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Phase transition and magnetization of a hexagonal prismatic nanoisland with a ferrimagnetic spin configuration
Author’s Accepted Manuscript Phase transition and magnetization of a hexagonal prismatic nanoisland with a ferrimagnetic spin configuration Wei Jiang, Ya-Ning Wang www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(16)30689-8 http://dx.doi.org/10.1016/j.jmmm.2016.10.112 MAGMA62025
To appear in: Journal of Magnetism and Magnetic Materials Received date: 17 May 2016 Revised date: 26 August 2016 Accepted date: 21 October 2016 Cite this article as: Wei Jiang and Ya-Ning Wang, Phase transition and magnetization of a hexagonal prismatic nanoisland with a ferrimagnetic spin configuration, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.10.112 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.
Phase transition and magnetization of a hexagonal prismatic nanoisland with a ferrimagnetic spin configuration
Wei Jiang*, Ya-Ning Wang
School of Science, Shenyang University of Technology, Shenyang 110870, China
*
Corresponding
author.
Wei
Jiang,
Tel.:
+86-13840076827;
[email protected]
ABSTRACT
Magnetic properties of a nanoisland with a ferrimagnetic spin configuration, described by the transverse Ising model, are studied by the effective-field theory with correlations. The hexagonal prismatic nanoisland consists of the bilayer with core-shell structure. The phase transition and the magnetization of system have been calculated for different values. A lot of novel features, such as the reentrant phenomenon, have been found in the phase transition diagrams of the nanoisland. They are heavily dependent on the exchange coupling, the single-ion anisotropy and the transverse field. These theoretical results may have guiding significance for preparing nanoisland experimentally.
Graphical Abstract
Phase diagrams and magnetization of the hexagonal prismatic nanoisland have been examined by the effective-field theory with correlations.
Keywords: Nanoisland; Ferrimagnetic; Magnetization; Phase transition
1. Introduction During recent years, the magnetic nanomaterials have been the focus on attention due to special structure and properties, related with their wide applications [1-11]. Nanoisland, one of the magnetic nanomaterials, can remarkably improve the specific surface area. Thus, the study of nanoisland is a significant issue on fundamental theoretical and experimental viewpoints. In experimental, great progress has been made in the study of nanoisland. Yamada et al have studied the atomic and magnetic order in bilayer Fe nanoislands grown on a Cu (111) substrate under the specified conditions. Besides, they found electric field pulses can cause a displacement of the Fe atoms, which proves of magnetoelectric coupling at the metallic Fe surface [12]. Nanoislands of α'-FeRh were prepared by the deposition of a 10 nm film onto (001)-MgO. This island structure leads to stabilized ferromagnetic (FM) ordering at low temperature [13]. High-density magnetic storage or quantum computing could be achieved using low dimensional rare-earth iron structure, such as the magnetic coupling between 4f single atoms and a 3d magnetic nanoisland [14]. The piezoelectric and magnetoelectric in three-dimensional multiferroic nanostructure
have been studied by using phase-field simulations. This structure consists of vertically magnetized nanomagnet with an in-plane long axis and a juxtaposed ferroelectric nanoisland [15]. Bruggemann et al have investigated cooling of a magnetic quantum dot by a spin-polarized tunneling to charge current exploiting the magnetomechanical coupling. The spin-polarized current polarizes the magnetic nanoisland, thereby lowering its magnetic energy [16]. Low-energy proton irradiation was applied to pattern ferromagnetic nanoislands through the local phase transformation. Such an array is strong enough to overcome the so-called superparamagnetism limit in magnetic recording [17]. Ordered FePdCu nanoisland arrays were formed by annealing on the top of SiO2 nanosphere. This is a method of providing nanoislands with well-controlled size and positions [18]. The initial nucleation and growth of Co nanoislands on highly oriented pyrolytic graphite (HOPG) were experimentally investigated by scanning tunneling microscopy with atomic resolution at room temperature [19]. The research on nanoisland progress fast experimentally, however, only a few researchers have carried out the theoretical study of nanoisland. Thus theoretical investigations are urged to explore the nanoislands. Choi et al have systematically investigated shape-dependent magnetic moment and island formation energy of Fe adatoms on a Cu (111) surface by the density functional theory based first-principles calculations. The magnetic moments of Fe nanoisland atoms rest with two key factors, the numbers of Fe-Fe bonding and the angles between the Fe-Fe bonding [20]. The phase diagrams and temperature dependences of magnetizations in transverse Ising
nanoislands which are consisted of the two layers are examined by using the effective-field theory with correlations (EFT). A lot of unexpected characteristic phenomena are obtained for the magnetic properties in those systems [21-24]. Further, the magnetic properties of hexagonal prismatic nanoislands with transverse field and magnetic anisotropy have not been examined. We have successfully studied magnetic properties of various nanoscaled transverse Ising model systems by the effective-field theory with correlations in a series of recent works [25-31]. The results show the physical parameters including surface coupling, interlayer coupling, transverse fields and the single-ion anisotropy have important effects on the phase diagrams, the magnetizations and the thermodynamics properties. Therefore, in this paper, we intend to continue to study the magnetic properties of a hexagonal prismatic nanoisland within the EFT. The paper is organized as follows: in Section 2, the models and formulations for the system are given. In Section 3, we examine the numerical results of the phase diagrams and the magnetization. Finally, some conclusions are given in Section 4.
