Continuum of compensation points in the mixed spin Ising ferrimagnet with four-spin interaction and next-nearest neighbor coupling

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Continuum of compensation points in the mixed spin Ising ferrimagnet with four-spin interaction and next-nearest neighbor coupling

M. Azhari, N. Benayad & M. Mouhib

To cite this article: M. Azhari, N. Benayad & M. Mouhib (2017): Continuum of compensation points in the mixed spin Ising ferrimagnet with four-spin interaction and next-nearest neighbor coupling, Phase Transitions, DOI: 10.1080/01411594.2016.1227985

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RESEARCH ARTICLE

Continuum of compensation points in the mixed spin Ising

ferrimagnet with four-spin interaction and next-nearest neighbor coupling

M. Azhari, N. Benayad and M. Mouhib

Laboratory of High Energy Physics and Condensed Matter, Faculty of Sciences, Hassan II University–Casablanca, A€ın Chock, Casablanca, Morocco

ARTICLE HISTORY Received 25 May 2016 Accepted 11 August 2016 ABSTRACT

We investigate the magnetic properties of the mixed spin-1/2 and spin-1 Ising ferrimagnetic system with four-spin interactionJ4and next-nearest neighbor (NNN) couplingJ⁰. We perform exact ground-state calculations and use thefinite cluster approximation, based on a single cluster theory, to derive the state equations for the two-dimensional square lattice. The main attention has been paid to the study of the phase diagram for both the transition and compensation temperatures. We find a number of characteristic behaviors. The model with only NNNs induces one compensation point while the four-spin interaction does not. The investigation of the model with both interactions shows a number of characteristic behaviors. In particular, the presence of the four-spin interaction, according to J4 and J⁰, may lead to one, two or possibly a continuum of compensation points. This phenomenon may have important applications in technology such as thermomagnetic writing and erasing at the compensation point.

KEYWORDS

Mixed spin Ising ferrimagnet;

four-spin interaction; next- nearest neighbor coupling;

phase diagram; continuum of compensation temperatures

1. Introduction

The synthesis of new stable crystalline room-temperature magnets that are able to keep magnetiza- tion in the absence of an applied magnetic

fi

eld is a major issue for many industrial applications.

This is not only because of their potential device application, including those in areas such as ther- momagnetic recording, electronic and computer technologies [1] as well as biomedical materials [2]

and catalysts [3], but also due to their common uses as research tools in multi-interdisciplinary areas of chemistry [4], geology, biology and medicine [5]. In recent years both chemists and physicists started to collaborate very closely on chemistry and material sciences in order to produce materials soluble in organic solvents, biocompatible, optically transparent, with spontaneous moments at room temperature [1,4,6]. Generally, the ferrimagnetic ordering seems often to play a prominent rel- evance in these materials. There is currently a great deal of interest in the synthesis of single-chain magnets (SCMs) with ferrimagnetic arrangement, such as Co

^II

(hfac)

2

(NITPhOMe) [7], Co

^II–Cu^II

bimetallic single-chain magnet [8] and [Fe

^II

(ClO

₄

)

₂

{Fe

^III

(bpca)

₂

}](ClO

₄

) [9], because they can act as elementary binary units (bits) used for information storage, providing potential applications in high-density information storage [10,11] or as qubits in quantum computers [12]. As purely one- dimensional (1D) systems are known to have a long-range order only at T

D

0 K, these SCM materi- als promote long relaxation times and the system can behave like a magnet [13]. In the search for

CONTACT N. Benayad [email protected]

© 2017 Informa UK Limited, trading as Taylor & Francis Group http://dx.doi.org/10.1080/01411594.2016.1227985

new and improved molecular materials, synthesis has been expanded toward two-dimensional (2D) and three-dimensional (3D) ferrimagnets, such as 2D organometallic ferrimagnets [13]. 2D networks of the mixed-metal material {[P(Phenyl)4][MnCr(oxalate)3]}n [14], 3D ferrimagnets with T

