Author’s Accepted Manuscript
Mixed spin (1/2,1) transverse Ising nanoparticles M. Mouhib, N. Benayad, M. Azhari
PII: S0304-8853(16)31042-3
DOI: http://dx.doi.org/10.1016/j.jmmm.2016.06.022 Reference: MAGMA61543
To appear in: Journal of Magnetism and Magnetic Materials Received date: 14 January 2016
Revised date: 22 May 2016 Accepted date: 9 June 2016
Cite this article as: M. Mouhib, N. Benayad and M. Azhari, Mixed spin (1/2,1) transverse Ising nanoparticles, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.06.022
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form.
Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
www.elsevier.com/locate/jmmm
1
Mixed spin (1/2,1) transverse Ising nanoparticles
M. Mouhib, N. Benayad*, M. Azhari
Laboratory of High Energy Physics and Scientific Computing, HASSAN II University–Casablanca, Faculty of Sciences-Aïn Chock,
B.P : 5366 Maarif , Casablanca 20100, Morocco.
*Corresponding author: E-mail: n.benayad@fsac.ac.ma ; noureddine_benayad@yahoo.fr Phone: + 212 5 22 23 06 84, Fax: + 212 5 22 23 06 74
_________________________________________________________________
Abstract
Mixed spin 2D-nanoparticles (circle and square) described by the transverse Ising model are investigated by the use of the finite cluster approximation. The effects of the exchange interactions and the transverse field parameters on the phase diagrams are systematically discussed, in particular, it was shown that the transition temperature is not so sensitive to shell exchange interaction, when the spins S are strongly correlated. A number of interesting phenomena have been found such as reentrant behavior. This latter, which is due to the competition between the exchange interaction and core-transverse field, disappears completely for any no-zero shell transverse field.
Keywords: mixed spin, circle and square 2D-nanoparticle, Transverse Ising model, Finite cluster approximation, Phase diagrams.
___________________________________________________________________________
1. Introduction
During the last few years, research on magnetic nanomaterials such as nanoparticles,
nanorods, nanobelts, nanowires and nanotubes, has been attracting considerable attention, and
is considered the most actively studied topic in statistical mechanics and condensed matter
physics. This is due to their promising applications in nanotechnology [1,2] such as
permanent magnets [3], information storage devices [4,5], biosensors [6] and some medical
applications [7]. At present, the scientist can produce such kinds of fine nanoscaled materials
[8,9], and the magnetization of certain nanomaterials such as γ-Fe
2O
3nanoparticles has been
experimentally measured [10]. In particular, magnetic nanowires and nanotubes such as ZnO
[11], FePt, and Fe
3O
4[12] can be synthesized by various experimental techniques and they
have many application in nanotechnology [13,14]. They are also utilized as raw materials in
fabrication of ultra-high density magnetic recording media [15-17]. On the other hand, the
2
magnetic properties of a nanometric scale material depend highly on the size and dimensionality of the nanomaterial.
From the theoretical point of view, Ising nanomagnetic systems have been studied by a variety of techniques, including mean field theory [18-20], effective field theory with correlations [21-24], Monte Carlo simulation [25-28], variational cumulant expansion method [29,30], and Green’s function techniques [31].
The transverse Ising model (TIM) is believed to describe phase transitions occurring in several physical systems, such as cooperative Jahn-Teller system [32] (like DyVO
4and TbVO
4) and also in some real magnetic materials with strong uniaxial anisotropy in a transverse field [33]. The model has also been successfully applied to the investigation of materials with nanostructures, i.e. nanoparticles, nanowires and nanotubes [34-36].
Recently, the study of two sublattice mixed spin systems have become one of the actively studied subject in statistical physics. They are of interest for the following main reasons: The first one is that they have less symmetry than their single-spin counterparts. The second one is that they are well adapted to study a certain type of ferrimagnetism [37], and finally many new phenomena are observed in these systems, which cannot be seen in the single-spin Ising model [38]. However, there are few studies in the literature that include the transverse field on the magnetism of nanoscaled particle systems with core-shell structure [39,40]. As far as we know, no works have been carried out with respect to magnetic properties of two dimensional (2D) transverse Ising nanoparticles consisting of a mixed spins.
The aim of this paper is to investigate the mixed spin transverse Ising nanoparticles.
These nanoparticles, consisting of spin 1/2 and spin 1, have two different shapes (circle and square). In particular, we study the effect of the core-shell coupling, as well as the influence of the transverse fields acting on the core and shell sites on the phase diagrams. To this end, we use the finite cluster approximation (FCA) [41,42] within the framework of a single-site cluster theory.
The outline of this work is as follows: in section 2, we present the model and the theoretical framework. The state equations are derived. The main results and discussion are arranged in section 3. Finally, the last section is devoted to a brief conclusion.