2. Models and formulations First we will introduce nanoisland structure as shown in Fig.1. It consists of two of the same layer. Each layer has a core atom C with a spin-2/5 (big pink ball) and six shell atoms S with spin-2 (small blue balls). The solid line represents the ferromagnetic exchange coupling J1 (>0) between the magnetic atom on the surface shell. The dashed line denotes the ferromagnetic exchange coupling J (>0) between the magnetic atom in bilayer. The dotted line is the ferrimagnetic exchange coupling
J 2 (<0), which comes from the shell spins coupled to the core spin. This system can be
abstracted by the transverse Ising model with the anisotropy. Second the effective-field theory with correlations is applied to nanoisland and the calculation formula is given. The Hamiltonian of the present system is written as H J1 iz zj J iz kz J Smz S nz J 2 iz S mz ( ix S mx ) i, j
i,k
m, n
i ,m
i
m
(1)
D1 ( ) D2 ( S ) h( S ) z 2 i
z 2 m
i
z i
m
i
z m
m
Where iz and S mz are spin-2 and spin-2/5, respectively. The transverse and longitudinal magnetic field expressed by and h . D1 and D2 are the single-ion anisotropies, which come from the shell atoms S and the core C. By the effective-field theory with correlations, the core magnetization M c and shell magnetization M s at per site can be calculated as follows: M c (n1e J2s n2 e J2s )6 (n3e J c n4 e J c ) f c x
(2)
x 0
M s (n1e J1s n2 e J1s )2 (n3e J2c n4 e J2c ) (n1e J s n2 e J s ) f s x
x 0
(3)
Where x is the differential operator, and c2 and s2 as coupled equations: c 2 (n1e J n2 e J )6 (n3e J n4 e J ) gc x 2 s
2 s
c
c
(4)
x 0
s 2 (n1e J n2 e J )2 (n3eJ n4 e J ) (n1eJ n2 e J ) g s x 1 s
1 s
2 c
2 c
s
s
x 0
(5)
Where n1 , n2 , n3 and n4 are respectively defined as n1 (1 2)(1 M s s ) , n2 (1 2)(1 M s s ) , n3 (1 2)(1 M c c ) and n4 (1 2)(1 M c c ) . Meanwhile, the four
functions fc ( x) , f s ( x) , gc ( x) and g s ( x) are defined by f c ( x)
1
6 m 1
exp(m )
6
{ m Smz m exp( m )} m 1
(6)
f s ( x)
g c ( x)
g s ( x)
1
i 1 exp(i ) 5
5
{ i iz i exp(i )}
6
1
6
exp(m ) m 1 1
5 i 1
exp(i )
(7)
i 1
{ m ( Smz )2 m exp(m )}
(8)
m 1
5
{ i ( iz ) 2 i exp(i )}
(9)
i 1
Here 1 kB T and T is a temperature, and k B is the Boltzmann constant. The eigenvalues and eigenfunctions of the Hamiltonian are showed by m (i ) and m (i ) , depending on the value of spin. They can only be obtained by numerical
calculation through computer programming. Further, the total average magnetization
M
and susceptibility
are
respectively defined by M (M c 6 M s ) 7
M h h0
(10) (11)
3. Numerical results and discussions In this section, we investigate the phase diagrams and the behaviors of magnetizations in the hexagonal prismatic nanoisland by solving the formulations given in Section 2. In the following calculations, the ferromagnetic exchange coupling J1 on the surface shell is fixed J1 1.0 as a unit of the system. The effects of the
exchange couplings, the transverse field, single-ion anisotropies on the magnetic properties in this model have been discussed in detail. From the processes, we have found some unexpected characteristic phenomena in the system. 3.1 Phase transition temperature In this part, we shall discuss the effects of various parameters on the phase transition temperature kBTc for the hexagonal prismatic nanoisland. The intralayer
antiferromagnetic J 2 dependence of the phase transition temperatures are plotted in Fig. 2(a)-(c). In Fig. 2(a), six kBTc curves are presented with the ferromagnetic exchange coupling between bilayer J =0.6, 0.8, 0.9, 1.0, 1.3 and 1.5. In the figure, multi-transition temperatures can be found for some typical values J 2 , which namely the reentrant phenomena. For example, the curve exists three transition temperatures k BT =0.32, 0.78 and 1.59 with J =1.0 and J 2 =-0.53. The reentrant phenomena have
been found in a variety of disordered magnetic systems experimentally and theoretically [32-34]. This interesting behavior depends on the competition among the parameters J 2 , J , D1 and . The similar phenomenon can be observed in Fig. 2(b), but value of D1 is changed. For example, the curves exist two transition temperatures k BT =0.58, 0.94 with D1 =-2.2, J 2 =-0.18 and k BT =0.6, 1.33 with D1 =-2.9, J 2 =-0.32.
In the figure, strong D1 increase the number of phase transition temperature, but the reentrant phenomena disappear for D1 =-1.2. The value of kBTc may reduce to zero at the critical value J 2c , such as J 2c =-0.449, -0.328, -0.189, -0.052 for D1 =-3.5, -2.9, -2.2, -1.6, respectively. Fig.2(c) shows another case of the phase diagrams in the hexagonal nanoisland with J =1.2, D1 =-3.5, D2 =-0.4 and five fixed transverse field . For =1.0, 1.5 and 2.0, the phase transition temperature kBTc increases noticeably with increasing value of J 2 . As is seen from the curves labeled =0.2 and =0.5, the tendency of the transition temperature curves are different from others, and they have points of intersection with other curves. It may exhibit the reentrant phenomena, which have multi-transition points for some fixed values of J 2 . The interlayer ferromagnetic exchange coupling J dependences of the phase transition temperature kBTc are plotted in Fig.3 (a)-(c). Fig.3 (a) shows the numerical results of the phase diagram in the nanoisland for selecting some typical values of J 2 .
The kBTc curves exhibit multi-transition point when J 2 <1.37, especially the system has three phase transition temperatures kBT =0.34, 0.84, 1.48 for J 2 =-0.5 and J =1.04. The reentrant phenomena disappear for J 2 >1.37. In Fig. 3(b), various shapes of the phase diagram are observed with the different values of D1 . As seen from the figure, the reentrant phenomena also can be found in this system when D1 >2.5. For example, the curves have multiple phase transition temperatures, which kBT =0.65, 2.52 for J =0.4, D1 =-2.8 and kBT =0.34, 0.72, 2.35 for J =0.95, D1 =-4.2. When D1 <2.5, the curves have the same behavior which belong to the monotone increasing function and the reentrant phenomena disappear, such as D1 =-0.8, -1.5 and -2.0. In Fig.3(c), the phase diagram is given by selecting the five values of transverse field . The phase transition temperature curves fall from its maximum value and reaches zero for critical value. From the figure, we can easily see that the value of the critical value J c increases with increasing transverse field except for =0.2 ( J c =0.836). Further,
three kBTc are obtained in the system with a small value of ( <0.4), such as k BT =0.19, 0.45, 2.97 for J =0.85 and =0.2.