C D

240 K and T

C D

190 K [15] and the amorphous V(TCNE)

x

.

y

(solvent) with temperatures ordering as high as 400 K [16]. One of the interesting properties of such ferrimagnetic systems is

‘compensation points’

or N-type behavior in N eel classification nomenclature. This property is very useful in magnetic memory and spin analyzing applications [17]. In recent years, the existence of more than one compensation point, which cannot be explained by N eel theory, has been observed experimentally [18,19] and predicted theoretically [20]. Therefore, it reopened the interest in study- ing the ferrimagnetic systems. Intensive efforts are required to study these materials in order to clar- ify their very interesting and sometimes unusual behaviors.

The two-sublattice-mixed-spin Ising systems are well adapted to study a certain type of ferrimag- netism [21]. They have less translational symmetry than their single spin counterparts. Experimen- tally, it has been shown that the MnNi(EDTA)6H

2

O complex [22] is a good example of a mixed system. The magnetic properties of these systems have been investigated by various methods. In par- ticular, the mixed spin-1/2 and spin-1 Ising model with only nearest neighbor coupling and crystal

field has been studied using high(low)-temperature series expansions [23], numerical transfer matrix

techniques supplemented by Monte Carlo simulations (MCS) [24,25] and renormalization-group technique [26]. They found no evidence neither for a compensation point nor a tricritical point. On the other hand, they showed that compensation points are induced by the presence of an interaction between the next-nearest neighbor (NNN) of the spin-1/2. The inclusion of further-neighbor inter- actions would allow for a better modeling of real magnetic systems [27] and of course of all other systems that can be mapped onto the Ising models such as models of microemulsions [28].

Recently, there has been considerable interest in experimental and theoretical research of Ising model with multispin interactions. These models are interesting because they found their theoretical explanation in the theories of super exchange interaction, the magnetoelastic effect and the spin- phonon coupling [29,30]. Moreover, it was pointed out that the models with the higher-order exchange interactions may exhibit rich phase diagrams and can describe phase transition in some physical systems. Additionally, they show physical behavior not detected in the usual spin systems.

For example, the non-universal critical phenomena [31] and deviation from T

^3/2

Block law at low temperature [32,33]. From the theoretical point of view, the monoatomic Ising models with multi- spin interactions have been investigated in detail within different methods, such as mean

field

approximation [34,35], effective

field theory [36,37], some more accurate treatments such as series

expansion [38,40], renormalization group methods [41,43], MCS [44,46] and also exact calculations [47,51].

Experimentally, an interesting fact for the models with multispin interactions has been reported.

Indeed, it can be used to describe various physical systems such as classical

fluid [52], solid ³

He [53], lipid bilayers [54], and rare gases [55]. Moreover for some materials it has been shown that the multispin interactions play a signi

fi

cant role; and they are comparable or even much important than the bilinear ones. The models with pair and quartet interaction have been applied successfully to study and explain the existence of

fi

rst-order phase transition in squaric acid crystal H

2

C

2

O

4

[56,57]. Such models have also been used to describe thermodynamical properties of hydrogen- bonded ferroelectric PbHPO

4

, PbDPO

4

[58], some copolymers [59] and optical conductivity [60]

observed in cuprate ladder La

_x

Ca

_14-X

Cu

24

O

41

. On the other hand, some experimental studies on La

6

Ca

8

Cu

24

O

41

[61,62] and La

4

Sr

10

Cu

24

O

41

[63] reveal that they could be explained by the introduc- tion of the four-spin interaction. It is worthy to note here that this later plays an important role in the 2D antiferromagnet La

2

CuO

2

[64], the parent material of high-T

C

superconductors.