2. Theoretical framework
We consider two kinds of 2D-nanoparticles (circle and square), as depicted in Fig. 1. They consist of the surface shell and the core. The core is surrounded by the surface shell. Each site is occupied by a spin operator ⃗ or ⃗⃗⃗⃗ which are distributed in concentric and alternate rings.
The Hamiltonian of the system is given by
∑
〈 〉
∑
〈 〉
∑
〈 〉
∑
〈 〉
∑ ∑
(α= x or z) are components of spin-1/2 and spin-1 operators, respectively.
J
sis the exchange interaction between two nearest neighbor magnetic atoms at the surface shell. This latter is coupled to the next shell in the core with the exchange interaction J
1. J is the exchange interaction between spins in the core. J
2is the exchange interaction between the central site and the sites belonging to the first ring. The terms and represent the transverse field acting on the core and shell sites, respectively.
The method that we adopt in the study of the mixed spin transverse Ising nanoparticles
(circle and square) consisting of spin 1/2 and spin 1 described by the Hamiltonian (1) is the
finite cluster approximation (FCA) [41,42] 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
3
disordered spin Ising systems [43,46]. It has also been used for transverse Ising models [47- 50] and semi-infinite Ising systems [51-55].
Within the core-shell concept, for the nanoparticle with a circle shape depicted in Fig .1 (a), we have to determine the magnetizations at each site in the core and at the surface shell.
Indeed, we denote by 〈 〉, 〈 〉 (α= z or x) the magnetizations per site in the core corresponding to the central site and the site belonging to the ring consisting of spin S, respectively. At the surface shell of the system, we have to define two magnetizations 〈 〉 and 〈 〉 which correspond to the coordination numbers z=3 and z=4, respectively.
Using the framework of the FCA, we evaluate 〈 〉 , 〈 〉 , 〈 〉 and 〈 〉 the mean values of , , and for a given configuration c of all other spins (i.e when all other spins and are kept fixed).
Following the same procedure as in the [56], they are given by 〈 〉
(2) 〈 〉
(3) 〈 〉
(4) 〈 〉
(5)
where
√
( √ ) (6)
√
( √ )
( √ )
(7)
( √ )( √ )
(8) with
∑
∑
∑
∑
∑
∑
Here, ( or ) means the trace performed over ( or ). As usual ⁄ where T is
the absolute temperature. n=1,2 correspond to the magnetization and the quadrupolar moment
respectively. Therefore, the longitudinal magnetizations at the shell and in the core are then
given by
4
〈〈 〉 〉 〈 〉 (9)
〈〈 〉 〉 〈 〉 (10)
〈〈 〉 〉 〈 〉 (11)
〈〈 〉 〉 〈 〉 (12)
〈〈 〉 〉 〈 〉 (13)
where 〈 〉 denotes the average over all spin configurations. is the longitudinal quadrupolar moment. The corresponding results for the transverse components , , , and may be obtained from the longitudinal components by the same equations as those of (9) – (13), only replacing the functions and by the new functions and defined by
√( √ ) (14)
√ ( √ ) ( √ )
(15)
( √ ) ( √ )
(16)
Now, we have to average the right-hand sides of Eqs. (9)-(13) over all spin configurations, within the FCA framework. This latter has been designed to treat all spin self-correlations exactly while still neglecting correlations between different spins. We can easily observe that any functions such as f( , ) and g(S) of and can be written as the linear superpositions (17)
(18) with appropriate coefficients f
i(i=1,…,6) and g
j(j=1,2,3). Applying this to all spins σ
iand S
jin expressions between brackets in eqs. (9)-(13), we obtain
〈 〉 ∑ ∑ ∑ ∑
[ ]
〈 〉 ∑ ∑ ∑ ∑ ∑ ∑
[ ]
〈 〉 ∑ ∑ ∑ ∑
[ ( ) ]
〈 〉 ∑ ∑ ∑ ∑ ∑ ∑
[ ]
5
〈 〉 ∑ ∑ ∑ ∑ ∑ ∑
[ ]
where [ ] denotes the term containing p different factors of , n different factors of , and l different factors of with (k≠j). [ ] denotes the term containing n different factors of , and l different factors of . These factors are selected from sets { }, { }, { } and { } for 〈 〉 , 〈 〉 ,
〈 〉 and 〈 〉 , respectively.