As discussed in [35-38], it is found that the anisotropy acts a major role in the low-dimension magnetic systems. Fig. 4(a)-(c) shows the variation of phase diagrams versus the shell single-ion anisotropy D1 by changing the value of J 2 , D2 and . Fig. 4(a) reveals the shell single-ion anisotropy D1 dependences of the phase transition temperatures when the value of intralayer antiferromagnetic exchange coupling J 2 is fixed some values. We should notice that the number of multi-transition temperatures is decreased with the value of D2
increases. The
system may exhibit the three kBTc for D2 =-0.6 and D2 =-0.8. Further, the critical value of D1c
increases with the increasing
J2
when the phase transition
temperature is zero, such as D1c =-1.595, -2.255, -3.258, -4.22, -5.12 for J 2 =-0.05, -0.2, -0.4, -0.6, -0.8, respectively. Fig. 4(b) shows the phase diagram in the hexagonal nanoisland with six fixed values of D2 . From this figure, the reentrant phenomena are shown by changing D2 from D2 =-0.4 to D2 =-3, and the number of phase transition point decreases with increasing value of D2 , such as kBT =0.46, 0.95, 2.23 for D1 =-5, D2 =-0.4 and k BT =1.09, 3.08 for D1 =-3, D2 =-1.0. In Fig.4(c), the phase diagram is
given for selecting some typical value of the transverse magnetic field . From our calculation the tendency of the kBTc curves are changed for <1.0. The critical value of D1c decreases with the increasing for >1.0, but increases for <1.0. For example, the critical value D1 are D1c =4.211 for =2 and D1c =3.097 for =3, but D1c =5.12 for =0.6 and D1c =5.306 for =0.8. Further, the curve intersects with each other (apart from =2 and 3) and exhibits the reentrant phenomenon, which is a new type of phase diagram in nanoisland. The core single-ion anisotropy D2 dependences of the phase transition temperature kBTc are plotted in Fig. 5(a)-(c). In Fig. 5(a), the kBTc of this system are plotted as a function of D2 with J =1.2, D1 =-3.5 and =0.2 for the four values of J 2 . It is clearly seen that the shape of this phase diagram is completely different from that we have shown in Fig.2-4, where the critical value D2c are concentrated. The region of reentrant part is smaller as value of J 2 decreases. Moreover, three kBTc are shown at a narrow region of 0.589< D2 <0.599 for J 2 =-0.8. Fig. 5(b) shows the phase diagram with fixed values of J 2 =-0.8, J =1.2 and =0.2 when the value of D1 is changed from D1 =-1.5 to D1 =-2.5. The phase transition temperature curves make the system present two parts of the ferrimagnetic and paramagnetic area. The system is in the paramagnetic phase when the temperature beyond the phase transition temperature.
The ferrimagnetic region is larger with decreasing of D1 while the paramagnetic region becomes smaller. It should be noticed the curves shows the existence of two phase transition temperatures for D1 =-2.0 and D1 =-2.5. Fig.5(c) illustrate the D2 versus kBTc with changing the value of from 0.5 to 2.5 for J 2 =-0.8, J =1.2 and D1 =-3.5.
Each curves reduce to zero at each critical value of D2 , such as D2c =-0.438, -0.576, -0.665, -0.674 for =2.5, 2.0, 1.5, 1.0, respectively. It has been found two phase transition temperatures for <1.92, but the reentrant behavior disappears for >1.92. 3.2 Magnetization and magnetic susceptibility In order to confirm the predictions of multi-transition temperatures obtained from the phase diagrams, we also study the magnetization and magnetic susceptibility in the present system. The numerical results is given in Fig.6 (a)-(b) with J 2 =-0.47, J =1.2, D1 =-3.5 and D2 =-0.4. Fig. 6(a) plots the temperature dependence of the total
average magnetization M with five typical values of . The M curves show different behaviors. The curves exist two transition temperatures kBT =0.79, 1.99 for =0.2 and k BT =0.61, 1.87 for =0.5. It is only one phase transition temperature on
the curve for =0.7, 1.0 and 1.2, which belong to Q-type. The saturation magnetization M =-0.123, -0.093, -0.042 for M =0.7, 1.0 and 1.2, respectively. The phase transition temperature (maximum of multi-transition temperatures) decreases with increasing value of which shows similar temperature effect. The magnetic susceptibilities versus the temperature are described in Fig. 6(b). The dash dot, dot and solid lines correspond to =0.2, =0.7 and =1.0, respectively. The curve appears singularity at kBT =0.79, 1.99 for =0.2, kBT =1.31 for =1.0 and kBT =1.78 for =0.7, which are in agreement with those of in Fig.6 (a). The temperature at the singularity of magnetic susceptibility corresponds to the phase transition
temperature, which is in agreement with that of Fig.2(c). In order to confirm the results of Fig.3(c), we calculate the magnetization and magnetic susceptibility in Fig. 7(a)-(b) with fixed values J 2 =-0.8, J =0.8, D1 =-3.5 and D2 =-0.4. In Fig. 7(a) temperature dependency of the total average magnetization M is plotted with several values of the transverse field . The two curves labeled =1.0 and =1.5 with J =0.8 show the reentrant behavior as predicted in Fig.3(c).