It is worth to notice that in our recent works, we investigated the thermodynamic properties of

the ferromagnetic mixed Ising model consisting of spin-1/2 and spin-1 with four-spin interactions

and NNN coupling, by using MCS [65] and

finite cluster approximation (FCA) [66]. In these two

papers, according to the value of the NNN couplings, it has been shown that the system behaves

qualitatively and quantitatively different from that obtained when the interactions between NNN are ignored. For instance, it has been shown that for J

⁰<

0, the system exhibits, at zero temperature, a phase transition at a

finite critical value (J4

/J

2

)

C

; while for J

⁰>

0 and J

4

/J

2D ¡4 the system keeps the

coexistence of two ground states up to a J

⁰

-dependent

finite temperature. Thus, it is very important

to investigate the ferrimagnetic version of this model, particularly the magnetic properties and the existence of multiple compensation points.

The purpose of this paper is to investigate the influence of NNN coupling on the phase diagram and, in particular, the existence of compensation behavior in ferrimagnetic mixed-spin Ising model with four-spin interaction. Since these types of interactions have important effect on 2D Ising real systems, this makes the suggested model extremely interesting from the theoretical and experimental viewpoints.

This paper is organized as follows. The second section deals with the description of the Hamilto- nian of the model and its ground states. Then, the third section is devoted to the theoretical frame- work and the state equations, whereas the fourth section focuses on the results and the discussions.

Finally, some conclusions are summarized in the

fi

fth section.

2. Formulation of the model and its ground states

The ferrimagnetic mixed-spin Ising system with four-spin interactions and NNN coupling can be described by the following Hamiltonian:

H

D

J

2

X

ij h i

s_i

S

j

J

4

X

i;j;k;l

f g

s_is_k

S

j

S

l

J

⁰X

ik h i

s_is_k

(1)

The underlying lattice is composed of two interpenetrating sublattices. One occupied by spins with spin moment

sD §1/2 and the other one is occupied by spins with moment

S

D

0,

§1. The

Figure 1.(a) Parts of the square latticeand£correspond tosandS-sublattice sites, respectively. (b) Neighbors of spins0(S0) with which directly interacts.

bilinear interaction J

2

is positive. The

first summation is carried out only over nearest-neighbor pair

of spins. The second and the third summations represent the four-spin and NNN interactions, respectively, where the summations concern all alternate squares shaded in Figure 1(a).

The structure of the ground-state phase diagram of the Hamiltonian (1) can be exactly calculated by comparing the energies of corresponding configurations. The resulting ground-state phase dia- grams in the (J

4

/J

2

, J

⁰

/J

2

) plane are shown in Figure 2. As mentioned above, our study is focused on the ferrimagnetic state I.

3. Theoretical framework

The theoretical framework that we adopt in the study of the mixed spin-1/2 and spin-1 Ising model with four-spin and NNN interactions, described by the Hamiltonian (1), is the

finite cluster approxi-

mation (FCA) [67,68] based on a single-site cluster theory. We have to mention that this method has been successfully applied to a number of interesting pure and disordered spin Ising systems [69,74].

It has also been used for transverse Ising models [75,79] and semi-infinite Ising systems [80,84]. In all these applications, it was shown that the FCA improves qualitatively and quantitatively the results obtained in the frame of the mean-field theory. In this approach, attention is focused on a

Figure 2.Ground-state phase diagram.

cluster comprising just a single selected spin

s₀

(S

0

) and its neighbor spins {s

₁

,

s₂

, S

1

, S

2

, S

3

, S

4

} ({S

1

, S

2

,

s₁

,

s₂

,

s₃

,

s₄

}) with which it directly interacts (see Figure 1(b)).

We split the total Hamiltonian (1) into two parts, H

D

H

0C

H

⁰

, where H

0

includes all parts of H associated with the lattice site 0. In the present system, H

0

takes the form

H

0sD J2

X⁴

iD1

S

_iC

J

4ð

S

1

S

2s₁C

S

3

S

4s₂ÞC

J

⁰ðs₁Cs₂Þ

" #

s₀

(2)

H

0SD J2

X⁴

iD1

s_iC

J

4ðs₁s₂

S

1Cs₃s₄

S

2Þ

" #

S

0

(3)

where the lattice site 0 belongs to

s

or S-sublattice.

