.Z. Wei, Z.D. Zhang, Phys. Rev. B 68 (2003) 134432. W. Jiang, V.C. Lo, B.D. Bai, J. Yang, Physica A 389 (2010) 2227. Y. Yuksel, H. Polat, J. Magn. Magn. Mater. 322 (2010) 3907. T. Kaneyoshi, J. Phys. Soc. Jpn. 70 (2001) 884. S. Ohkoshi, Y. Abe, A. Fujishima, K. Hashimoto, Phys. Rev. Lett. 82 (1999) 1285.