We should notice that the curves labeled =0.2 and =0.6 appear different behavior, which has a compensation temperature kBTcomp . Namely, sublattice magnetization is equal and the sign is opposite ( M s M c 0 ) which result in the total average magnetization M =0. This compensation behavior is unique for the ferrimagnetic system. For =0.6, the phase transition temperatures are kBT =0.44, 2.72 and the compensation temperature is kBTcomp =1.21. The magnetic susceptibilities versus the temperature are described in Fig. 7(b). The dash dot, dot and solid lines correspond to =0.2, =1.0 and =1.5, respectively. From the figure, the phase transition temperatures are obtained at kBT =0.58 and 2.84, kBT =2.48, kBT =1.11 and 1.55 for =0.2, =1.0, =1.5, respectively. In Fig.8(a)-(b), we plot the total magnetization and magnetic susceptibility of the system with J 2 =-0.8, J =1.2, D1 =-5.1 and D2 =-0.4. In Fig.8(a), the maximum phase transition temperature decreases with increasing value of , such as kBT =2.29 for =0.2, kBT =2.21 for =0.4, kBT =2.08 for =0.6 and kBT =1.81 for =0.8, respectively. We can also notice that the phase transition temperature is lower with the increasing in the inset. In Fig.8(b), the dash dot line delineates =0.8, the dot line expresses =0.2 and the solid line stands for =0.4. The phase diagram with =0.8 and D1 =-5.1 in Fig.4(c) has three phase transition temperatures. Here,
the total magnetization curve M reduces to zero three times and the magnetic susceptibility curve shows three singularity at kBTc =0.61, 1.26 and 1.81 respectively. So the numerical results obtained in Fig.8 (a)-(b) are consistent with those of in Fig.4(c). In comparison with the results of Fig.5(c), we examine the temperature dependences of total average magnetizations M and the magnetic susceptibility of the system with J 2 =-0.8, J =1.2, D1 =-3.5 and D2 =-0.68 by selecting the typical values of in Fig.9(a)-(b). For =1.0, 1.1 and 1.2, we can see that the magnetization curves have no reentrant behavior in Fig. 9(a). The shape of M changes from P-type ( =1.0) to N-type ( =1.1, 1.2). At zero temperature the magnetization curves have the saturation magnetization M =0.04, -0.1, -0.13 for =1.0, 1.1 and 1.2, respectively. The magnetization curves have two phase transition temperatures kBT =0.92, 2.24 for =1.5 and kBT =1.29 and 1.89 for =1.6. Further, the curve labeled =1.5 shows a compensation temperature. In Fig. 9(b), the magnetic susceptibility plots are given by changing the value of . The dash dot, solid and dot line stands for =1.0, 1.5 and 1.6, respectively. The maximum phase transition temperature increase with smaller value of , such as kBT =1.89, 2.24, 2.89 for =1.6, 1.5, 1.0, respectively.
4. Conclusion In this work, we have investigated the magnetic properties of the hexagonal prismatic nanoisland with the ferrimagnetic spin configuration based on the effective-field theory with correlations. The effects of surface parameters on phase transition temperatures have been found from the numerical results. The reentrant
phenomenon has been found in the system. The phase transition temperature curves display several different behaviors depending on the competition among the exchange coupling, the transverse magnetic field and single-ion anisotropy. The total average magnetization M and the magnetic susceptibility of this system have been examined. The results of the total magnetization have multi-transition temperatures and compensation behavior. We hope that the results in a hexagonal prismatic nanoisland can give some keys for the experimental research and technological applications in the future.
Acknowledgments This project was supported by Natural Sciences Foundation of Liaoning province, China (Grant no. 2013020109) and Program for Liaoning Innovative Research Team in University (Grant no. LT2014004).
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Fig.1 A sketch of hexagonal prismatic nanoisland. The big pink (small blue) balls delegate magnetic atom in the core (shell). The lines connecting the big pink and small blue balls represent the three nearest-neighbor exchange interactions ( J1 , J 2 and J ). Fig.2 Phase diagrams ( kBTc versus J 2 ) for the hexagonal nanoisland. (a) D1 =-3.5, D2 =-0.4, =0.6 and different values of J ; (b) J =1.2, D2 =-0.4, =0.6 and
different values of D1 ; (c) J =1.2, D1 =-3.5, D2 =-0.4 and different values of . Fig.3 Phase diagrams ( kBTc versus J ) for the hexagonal nanoisland. (a) D1 =-3.5, D2 =-0.4, =0.6 and different values of J 2 ; (b) J 2 =-0.8, D2 =-0.4, =0.6 and
different values of D1 ; (c) J 2 =-0.8, D1 =-3.5, D2 =-0.4 and different values of . Fig.4 Phase diagrams ( kBTc versus D1 ) for the hexagonal nanoisland. (a) J =1.2, D2 =-0.4, =0.6 and different values of J 2 ; (b) J 2 =-0.8, J =1.2, =0.6 and different values of D2 ; (c) J 2 =-0.8, J =1.2, D2 =-0.4 and different values of . Fig.5 Phase diagrams ( kBTc versus D2 ) for the hexagonal nanoisland. (a) J =1.2, D1 =-3.5, =0.2 and different values of J 2 ; (b) J 2 =-0.8, J =1.2, =0.2 and different values of D1 ; (c) J 2 =-0.8, J =1.2, D1 =-3.5 and different values of .
Fig.6 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.47, J =1.2, D1 =-3.5 and D2 =-0.4. The numerical values on the curves are the transverse field . Fig.7 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.8, J =0.8, D1 =-3.5 and D2 =-0.4. The numerical values on the curves are the transverse field . Fig.8 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.8, J =1.2, D1 =-5.1 and D2 =-0.4. The numerical values on the curves are the transverse field . Fig.9 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.8, J =1.2, D1 =-3.5 and D2 =-0.68. The numerical values on the curves are the transverse field .
Highlights
The hexagonal prismatic nanoisland with the spin-5/2 core and spin-2 shell.
Phase diagrams and magnetization of the system have been examined.