The problem consists in evaluating the sublattice magnetizations and the quadrupolar moment.

To this end, we denote by

hs₀ic

and

hSⁿ₀ic

(n

D

1, 2), respectively, the mean value of

s₀

and S

ⁿ₀

for a given configuration c of all other spins (i.e. when all other spin

si

and S

_j

(i, j

6¼

0) are kept

fixed).

hs₀ic

and

hSⁿ₀ic

are given by

s₀

h i_CD^Trs0s₀

exp

ðbH0sÞ

Trs0

exp

ðbH0sÞ

(4)

S

ⁿ_{0 C}D^TrS0

S

ⁿ₀

exp

ðbH_0SÞ

TrS0

exp

ðbH_0SÞ

(5)

where Tr

s0

(or Tr

_S0

) means the trace performed over

s₀

(or S

0

) only. As usual

bD

1/T, where T is the absolute temperature. The sublattice magnetizations

m,

m and the quadrupolar moment q are then given by

mh is_{0 C}

D ^Trs0s₀

exp

ðbH0sÞ

Trs0

exp

ðbH0sÞ

(6) m

h i

S

0 _C

D ^TrS0

S

0

exp

ðbH_0SÞ

TrS0

exp

ðbH0SÞ

(7) q

D

S

²_{0 C}E

D ^TrS0

S

²₀

exp

ðbH_0SÞ

TrS0

exp

ðbH0SÞ

(8)

where

h^…i

denotes the average overall spin configurations. Performing the inner traces in (6), (7) and (8) over the states of the selected spin

s₀

(S

0

), we obtain the following exact relations:

mD

1 2 tanh K

2

fð

S

1C

S

2C

S

3C

S

4ÞCað

S

1

S

2s₁C

S

3

S

4s₂ÞCg sð ₁Cs₂Þg

(9) m

D

2sinh K

½ fðs₁Cs₂Cs₃Cs₄ÞCa sð ₁s₂

S

1Cs₃s₄

S

2Þg

1

C

2cosh K

½ fðs₁Cs₂Cs₃Cs₄ÞCa sð ₁s₂

S

1Cs₃s₄

S

2Þg

(10) q

D

2cosh K

½ fðs₁Cs₂Cs₃Cs₄ÞCa sð ₁s₂

S

1Cs₃s₄

S

2Þg

1

C

2cosh

½

K

fðs₁Cs₂Cs₃Cs₄ÞCa sð ₁s₂

S

1Cs₃s₄

S

2Þg

(11) where K

DbJ2

,

aD

J

4

/J

2

and

gD

J

⁰

/J

2

.

It is very difficult task to calculate the average on the right-hand sides of Equations (9)–(11) over-

all spin configurations. We can easily observe that any function such as

ƒ(s,

S) of

s

and S can be

written as the linear superposition

f

ðs;

S

ÞD

f

1C

f

2sC

f

3

S

C

f

4

S

²C

f

5sSC

f

6sS²

(12) with appropriate coefficients f

_i

(i

D

1,

…, 6). After applying this to all spinssi

and S

_j

in expressions between brackets in Equations (9)–(11), we average overall spin configurations. In this paper, we use the simplest approximation in which we treat all spin self-correlations exactly while still neglect- ing correlations between quantities pertaining to different sites. This leads to the following coupled equations.

