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Remarkable features of magnetic properties in transverse Ising nanoislands
Journal of Magnetism and Magnetic Materials 374 (2015) 321–326
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Remarkable features of magnetic properties in transverse Ising nanoislands T. Kaneyoshi 1 1–510, Kurosawadai, Midoriku, Nagoya, 458-0003, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 3 February 2014 Available online 27 August 2014
The phase diagram and magnetizations in two transverse Ising nanoislands with the same structures which are consisted of the two layers are examined by using the effective-field theory with correlations. The effects of interlayer coupling and two transverse fields at the center and the perimeter atoms on them are examined. We find remarkable features in them which come from the frustration induced by the interlayer coupling and two transverse fields. & 2014 Elsevier B.V. All rights reserved.
Keywords: Nanoisland Phase diagrams Magnetizations Transverse Ising model
1. Introduction Nowadays, magnetic nanomaterials, such as nanoparticles, nanoislands, nanofilms, nanowires, nanotubes and so on, have attracted considerable attention experimentally and theoretically, related with their technological and biomedical applications. The magnetic properties of nanomaterials are due to the distinct contributions of inside, perimeter and surface atoms. The remarkably different contributions of surface and perimeter spins govern the nanomagnetism of a material and points to the new way of finding important possible technological applications. From the academic point of view, the magnetism of a single-domain nanoparticle (or nanoisland) and a nanoscaled thin film can be addressed to the research. The thickness dependence of transition temperature, the temperature dependence of magnetization, the critical exponent β and the Neel hyperbola for paramagnetic susceptibility [1] in these materials have been obtained by the use of some elaborate experimental techniques [2–4], in addition to hysteresis loops which are important for the manufacture of magnetic recording media. From the experimental point of view, at the present time, it is almost impossible to study the morphology of the very same particle in conjunction of magnetism. The experimental topography of a nanoparticle displays a discontinuous (or island-like) architecture composed of well-separated and nonuniform particles with an average island diameter [5,6]. As far as we know, the contribution to the magnetic properties from such islands has not been taken into account theoretically. In a series of recent works [7–12], we have examined the phase diagrams and the magnetizations of nanoscaled Ising (or transverse
E-mail address: [email protected] 1 Prof. Emeritus at Nagoya University, Japan. http://dx.doi.org/10.1016/j.jmmm.2014.08.062 0304-8853/& 2014 Elsevier B.V. All rights reserved.
Ising) thin films consisting of a few magnetic layers by the use of the effective-field theory with correlations (EFT) [13,14]. The EFT corresponds to the Zernike approximation (ZA) [15] and it is believed to give more exact results than those of the mean field approximation. In them, when the value of r (r¼Jr /J) is taken as a very small or a very large value, some interesting phenomena have been found in the magnetic properties of the systems with thickness L (from L¼2 to L¼10), such as the appearance of a broad maximum in the variation of transition temperature (TC) as a function of r for the site dilution, while such a phenomenon has not been obtained for the bond dilution. Here, Jr is the interlayer coupling between the surface and the next inner layer, when the value of the inner layer coupling is fixed at J. In particular, a new type of frustration has been obtained in the systems [10–12], when a uniform transverse field is applied and a large value of r is selected. Furthermore, such a frustration has been also found in the transverse Ising nanoisland consisting of two layers with eighteen atoms in total [16]. The aim of this work is, within the theoretical framework of the EFT, to investigate the effects of interlayer coupling and two transverse fields (ΩS and Ω) on the magnetic properties (phase diagram, magnetizations) in two nanoscaled transverse Ising islands with the same structural aspect as that in Ref. [16], since the effects of a uniformly applied transverse field (ΩS ¼Ω) have been examined in Ref. [16]. Here, ΩS is the transverse field at the perimeter spin and Ω is the transverse field at the central (inside) spin.