Multi-transition tenperatures has been found for the certain values parameters.
Graphical Abstract
PII: DOI: Reference:
S0304-8853(16)30689-8 http://dx.doi.org/10.1016/j.jmmm.2016.10.112 MAGMA62025
To appear in: Journal of Magnetism and Magnetic Materials Received date: 17 May 2016 Revised date: 26 August 2016 Accepted date: 21 October 2016 Cite this article as: Wei Jiang and Ya-Ning Wang, Phase transition and magnetization of a hexagonal prismatic nanoisland with a ferrimagnetic spin configuration, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.10.112 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.
Phase transition and magnetization of a hexagonal prismatic nanoisland with a ferrimagnetic spin configuration
Wei Jiang*, Ya-Ning Wang
School of Science, Shenyang University of Technology, Shenyang 110870, China
*
Corresponding
author.
Wei
Jiang,
Tel.:
+86-13840076827;
[email protected]
ABSTRACT
Magnetic properties of a nanoisland with a ferrimagnetic spin configuration, described by the transverse Ising model, are studied by the effective-field theory with correlations. The hexagonal prismatic nanoisland consists of the bilayer with core-shell structure. The phase transition and the magnetization of system have been calculated for different values. A lot of novel features, such as the reentrant phenomenon, have been found in the phase transition diagrams of the nanoisland. They are heavily dependent on the exchange coupling, the single-ion anisotropy and the transverse field. These theoretical results may have guiding significance for preparing nanoisland experimentally.
Graphical Abstract
Phase diagrams and magnetization of the hexagonal prismatic nanoisland have been examined by the effective-field theory with correlations.
Keywords: Nanoisland; Ferrimagnetic; Magnetization; Phase transition
1. Introduction During recent years, the magnetic nanomaterials have been the focus on attention due to special structure and properties, related with their wide applications [1-11]. Nanoisland, one of the magnetic nanomaterials, can remarkably improve the specific surface area. Thus, the study of nanoisland is a significant issue on fundamental theoretical and experimental viewpoints. In experimental, great progress has been made in the study of nanoisland. Yamada et al have studied the atomic and magnetic order in bilayer Fe nanoislands grown on a Cu (111) substrate under the specified conditions. Besides, they found electric field pulses can cause a displacement of the Fe atoms, which proves of magnetoelectric coupling at the metallic Fe surface [12]. Nanoislands of α'-FeRh were prepared by the deposition of a 10 nm film onto (001)-MgO. This island structure leads to stabilized ferromagnetic (FM) ordering at low temperature [13]. High-density magnetic storage or quantum computing could be achieved using low dimensional rare-earth iron structure, such as the magnetic coupling between 4f single atoms and a 3d magnetic nanoisland [14]. The piezoelectric and magnetoelectric in three-dimensional multiferroic nanostructure
have been studied by using phase-field simulations. This structure consists of vertically magnetized nanomagnet with an in-plane long axis and a juxtaposed ferroelectric nanoisland [15]. Bruggemann et al have investigated cooling of a magnetic quantum dot by a spin-polarized tunneling to charge current exploiting the magnetomechanical coupling. The spin-polarized current polarizes the magnetic nanoisland, thereby lowering its magnetic energy [16]. Low-energy proton irradiation was applied to pattern ferromagnetic nanoislands through the local phase transformation. Such an array is strong enough to overcome the so-called superparamagnetism limit in magnetic recording [17]. Ordered FePdCu nanoisland arrays were formed by annealing on the top of SiO2 nanosphere. This is a method of providing nanoislands with well-controlled size and positions [18]. The initial nucleation and growth of Co nanoislands on highly oriented pyrolytic graphite (HOPG) were experimentally investigated by scanning tunneling microscopy with atomic resolution at room temperature [19]. The research on nanoisland progress fast experimentally, however, only a few researchers have carried out the theoretical study of nanoisland. Thus theoretical investigations are urged to explore the nanoislands. Choi et al have systematically investigated shape-dependent magnetic moment and island formation energy of Fe adatoms on a Cu (111) surface by the density functional theory based first-principles calculations. The magnetic moments of Fe nanoisland atoms rest with two key factors, the numbers of Fe-Fe bonding and the angles between the Fe-Fe bonding [20]. The phase diagrams and temperature dependences of magnetizations in transverse Ising
nanoislands which are consisted of the two layers are examined by using the effective-field theory with correlations (EFT). A lot of unexpected characteristic phenomena are obtained for the magnetic properties in those systems [21-24]. Further, the magnetic properties of hexagonal prismatic nanoislands with transverse field and magnetic anisotropy have not been examined. We have successfully studied magnetic properties of various nanoscaled transverse Ising model systems by the effective-field theory with correlations in a series of recent works [25-31]. The results show the physical parameters including surface coupling, interlayer coupling, transverse fields and the single-ion anisotropy have important effects on the phase diagrams, the magnetizations and the thermodynamics properties. Therefore, in this paper, we intend to continue to study the magnetic properties of a hexagonal prismatic nanoisland within the EFT. The paper is organized as follows: in Section 2, the models and formulations for the system are given. In Section 3, we examine the numerical results of the phase diagrams and the magnetization. Finally, some conclusions are given in Section 4.