mDm½2A1C

4

ð

A

2C

A

3ÞqCð

2A

4C

8A

5C

2A

6Þq²C

4

ð

A

7C

A

8Þq³C

2A

9

q

⁴C

m

½

4A

10Cð

4A

11

C

8A

12ÞqCð

4A

13C

8A

14Þq²C

4A

15

q

³C

mm

²

4A

16C½

4A

17C

8A

18ÞqCð

4A

19C

8A

20Þq² C

4A

21

q

³Cmm²½

2A

22C

8A

23C

2A

24Cð

4A

25C

8A

26C

4A

27C

8A

28ÞqCð

2A

29C

8A

30

C

2A

31Þq²C

m

³½

4A

32C

4A

33

q

Cmm⁴½

2A

34Cm²

m

³½

4A

35C

4A

36

q (13) m

Dm

4B

1C

8B

2

q

C

4B

3

q

²

C

m

½

2B

4C

2B

5

q

Cm³

4B

6C

4B

7

q

C

4B

8

q

²

C

mm

²½

2B

9C

2B

10

q

Cmm²½

4B

11C

mm

⁴½

2B

12C

2B

13

q

C

m

²m³½

4B

14

(14) q

D

C

1C

2C

2

q

C

C

3

q

²

Cm²

6C

4C

12C

5

q

C

6C

6

q

²

C

m

²½

C

7 Cmm½

4C

8C

4C

9

q

Cm⁴

C

10C

2C

11

q

C

C

12

q

²

C

mm

³½

4C

13C

4C

14

q

Cm²

m

²½

6C

15C

m

²m⁴½

C

16

(15) The non-zero coefficients quoted in Equation (13) are very lengthy. They can be obtained from the authors on request, while those quoted in Equations (14) and (15) can be taken from Appendix in [85], where the functions g(a

C

ba) and h(a

⁰C

b

⁰a) must be changed, respectively, to

g(¡a

C

ba) and h(¡a

⁰C

b

⁰a) with

D

D

0.

If we replace m and q in (13) by their expressions taken from (14) and (15), we obtain an equa- tion for

m

of the form

mD

am

C

bm

³C. . .

(16)

In order to obtain the second-order transition temperature, we neglect higher-order terms in the magnetizations in Equations (13)

–

(15). Therefore, the critical temperature is analytically obtained through a determinantal equation, i.e.

1

D

2A

1C

4

ð

A

2C

A

3Þq0Cð

2A

4C

8A

5C

2A

6Þq²₀C

4

ð

A

7C

A

8Þq³₀C

2A

9

q

⁴₀ C

4B

1C

8B

2

q

0C

4B

3

q

²₀

1

ð

2B

4C

2B

5

q

0Þ

4A

10Cð

4A

11C

8A

12Þq0Cð

4A

13C

8A

14Þq²₀C

4A

15

q

³₀

;

(17)

where q

0

is the solution of

q

0D

C

1C

2C

2

q

0C

C

3

q

²₀:

(18)

Equation (17) means that its right-hand side corresponds to the coefficient a in (16).

In the vicinity of the second-order transition, the sublattice magnetization

m

is given by

m²D

1 a

b

:

(19)

The right-hand side of (19) is positive since we are in the long-ranged ordered regime. This means that the signs of 1

¡

a and b are the same. When b changes sign (1

¡

a keeping its sign),

m²

becomes negative. It means that we are not in the vicinity of a second-order transition line. So, the transition is of the

fi

rst order. Therefore the point at which

a K;

ð a;gÞD

1andb K;

ð a;gÞD

0

;

(20)

characterizes the tricritical points.

To obtain the expression for b, one has to solve (13)–(15) for small

m

and m. The solution is of the form

q

D

q

0C

q

1m²C

q

2

m

²C

q

3m

m

;

(21) where q

1

, q

2

and q

3

are given by

q

1D

6C

4C

12C

5

q

0C

6C

6

q

²₀

1 2C

2

2C

3

q

0

;

q

2D

C

7

1 2C

2

2C

3

q

0

;

q

3D

4C

8C

4C

9

q

0

1 2C

2

2C

3

q

0

:

After some algebraic manipulations, (14) and (21) can be written in the following forms

m

D

A

B

mC

8B

2

q

4

B

C

4B

3

q

0

q

4

B

C

C B

C

DA

B

² C

2AB

5

q

4

B

² C

4B

11

A

²

B

³

m³C. . .