For example:
p=1, n=0, l=1:
p=2 , n=1 , l=1:
n=3 , l=1:
( )
The magnetizations at the shell and in the core are given by Eqs. (9)-(13) using the expansions (19)-(23). They constitute a set of exact relations according to which we can study the nanoparticle with a circle shape. However, in order to carry out the thermal average over all spin configurations, we have to deal with multispin correlations appearing via the right-hand side of Eqs. (19)-(23). The problem becomes mathematically intractable if we try to treat them exactly in the spirit of the FCA. In this paper, we use the simplest approximation in which we treat all spin self-correlations exactly while still neglecting correlations between quantities pertaining to different sites even if they are the same type. This leads to the following coupled equations
(24) (25)
(26)
(27)
(28)
6
The non-zero coefficients quoted in Eqs. (24)-(28), are listed in the appendix.
At high temperature, the longitudinal magnetization moments (i=1,2,3) and m
zare all equal to zero. Below a transition temperature T
c, we have spontaneous ordering ( (i=1,2,3)≠0 and m
z≠0), while the corresponding transverse magnetizations (i=1,2,3) and m
xare expected to be non-zero at all temperatures. If we substitute (i=1,2,3) and q
zinto (27) with their expressions taken from (24), (25), (26) and (28), we obtain an equation for m
zof the form
m
z=a.m
z+ b.(m
z)
3+…. (29) The second-order transition is determined by the condition a=1, for this reason we neglect higher-order terms in the magnetizations in (24)-(28). Therefore, the order-disorder transition temperature is analytically determined by the equation
(30) with
[ ] [ ]
[ ] [ ]
where q
0is the solution of
(31) Eq. (30) means that the right-hand-side corresponds to the coefficient a in Eq. (29).
The magnetization m
zin the vicinity of the second-order transition is given by
(32) The right-hand side of (32) must be positive. If this is not the case, the transition is of the first- order and the point at which a=1 and b=0 characterizes the tricritical point. To obtain the expression for b, one has to solve Eqs (24)-(28) for small , , and . The solution is of the form
(33) where
,
,
,
7
,
After some algebraic manipulations, Eqs (24), (25), (26) and (33) can be written in the following forms
=A.m
z+ D.(m
z)
3+…… (34)
=B.m
z+ E.(m
z)
3+…… (35)
=C.m
z+ F.(m
z)
3+…… (36)
(37)
Where { } { }
By substituting , , and in Eq.(27), with their expressions taken from (34)-(37), we obtain the equation (29), where b is given by
(38) For the nanoparticle with a square shape depicted in Fig. 1(b), we define three magnetizations
, , at the surface shell, three magnetizations ( and at the second shell and at the center) in the core, and two quadrupolar moments and . Using FCA, we determine all these quantities from coupled equations, similar to those (Eqs. (24)-(28)) written for the circle nanoparticle, so that they will not be presented here.
In the present work, we consider that the shells at the surface and in the core interact with each other via the same coupling (J
1= J
2), and all interactions as well as the temperature are normalized with exchange interaction J
2.
3. Results and discussions
Let us introduce the magnetizations
and at the shell and in the core as well as
the total magnetization per site . Their expressions are given by
8
,
,
(39) for circle nanoparticle,
,
,
(40) for square nanoparticle.
In order to examine the order-disorder transition temperature of the shell and the core, we investigate the thermal dependence of the longitudinal magnetizations. By analyzing the coupled equations (24)-(28), we find that all magnetizations ( , , and ) vanish at a unique order-disorder transition temperature, which a solution of Eq. (30). This behavior is observed for any strength of the Hamiltonian parameters as is shown in Fig. 2 and Fig. 3.
Therefore, the shell and the core undergo a transition at the same temperature. The same behavior is also observed for the nanoparticle with square shape. It is worthy of notice that the similar behavior have been observed for monoatomic spin -1/2 Ising 2D nanoparticles [21].
First let us investigate the system, described by the Hamiltonian (1), in the absence of shell and core transverse fields . In Fig. 4, we present in ⁄ ⁄ plane the phase diagrams for circle and square nanoparticles for selected values of the shell coupling J
s. As seen in this figure, for fixed value of the shell coupling J
s, the order-disorder transition temperature increases gradually with the increase of J. For relatively low strength of this latter interaction, the transition temperature increases with J
s. This dependence becomes less and less sensitive for high values of the interaction J. We note that this behavior is more pronounced for the nanoparticle with a square shape.
It is a well-known fact that, by increasing transverse field and values at the surface shell and in the core respectively the system is dragged to the disordered phase, due to the distribution of magnetic transverse fields on the lattice sites. On the other hand, the interactions J
sand J enforce the system to stay in the ordered phase. Thus a competition takes place between these two factors, in addition to the thermal agitations.