2. Model and formulation We consider the 3D nanoisland (or nanoparticle), as depicted in Fig. 1, in which they are consisted of the surface shell and the core. The core (white circles) is surrounded by the surface shell (black
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The surface exchange interaction JS is often defined as JS ¼ J ð1 þ ΔS Þ;
ð2Þ
in order to clarify the effect of surface shell on physical properties in the system. In particular, when the value of JS is given by JS ¼0.0 (or ΔS ¼ 1.0) in (3), the system shown as Fig. 1(B) represents the system as shown in Fig. 2, in which the atoms at the corner of each square (half black and half white circle) represent the nonmagnetic atoms. The system represents one of the transverse Ising islands with the same structural aspect as that in Fig. 1(A). Within the framework of the EFT [13,14], we can obtain the longitudinal magnetizations mS ( mS ¼ o σZi 4 ) at the surface shell and the longitudinal magnetization mC ( mC ¼ o σZm 4 ) in the center for the system shown in Fig. 1(B), as coupled equations: 2 mS ¼ coshðAÞ þmS sinhðAÞ coshðCÞ þmC sinhðCÞ ð3Þ coshðBÞ þmS sinhðBÞ f S ðxÞjx ¼ 0 4 mC ¼ cosh ðCÞ þ mS sinh ðCÞ ½ cosh ðBÞ þ mC sinh ðBÞfðxÞjx ¼ 0
ð4Þ
where A, B and C are defined by A ¼ JS D, B ¼ Jr D and C ¼J D. D¼ d/dx is the differential operator. Here, the functions fS (x) and f (x) are defined by f S ðxÞ ¼ ðx=yS Þ tanh ðβ yS Þ and fðxÞ ¼ ðx=yÞ tanh ðβ yÞ
ð5Þ
with yS ¼ ðx2 þ ΩS Þ1=2 2
and Fig. 1. Schematic representation of two transverse Ising nanoislands. In Fig. (A), the above and below layers are consisted of 9 spin-1/2 atoms, where 8 black circles and 1 white circle represent the same magnetic atoms. In Fig. (B), the above and below layers are consisted of 5 spin-1/2 atoms, where 4 black circles and 1 white circle represent the same magnetic atoms. The lines connecting the black and white circles in each figure represent the three nearest-neighbor exchange interactions (JS, J and Jr).
circles). Each site on the figure is occupied by a Ising spin. Each spin is connected to the nearest neighbor spins with an exchange interaction. The surface spins are coupled to the center spin with an exchange interaction J. The atoms on the surface shell are connected by the exchange interaction JS. Each spin on the upper layer is connected to the corresponding spin on the lower layer with an exchange interaction Jr. The choice corresponds to the experimental facts [5,17], as discussed in [16]. In this work, let us formulate the system depicted as Fig. 1(B), since the phase diagram and the magnetizations of the system shown in Fig. 1 (A) has been formulated in [16] and the numerical results of them have been examined for the case of ΩS ¼ Ω. The Hamiltonian of the system is given by
y ¼ ðx2 þ Ω Þ1=2 ; 2
ð6Þ
where β ¼1 / kBT and T is a temperature. X Furthermore, the transverse magnetizations mX S ¼ o σi 4 and X X mC ¼ o σm 4 are also given by the same equations as those of Eqs. (3,4), only replacing the functions fS(x) and f(x) in Eqs. (3,4) by the new functions hS(x) and h(x), respectively. The new functions hS(x) and h(x) are defined by hS ðxÞ ¼ ðΩS =yS Þ tanh ðβ yS Þ and hðxÞ ¼ ðΩ=yÞ tanh ðβ yÞ
ð7Þ
H ¼ J S ∑σ i Z σ j Z J∑σ m Z σ n Z J r ∑σ i Z σ m Z ðmnÞ
ðijÞ
ΩS Σσ i X ΩΣσ m X ; ðiÞ
ðmÞ
ðimÞ
ð1Þ
α
where σi (α ¼z, x) is the Pauli spin operator with σZi ¼ 71. ΩS and Ω represent the transverse fields at the surface shell and in the core, respectively, since the transverse fields may be different at the surface and in the core. The first (ij) and second (mn) terms in the Hamiltonian (1) represent the contributions from the surface shell and the core, respectively. The third term shows the contribution from the interlayer interaction.
Fig. 2. Schematic representation of a transverse Ising nanoisland. The above and below layers are consisted of 9 spin-1/2 atoms, where 8 black circles and 1 white circle represent the same magnetic atoms and the half black and half white atoms represent a nonmagnetic atom. The lines connecting the black and white circles represent the two nearest-neighbor exchange interactions (J and Jr).