2. Models and formulations First we will introduce nanoisland structure as shown in Fig.1. It consists of two of the same layer. Each layer has a core atom C with a spin-2/5 (big pink ball) and six shell atoms S with spin-2 (small blue balls). The solid line represents the ferromagnetic exchange coupling J1 (>0) between the magnetic atom on the surface shell. The dashed line denotes the ferromagnetic exchange coupling J (>0) between the magnetic atom in bilayer. The dotted line is the ferrimagnetic exchange coupling
J 2 (<0), which comes from the shell spins coupled to the core spin. This system can be
abstracted by the transverse Ising model with the anisotropy. Second the effective-field theory with correlations is applied to nanoisland and the calculation formula is given. The Hamiltonian of the present system is written as H J1 iz zj J iz kz J Smz S nz J 2 iz S mz ( ix S mx ) i, j
i,k
m, n
i ,m
i
m
(1)
D1 ( ) D2 ( S ) h( S ) z 2 i
z 2 m
i
z i
m
i
z m
m
Where iz and S mz are spin-2 and spin-2/5, respectively. The transverse and longitudinal magnetic field expressed by and h . D1 and D2 are the single-ion anisotropies, which come from the shell atoms S and the core C. By the effective-field theory with correlations, the core magnetization M c and shell magnetization M s at per site can be calculated as follows: M c (n1e J2s n2 e J2s )6 (n3e J c n4 e J c ) f c x
(2)
x 0
M s (n1e J1s n2 e J1s )2 (n3e J2c n4 e J2c ) (n1e J s n2 e J s ) f s x
x 0
(3)
Where x is the differential operator, and c2 and s2 as coupled equations: c 2 (n1e J n2 e J )6 (n3e J n4 e J ) gc x 2 s
2 s
c
c
(4)
x 0
s 2 (n1e J n2 e J )2 (n3eJ n4 e J ) (n1eJ n2 e J ) g s x 1 s
1 s
2 c
2 c
s
s
x 0
(5)
Where n1 , n2 , n3 and n4 are respectively defined as n1 (1 2)(1 M s s ) , n2 (1 2)(1 M s s ) , n3 (1 2)(1 M c c ) and n4 (1 2)(1 M c c ) . Meanwhile, the four
functions fc ( x) , f s ( x) , gc ( x) and g s ( x) are defined by f c ( x)
1
6 m 1
exp(m )
6
{ m Smz m exp( m )} m 1
(6)
f s ( x)
g c ( x)
g s ( x)
1
i 1 exp(i ) 5
5
{ i iz i exp(i )}
6
1
6
exp(m ) m 1 1
5 i 1
exp(i )
(7)
i 1
{ m ( Smz )2 m exp(m )}
(8)
m 1
5
{ i ( iz ) 2 i exp(i )}
(9)
i 1
Here 1 kB T and T is a temperature, and k B is the Boltzmann constant. The eigenvalues and eigenfunctions of the Hamiltonian are showed by m (i ) and m (i ) , depending on the value of spin. They can only be obtained by numerical
calculation through computer programming. Further, the total average magnetization
M
and susceptibility
are
respectively defined by M (M c 6 M s ) 7
M h h0
(10) (11)
3. Numerical results and discussions In this section, we investigate the phase diagrams and the behaviors of magnetizations in the hexagonal prismatic nanoisland by solving the formulations given in Section 2. In the following calculations, the ferromagnetic exchange coupling J1 on the surface shell is fixed J1 1.0 as a unit of the system. The effects of the
exchange couplings, the transverse field, single-ion anisotropies on the magnetic properties in this model have been discussed in detail. From the processes, we have found some unexpected characteristic phenomena in the system. 3.1 Phase transition temperature In this part, we shall discuss the effects of various parameters on the phase transition temperature kBTc for the hexagonal prismatic nanoisland. The intralayer
antiferromagnetic J 2 dependence of the phase transition temperatures are plotted in Fig. 2(a)-(c). In Fig. 2(a), six kBTc curves are presented with the ferromagnetic exchange coupling between bilayer J =0.6, 0.8, 0.9, 1.0, 1.3 and 1.5. In the figure, multi-transition temperatures can be found for some typical values J 2 , which namely the reentrant phenomena. For example, the curve exists three transition temperatures k BT =0.32, 0.78 and 1.59 with J =1.0 and J 2 =-0.53. The reentrant phenomena have
been found in a variety of disordered magnetic systems experimentally and theoretically [32-34]. This interesting behavior depends on the competition among the parameters J 2 , J , D1 and . The similar phenomenon can be observed in Fig. 2(b), but value of D1 is changed. For example, the curves exist two transition temperatures k BT =0.58, 0.94 with D1 =-2.2, J 2 =-0.18 and k BT =0.6, 1.33 with D1 =-2.9, J 2 =-0.32.
In the figure, strong D1 increase the number of phase transition temperature, but the reentrant phenomena disappear for D1 =-1.2. The value of kBTc may reduce to zero at the critical value J 2c , such as J 2c =-0.449, -0.328, -0.189, -0.052 for D1 =-3.5, -2.9, -2.2, -1.6, respectively. Fig.2(c) shows another case of the phase diagrams in the hexagonal nanoisland with J =1.2, D1 =-3.5, D2 =-0.4 and five fixed transverse field . For =1.0, 1.5 and 2.0, the phase transition temperature kBTc increases noticeably with increasing value of J 2 . As is seen from the curves labeled =0.2 and =0.5, the tendency of the transition temperature curves are different from others, and they have points of intersection with other curves. It may exhibit the reentrant phenomena, which have multi-transition points for some fixed values of J 2 . The interlayer ferromagnetic exchange coupling J dependences of the phase transition temperature kBTc are plotted in Fig.3 (a)-(c). Fig.3 (a) shows the numerical results of the phase diagram in the nanoisland for selecting some typical values of J 2 .
The kBTc curves exhibit multi-transition point when J 2 <1.37, especially the system has three phase transition temperatures kBT =0.34, 0.84, 1.48 for J 2 =-0.5 and J =1.04. The reentrant phenomena disappear for J 2 >1.37. In Fig. 3(b), various shapes of the phase diagram are observed with the different values of D1 . As seen from the figure, the reentrant phenomena also can be found in this system when D1 >2.5. For example, the curves have multiple phase transition temperatures, which kBT =0.65, 2.52 for J =0.4, D1 =-2.8 and kBT =0.34, 0.72, 2.35 for J =0.95, D1 =-4.2. When D1 <2.5, the curves have the same behavior which belong to the monotone increasing function and the reentrant phenomena disappear, such as D1 =-0.8, -1.5 and -2.0. In Fig.3(c), the phase diagram is given by selecting the five values of transverse field . The phase transition temperature curves fall from its maximum value and reaches zero for critical value. From the figure, we can easily see that the value of the critical value J c increases with increasing transverse field except for =0.2 ( J c =0.836). Further,
three kBTc are obtained in the system with a small value of ( <0.4), such as k BT =0.19, 0.45, 2.97 for J =0.85 and =0.2.