(22)

q

D

q

0C

q

4m²

(23)

where

A

D

4B

1C

8B

2

q

0C

4B

3

q

²₀;

B

D

1

ð

2B

4C

2B

5

q

0Þ;

C

D

4B

6C

4B

7

q

0C

4B

8

q

²₀

D

D

2B

9C

2B

10

q

0;

q

4D

q

1C

A B

2

q

2C

A

B q

3

By substituting m and q in Equation (13), with their expressions taken from Equations (22) and (23), we obtain Equation (16), where b is given by

b

D

4

ð

A

2C

A

3Þq4C

2 2A

ð 4C

8A

5C

2A

6Þq0

q

4C

3 4A

ð 7C

4A

8Þq²₀

q

4C

8A

9

q

³₀

q

4

C

A

B

ð

4A

11C

8A

12Þq4C

2 4A

ð 13C

8A

14Þq0

q

4C

12A

15

q

²₀

q

4C

4A

16C

4

ð

A

17C

8A

18Þq0

Cð

4A

19C

8A

20Þq²₀C

4A

21

q

³₀C

EA

²

B

² C

FA

³

B

³ C

GI; (24)

with

E

D

2A

22C

8A

23C

2A

24Cð4A25C

8A

26C

4A

27C

8A

28Þq0Cð2A29C

8A

30C

2A

31Þq²₀

F

D

4A

32C

4A

33

q

0;

G

D

4A

10Cð

4A

11C

8A

12Þq0Cð

4A

13C

8A

14Þq²₀C

4A

15

q

³₀

and

I

D

8B

2

q

4C

8B

3

q

0

q

4C

C

B

C

AD

C

2B

5

q

4

A

B

² C

4B

11

A

²

B

³ :

4. Results and discussions

In this section, we examine the magnetic properties of the mixed spin-1 and spin-1/2 Ising ferrimag- netic with four-spin interaction J

4

and NNN coupling J

⁰

. Particularly, we are interested in the phase diagram and in the location and the multitude of the compensation point: the temperature where the total magnetization vanishes below the critical point. We have to note that such a behavior is of technological significance since at this point only a small driving

field is required to change the sign

of the resultant magnetization.

Let us

first consider the system in the absence of the NNN interaction (J⁰ D

0). The system is reduced to two-sublattice mixed spin with only bilinear and four-spin interactions. The ground state for this model corresponds to the line J

⁰D

0 in Figure 2. In Figure 3 we plot the sublattice and the total magnetizations for various values of J

4

/J

2

, where the system, at the ground-state, is ferrimag- netic. Using the state Equations (13)–(15), and as can be observed in this

figure, the study of the

magnetic properties leads to the conclusion that a compensation point cannot be induced in this model by the four-spin coupling. Indeed, at absolute zero the sublattice (

m

, m) and the total m

T

magnetizations have their saturation values (

¡

0.5, 1 and 0.25, respectively). As the temperature is increased, the effects of thermal agitation begin to be felt. Thus, the spontaneous magnetizations decrease with increasing temperatures and vanish once at a J

4

-dependent critical temperature T

C

(J

4

/ J

2

). This means that the total magnetization, for all selected values of J

4

/J

2

, does not vanish below the critical temperature T

C

(J

4

/J

2

). This can be explained as follows: the four-spin coupling affects to the both spin-1/2 and spin-1 atoms, instead of the next-nearest neighbor coupling (NNNC), which act only to the spin-1/2 atoms. So that both averaged magnetizations are similarly affected by the four spin coupling, different from the NNNC. Accordingly, it may be difficult to

find a compensation

point, since the compensation phenomenon results from the cancellation of the spin-1/2 and spin-1 atomic magnetization.