In Fig. 5 we investigate the influence of the transverse fields and (acting on the shell and core sites, respectively) on the phase diagrams plotted in Fig. 4 for both circle and square nanoparticles. Since the second ring in the nanoparticles is composed exclusively of spin -1, it is interesting to investigate how the nanoparticles behave when these spins do not interact (J=0). Their phase diagrams in T- - space are represented in Fig.5. Let us first indicate that, from these figures, we observe qualitatively interesting features in the T- plane when no transverse field acts on the spins located at the surface shell. In particular, as seen in Fig. 6 we note the existence of a critical value ⁄ of the surface exchange interaction ( ⁄ =2.0 and ⁄ =1.0 for the circle and square nanoparticles, respectively), indicating two qualitatively different behaviors of the system which depend on the range of J
s. Thus for
⁄ ⁄ , the nanosystem exhibits at the ground state a transition at a finite critical value ⁄ (for instance, ⁄ =3,5517 , ⁄ and ⁄ =1,3337 , ⁄ for circle and square nanoparticles, respectively) . But for ⁄ ⁄ there is no critical core- transverse field, and therefore, at very low temperatures the ordered state is stable for any value of the core-transverse field . This means that, although the transverse field reduces the longitudinal magnetization and therefore acts against the order, it cannot destroy the order.
As clearly shown in Fig. 5, the nanosystem exhibits a reentrant behavior in the narrow ranges
of J
sand . This latter phenomenon is due to the competition between the exchange
interaction and the core-transverse field. It is worth noting that this reentrant behavior exists
only when ⁄ , since it disappears for any non zero value of the shell transverse field
for both nanoparticles. Reentrance in this sense can be explained by a competition of energy E
and entropy S in the free energy F=E-TS. The entropy (measuring the disorder of the state) is
usually less important at low temperatures. In the absence of the transverse fields and at low
9
temperatures, the energy term becomes dominant and ordered state of the physical system is preferred. However, in the presence of the transverse field, an effective reduction of the energy E as compared with a disordered state may not be possible for all interaction parameters. The core transverse field by itself energetically disfavors the state | | . For a given set of J
Sand Ω this conspires to a thermodynamic state with < > = 0, for sufficiently low and high temperature, but with < > > 0 for some intermediate temperature and hence
< > > 0. This qualitative expectation is indeed verified by our quantitative calculations. On the other hand, as seen in Fig. 5 we can note that in the absence of the core-transverse field ( =0), the circle and square nanoparticles do not exhibit similar behavior at low temperatures.
Indeed, we plot in Fig. 7(a) and Fig. 7(b), the phase diagrams of the circle and square nanoparticles respectively, when the transverse field acts only on the spins situated on the surface shell. This gives the sections of the transition surface ⁄ ⁄ with planes of fixed values of the surface exchange interaction. As seen in the figure, the order-disorder transition temperature falls with increasing surface transverse field, and vanishes at a critical value of . This latter value, for the circle nanoparticle, depends on the strength of the surface exchange interaction; whereas for the nanoparticle with square shape, it is not so sensitive to J
sat low temperatures.
Let us now study the effects of the exchange interaction J on the behavior of the nanoparticles plotted in Fig. 5. As can be expected, this interaction changes quantitatively and qualitatively the phase diagrams of the nanosystems under investigation. Indeed, the positive value of this exchange interaction rises the absolute value of lattice energy coming from the spin1-spin1 interaction, and then reinforces the longitudinal magnetization. Therefore it acts in favor of the order. Thus the domain of the order state becomes wider and the transition temperature increases with J, which is clearly shown in Fig. 8 for both nanoparticles. On the other hand, J has a significant influence on the existence of the reentrant behavior observed in the phase diagram drawn in Fig. 5. In fact, the analysis of the system with no-zero exchange interaction J, shows that the bulge in the phase boundary (reentrant phenomenon) disappears even in the absence of the surface transverse field ( =0). Such behavior is shown in Fig. 8 for the nanoparticles with circle or square shapes.
Finally, We note that all results has been obtained within the zeroth-order approximation of the FCA. We can improve them by using higher order approximation. For example, for the first- order for < σ >
c( <S >
c), we perform the traces over σ
o( S
o) and its first neighbors and all other spins of the system having fixed values. Then, we continue the calculations in the spirit of the FCA. This procedure can be used systematically for higher orders. This approach, applies to the Ising model, shows an improvement of the transition temperature [42]. It is worth noting here that the zeroth-approximation of the FCA is equivalent to the effective field theory (EFT). This later approach has been used for monoatomic “circle” and “square”
nanoparticle [34]. We note that both methods give the same behavior of M.vs.T.
10
Appendix
For abbreviation we consider new functions
√
( √ )
√ ( √ ( √ ))( √ ( √ ) )
The coefficients A
i(i=1,…,4), B
i(i=1,…,8), C
i(i=1,…,12), D
i(i=1,…,24) and E
i(i=1,…,24) appearing respectively in Eqs. (24), (25), (26), (27) and (28) are given by:
11