As discussed in [35-38], it is found that the anisotropy acts a major role in the low-dimension magnetic systems. Fig. 4(a)-(c) shows the variation of phase diagrams versus the shell single-ion anisotropy D1 by changing the value of J 2 , D2 and . Fig. 4(a) reveals the shell single-ion anisotropy D1 dependences of the phase transition temperatures when the value of intralayer antiferromagnetic exchange coupling J 2 is fixed some values. We should notice that the number of multi-transition temperatures is decreased with the value of D2
increases. The
system may exhibit the three kBTc for D2 =-0.6 and D2 =-0.8. Further, the critical value of D1c
increases with the increasing
J2
when the phase transition
temperature is zero, such as D1c =-1.595, -2.255, -3.258, -4.22, -5.12 for J 2 =-0.05, -0.2, -0.4, -0.6, -0.8, respectively. Fig. 4(b) shows the phase diagram in the hexagonal nanoisland with six fixed values of D2 . From this figure, the reentrant phenomena are shown by changing D2 from D2 =-0.4 to D2 =-3, and the number of phase transition point decreases with increasing value of D2 , such as kBT =0.46, 0.95, 2.23 for D1 =-5, D2 =-0.4 and k BT =1.09, 3.08 for D1 =-3, D2 =-1.0. In Fig.4(c), the phase diagram is
given for selecting some typical value of the transverse magnetic field . From our calculation the tendency of the kBTc curves are changed for <1.0. The critical value of D1c decreases with the increasing for >1.0, but increases for <1.0. For example, the critical value D1 are D1c =4.211 for =2 and D1c =3.097 for =3, but D1c =5.12 for =0.6 and D1c =5.306 for =0.8. Further, the curve intersects with each other (apart from =2 and 3) and exhibits the reentrant phenomenon, which is a new type of phase diagram in nanoisland. The core single-ion anisotropy D2 dependences of the phase transition temperature kBTc are plotted in Fig. 5(a)-(c). In Fig. 5(a), the kBTc of this system are plotted as a function of D2 with J =1.2, D1 =-3.5 and =0.2 for the four values of J 2 . It is clearly seen that the shape of this phase diagram is completely different from that we have shown in Fig.2-4, where the critical value D2c are concentrated. The region of reentrant part is smaller as value of J 2 decreases. Moreover, three kBTc are shown at a narrow region of 0.589< D2 <0.599 for J 2 =-0.8. Fig. 5(b) shows the phase diagram with fixed values of J 2 =-0.8, J =1.2 and =0.2 when the value of D1 is changed from D1 =-1.5 to D1 =-2.5. The phase transition temperature curves make the system present two parts of the ferrimagnetic and paramagnetic area. The system is in the paramagnetic phase when the temperature beyond the phase transition temperature.
The ferrimagnetic region is larger with decreasing of D1 while the paramagnetic region becomes smaller. It should be noticed the curves shows the existence of two phase transition temperatures for D1 =-2.0 and D1 =-2.5. Fig.5(c) illustrate the D2 versus kBTc with changing the value of from 0.5 to 2.5 for J 2 =-0.8, J =1.2 and D1 =-3.5.
Each curves reduce to zero at each critical value of D2 , such as D2c =-0.438, -0.576, -0.665, -0.674 for =2.5, 2.0, 1.5, 1.0, respectively. It has been found two phase transition temperatures for <1.92, but the reentrant behavior disappears for >1.92. 3.2 Magnetization and magnetic susceptibility In order to confirm the predictions of multi-transition temperatures obtained from the phase diagrams, we also study the magnetization and magnetic susceptibility in the present system. The numerical results is given in Fig.6 (a)-(b) with J 2 =-0.47, J =1.2, D1 =-3.5 and D2 =-0.4. Fig. 6(a) plots the temperature dependence of the total
average magnetization M with five typical values of . The M curves show different behaviors. The curves exist two transition temperatures kBT =0.79, 1.99 for =0.2 and k BT =0.61, 1.87 for =0.5. It is only one phase transition temperature on
the curve for =0.7, 1.0 and 1.2, which belong to Q-type. The saturation magnetization M =-0.123, -0.093, -0.042 for M =0.7, 1.0 and 1.2, respectively. The phase transition temperature (maximum of multi-transition temperatures) decreases with increasing value of which shows similar temperature effect. The magnetic susceptibilities versus the temperature are described in Fig. 6(b). The dash dot, dot and solid lines correspond to =0.2, =0.7 and =1.0, respectively. The curve appears singularity at kBT =0.79, 1.99 for =0.2, kBT =1.31 for =1.0 and kBT =1.78 for =0.7, which are in agreement with those of in Fig.6 (a). The temperature at the singularity of magnetic susceptibility corresponds to the phase transition
temperature, which is in agreement with that of Fig.2(c). In order to confirm the results of Fig.3(c), we calculate the magnetization and magnetic susceptibility in Fig. 7(a)-(b) with fixed values J 2 =-0.8, J =0.8, D1 =-3.5 and D2 =-0.4. In Fig. 7(a) temperature dependency of the total average magnetization M is plotted with several values of the transverse field . The two curves labeled =1.0 and =1.5 with J =0.8 show the reentrant behavior as predicted in Fig.3(c).