Second, we study the mixed spin Ising system with only next-nearest-neighbor interaction; in

order to investigate the phase diagram and compensation phenomena. The ground-state of the

model corresponds to the line J

4D

0 in Figure 2. Its phase diagram is shown in Figure 4, obtained from Equation (17). We note that for J

⁰>

2J

2

, the critical temperature increases with increasing val- ues of NNN interaction, which is physically reasonable. As is clearly seen from Figure 4, the system exhibits, at low temperatures, a reentrant in a narrow range of J

⁰

(

¡

2.44J

2<

J

⁰

2J

2

). This phenome- non is due to the competition between J

⁰

and J

2

. In Figure 5, we plot the temperature dependence of the sublattice and the total magnetization (m

T D

(

m C

m)/2) for selected values of J

⁰

/J

2

. At the ground state (T

D

0), all curves start from their saturation values (

mD ¡

1/2, m

D

1, m

TD

0.25). As clearly observed through them, the magnetic moments of the sublattices compensate each other completely (m

TD

0) at the compensation temperature T

Comp

. The location of this latter in the phase diagram is represented by the dotted line in Figure 4. This behavior occurs when the interaction J

⁰

is strong enough compared with the bilinear one J

2

. As seen in Figure 5(b), the total magnetization is not sensitive to J

⁰

below T

Comp

, and it becomes sensitive when the temperature belongs to the range

Figure 3.Temperature dependence of the sublattice and the total magnetizations when the value ofJ⁴/J²is changed,withJ⁰/J²D 0. No compensation is seen in this case.

Figure 4.Phase diagram in the (J⁰,T) plane forJ4/J2D0.

T

Comp<

T

<

T

C

. Therefore, we conclude that the compensation temperature seems to be indepen- dent of the NNN interaction.

Let us now examine the in

fl

uence of the four-spin interaction on the compensation behavior and the phase diagram (Figure 4), obtained for the Ising ferrimagnet with NNN interaction. Using the state Equations (13)

–

(15), we plot in Figure 6, the critical and compensation temperature against the NNN interaction J

⁰

, for particular values of J

4

. From these

figures, we note that the compensation

temperature does not exist until the J

⁰

interaction takes some minimum value. This latter depends on the strength of J

4

interaction. As shown in Figure 6, the four-spin interaction has remarkable effects on location, existence and multiplicity of the compensation points. Indeed, as previously mentioned, in the absence of four-spin interaction (J

4D

0), the compensation temperature (when it exists) seems to be independent on NNN interaction (Figure 4); whereas for non-zero J

4

, T

Comp

depends qualitatively and quantitatively on the sign and the strength of J

4

. In fact, as is illustrated in Figure 6(a), for a positive value of the ratio J

4

/J

2

, the compensation temperature increases with the increasing value of J

⁰

and the system exhibits only one compensation point. We note here that each critical line ends in a tricritical point. Thus, the system keeps its tricritical behavior when J

4

/J

2

belongs to the range 0.076

<

J

4

/J

2

3. However, for the negative value of the four-spin interaction (Figure 6(b)) the compensation temperature decreases with increasing values of J

⁰

. It is worth to notice that at around J

⁰»

5J

2

, this decrease becomes more and more abrupt as J

4

approaches its crit- ical value

¡

4J

2

. Furthermore, for a given value of J

4

belonging to

¡

4J

2<

J

4

2.1J

2

, the line of com- pensation points, in the phase diagram, exhibits a reentrance. This means that there exists a range of J

⁰

where the system exhibits two successive compensation points. For higher values of J

⁰

the system undergoes only one compensation temperature. The location of this latter seems to be insensitive to

Figure 5.Temperature dependence of (a) the sublattice and (b) the total magnetizations when the value ofJ⁰/J2is changed, withJ4/J2D0.