We should notice that the curves labeled =0.2 and =0.6 appear different behavior, which has a compensation temperature kBTcomp . Namely, sublattice magnetization is equal and the sign is opposite ( M s M c 0 ) which result in the total average magnetization M =0. This compensation behavior is unique for the ferrimagnetic system. For =0.6, the phase transition temperatures are kBT =0.44, 2.72 and the compensation temperature is kBTcomp =1.21. The magnetic susceptibilities versus the temperature are described in Fig. 7(b). The dash dot, dot and solid lines correspond to =0.2, =1.0 and =1.5, respectively. From the figure, the phase transition temperatures are obtained at kBT =0.58 and 2.84, kBT =2.48, kBT =1.11 and 1.55 for =0.2, =1.0, =1.5, respectively. In Fig.8(a)-(b), we plot the total magnetization and magnetic susceptibility of the system with J 2 =-0.8, J =1.2, D1 =-5.1 and D2 =-0.4. In Fig.8(a), the maximum phase transition temperature decreases with increasing value of , such as kBT =2.29 for =0.2, kBT =2.21 for =0.4, kBT =2.08 for =0.6 and kBT =1.81 for =0.8, respectively. We can also notice that the phase transition temperature is lower with the increasing in the inset. In Fig.8(b), the dash dot line delineates =0.8, the dot line expresses =0.2 and the solid line stands for =0.4. The phase diagram with =0.8 and D1 =-5.1 in Fig.4(c) has three phase transition temperatures. Here,
the total magnetization curve M reduces to zero three times and the magnetic susceptibility curve shows three singularity at kBTc =0.61, 1.26 and 1.81 respectively. So the numerical results obtained in Fig.8 (a)-(b) are consistent with those of in Fig.4(c). In comparison with the results of Fig.5(c), we examine the temperature dependences of total average magnetizations M and the magnetic susceptibility of the system with J 2 =-0.8, J =1.2, D1 =-3.5 and D2 =-0.68 by selecting the typical values of in Fig.9(a)-(b). For =1.0, 1.1 and 1.2, we can see that the magnetization curves have no reentrant behavior in Fig. 9(a). The shape of M changes from P-type ( =1.0) to N-type ( =1.1, 1.2). At zero temperature the magnetization curves have the saturation magnetization M =0.04, -0.1, -0.13 for =1.0, 1.1 and 1.2, respectively. The magnetization curves have two phase transition temperatures kBT =0.92, 2.24 for =1.5 and kBT =1.29 and 1.89 for =1.6. Further, the curve labeled =1.5 shows a compensation temperature. In Fig. 9(b), the magnetic susceptibility plots are given by changing the value of . The dash dot, solid and dot line stands for =1.0, 1.5 and 1.6, respectively. The maximum phase transition temperature increase with smaller value of , such as kBT =1.89, 2.24, 2.89 for =1.6, 1.5, 1.0, respectively.
4. Conclusion In this work, we have investigated the magnetic properties of the hexagonal prismatic nanoisland with the ferrimagnetic spin configuration based on the effective-field theory with correlations. The effects of surface parameters on phase transition temperatures have been found from the numerical results. The reentrant
phenomenon has been found in the system. The phase transition temperature curves display several different behaviors depending on the competition among the exchange coupling, the transverse magnetic field and single-ion anisotropy. The total average magnetization M and the magnetic susceptibility of this system have been examined. The results of the total magnetization have multi-transition temperatures and compensation behavior. We hope that the results in a hexagonal prismatic nanoisland can give some keys for the experimental research and technological applications in the future.
Acknowledgments This project was supported by Natural Sciences Foundation of Liaoning province, China (Grant no. 2013020109) and Program for Liaoning Innovative Research Team in University (Grant no. LT2014004).
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Fig.1 A sketch of hexagonal prismatic nanoisland. The big pink (small blue) balls delegate magnetic atom in the core (shell). The lines connecting the big pink and small blue balls represent the three nearest-neighbor exchange interactions ( J1 , J 2 and J ). Fig.2 Phase diagrams ( kBTc versus J 2 ) for the hexagonal nanoisland. (a) D1 =-3.5, D2 =-0.4, =0.6 and different values of J ; (b) J =1.2, D2 =-0.4, =0.6 and
different values of D1 ; (c) J =1.2, D1 =-3.5, D2 =-0.4 and different values of . Fig.3 Phase diagrams ( kBTc versus J ) for the hexagonal nanoisland. (a) D1 =-3.5, D2 =-0.4, =0.6 and different values of J 2 ; (b) J 2 =-0.8, D2 =-0.4, =0.6 and
different values of D1 ; (c) J 2 =-0.8, D1 =-3.5, D2 =-0.4 and different values of . Fig.4 Phase diagrams ( kBTc versus D1 ) for the hexagonal nanoisland. (a) J =1.2, D2 =-0.4, =0.6 and different values of J 2 ; (b) J 2 =-0.8, J =1.2, =0.6 and different values of D2 ; (c) J 2 =-0.8, J =1.2, D2 =-0.4 and different values of . Fig.5 Phase diagrams ( kBTc versus D2 ) for the hexagonal nanoisland. (a) J =1.2, D1 =-3.5, =0.2 and different values of J 2 ; (b) J 2 =-0.8, J =1.2, =0.2 and different values of D1 ; (c) J 2 =-0.8, J =1.2, D1 =-3.5 and different values of .
Fig.6 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.47, J =1.2, D1 =-3.5 and D2 =-0.4. The numerical values on the curves are the transverse field . Fig.7 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.8, J =0.8, D1 =-3.5 and D2 =-0.4. The numerical values on the curves are the transverse field . Fig.8 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.8, J =1.2, D1 =-5.1 and D2 =-0.4. The numerical values on the curves are the transverse field . Fig.9 Temperature dependencies of (a) total average magnetization M (b) magnetic susceptibility of the system with the fixed values of J 2 =-0.8, J =1.2, D1 =-3.5 and D2 =-0.68. The numerical values on the curves are the transverse field .
Highlights
The hexagonal prismatic nanoisland with the spin-5/2 core and spin-2 shell.
Phase diagrams and magnetization of the system have been examined.
Multi-transition tenperatures has been found for the certain values parameters.
Graphical Abstract