Figure 6.Phase diagrams in the (J⁰,T) plane for (a) positive and (b) negative values ofJ4/J2.

sufficiently large values of J

⁰

and depends only on the strength of J

4

. On the other hand, it is very interesting to investigate the magnetic properties of the system in the vicinity of the compensation line where the slope is vertical. As seen in Figure 6(b), we observe that this region is spread out on a large range of temperatures for any

¡4J2<

J

4 2.1J2

and 4.30J

2<

J

⁰ <

4.88J

2

. In this domain, a detailed numerical treatment of the state Equations (13)

–

(15) shows that the total magnetization can be considered m

T

0 with high accuracy (

§

10

^¡5

). Thus, the mixed-spin Ising ferrimagnet with four-spin interaction J

4

and NNN coupling J

⁰

leads to the conclusion that this system can exhibit a range of compensation points in appropriate ranges of J

4

and J

⁰

. We believe that such a phenomenon is fruitful for technological applications since at this range of compensation points only a small driv- ing

fi

eld is required to change the sign of the total magnetization.

Finally, in order to facilitate the reading of the existence of compensation points according to the values of this interactions J

⁰

and J

4

, we indicate in Figure 7, the domain where the system does not present any compensation point and the one where it undergoes one, two or a range of

Figure 7.Compensation behavior for theJ2–J4–J⁰model in the (J⁰,J4) plane.

Figure 8.Total magnetization versus temperature forJ4/J2D ¡3.25 and several values ofJ⁰/J2. The system can exhibit one, two or a continuum of compensation points.

compensation temperatures. In Figure 8, we show examples of the behaviors of total magnetization where all these above kinds of compensation behaviors and critical point can be clearly observed.

5. Conclusion

In this work, we have studied the magnetic properties of the ferrimagnetic mixed-spin (1/2, 1) Ising model with the four-spin interaction J

4

and NNN coupling J

⁰

on square lattice. We have been mainly interested in the calculation of the phase diagrams and the behavior of the compensation points.

This study shows very interesting features. First, let us summarize by stating the main results of this investigation.

In the absence of the NNN interaction (J

⁰D

0), it has been shown that the four-spin interaction does not induce a compensation point for any strength of J

4

. However, the behavior of the ferrimag- netic mixed spin Ising model with only J

⁰

interaction is qualitatively different from the previous sys- tem. Indeed, it has been shown that the magnetic moments of the sublattice compensate each other at the compensation point T

Comp

. Its location on the phase diagram shows that it appears when J

⁰

is strong enough compared with the bilinear one J

2

, and it seems to be insensitive to NNN interaction.

Moreover, we have examined the influence of the four-spin interaction J

4

on the obtained compen- sation behavior and phase diagram. It has been found that J

4

has remarkable effects on the location, existence and multiplicity of the compensation points. Indeed, we have shown that for non-zero J

4

, T

Comp

depends qualitatively and quantitatively on the sign and the strength of J

4

. For J

4>

0, there is only one compensation point (when it exists) increases with increasing J

⁰

. However, for negative val- ues of J

4

, T

Comp

decreases with increasing values of J

⁰

. It is worth to notice that for

¡4J2<

J

4 2.1J2

, the line of the compensation exhibits reentrance. The detailed analysis of this behavior leads to the conclusion that this system can exhibit one, two or a continuum of compensation points in appro- priate ranges of the interactions J

4

and J

⁰

.

We can conclude that studied in this paper may be simple but fruitful from both theoretical and material sciences points of view, since it shows interesting behaviors, in particular, the existence of a continuum of compensation points. This phenomenon can be very useful from the technological point of view, especially in thermomagnetic writing and erasing along the range of compensation points. As far as we know, this is the

fi

rst time that such a continuum is found. So, we hope that our work will stimulate further theoretical approaches and/or experimental measurements.

Disclosure statement

No potential conflict of interest is reported by the authors.

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