T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 374 (2015) 321–326
Let us here define the total longitudinal and transverse magnetizations (mT ¼ mZT and mX T ) per site as α 4mS þ mαc ; ð8Þ mαT ¼ 5 where α ¼z or x. By expanding the right-hand sides of coupled Eqs. (3,4), the transition temperature (or phase diagram) can be determined from ½1 ð2K 1 þ K 2 Þ½1 K 3 4K 4 K 5 ¼ 0;
ð9Þ
where the coefficients Kn (n ¼1–6) are given by K1 ¼ coshðAÞ sinhðAÞ coshðBÞ coshðCÞ f S ðxÞjx ¼ 0 2
K2 ¼ cosh ðAÞ sinhðBÞ coshðCÞ f S ðxÞjx ¼ 0 4
K3 ¼ cosh ðCÞ sinhðBÞ fðxÞjx ¼ 0 2
K4 ¼ cosh ðAÞ coshðBÞ sinhðCÞ fsðxÞjx ¼ 0 3
K5 ¼ cosh ðCÞ sinhðCÞ coshðBÞ fðxÞjx ¼ 0;
ð10Þ
The coefficients Kn and the coupled Eqs. (3) and (4) can be easily calculated by applying a mathematical relation exp(a D)F (x)¼ F(xþa). For the following discussions, let us here define the four parameters t, r h and q as t¼
kB T J Ω ΩS ; r ¼ r ; h ¼ and q ¼ J J J Ω
ð11Þ
3. Numerical results In this section, let us examine the magnetic properties (phase diagram and magnetization) in the nanoisland (or Fig. 1(B)) by solving the Eqs. (3), (4) and (9) given in Section 2 numerically. In particular, we are interested in the behaviors of them, when the parameter q is selected as qa 1.0. The reasons are based on the previous work [16], in which the case of q ¼1.0 has been examined. In order to compare the results of Fig. 1(B) with those of Fig. 1(A) in [16], one should notice that the exchange interaction JS (or ΔS) in Fig. 1(B) is assumed to be the same value as that in Fig. 1(A), for convincing.
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3.1. (A) Phase diagrams At first, let us show the two typical results of phase diagrams for the two systems (Fig. 1(A) and (B)) with q ¼1.0 (or ΩS ¼Ω) in Fig. 3. Fig. 3(A) shows the phase diagram (tC ¼kBTC/J versus r plots) in the two systems with ΩS ¼ Ω¼0.0, when the three values of ΔS are selected. The solid lines express the results for the system depicted as Fig. 1(B) and the dashed lines represent the previous results in [16] for the system depicted as Fig. 1(A). For each value of ΔS, the curves at first exhibit the rather rapid increase and then change to the slow variation, when the value of r is increased. Next, let us examine whether the unconventional phase diagram (or Fig. 5 in [16]) in the nanoisland of Fig. 1(A) can be found for the system of Fig. 1(B) with fixed values of ΔS ¼0.0 and q ¼1.0, when the finite transverse field (or h ¼1.0) is applied. Fig. 3(B) shows such results for the two nanoislands, in which the dashed lines are equivalent to the corresponding results with h¼0.0 in Fig. 3(A), namely the sold and dashed curves labeled ΔS ¼0.0. The solid curve labeled A at first increases with following to the dashed line labeled A, shows the existence of two transition temperatures (or the reentrant phenomenon) in the region between r ¼12.39 and r ¼6.335, and reduces to zero at the critical value of rC (rC ¼6.335), as shown in [16]. The same behaviors as those labeled A can be also found for the system of Fig. 1(B). They are represented as the solid and dashed curves labeled B. Then, the critical value rC for the solid curve labeled B is given by rC ¼7.183. In [16], the phase diagrams have been obtained for the case of q¼ 1.0. In this work, let us examine another case (or tC versus q plot) for the two nanoislands, changing the value of r. The results for the two systems with fixed values of ΔS ¼0.0 and h¼ 1.0 are depicted in Fig. 4, when the three values of r are selected. The solid and dashed lines in the figure represent the results of Fig. 1(B) and Fig. 1(A), respectively. In the figure, the tC curve labeled r ¼4.0 is larger than the tC curve labeled r ¼ 1.0 and the tC curve labeled r ¼6.0 exhibits the reentrant behavior, for each system. In the Figs. 3 and 4, one should notice that the tc results of Fig. 1 (B) are always larger than those of Fig. 1(A). The reason is that the system of Fig. 1(B) is more closely packed than that of Fig. 1(A), even though the exchange interaction JS in the two systems is selected as the same value. In fact, the behavior of mS1 in Fig. 1(A), which is loosely connected as a whole to mS2, gives a lower tC value than that of Fig. 1(B).
Fig. 3. (A) The phase diagrams (tC versus r plot) in the two systems (or Figs.1(A) and (B)) with zero transverse fields (ΩS ¼ Ω ¼0.0), when the three values of ΔS are selected. The solid and dashed lines represent the results of Fig. 1(B) and (A), respectively. (B) The phase diagrams (tC versus r plot) in the two systems with fixed values of ΔS ¼0.0 and q¼ 1.0, where the curves labeled A and B show the results of Fig. 1(B) and (A). The dashed line labeled h ¼ 0.0 is respectively equivalent to the corresponding one (or the curve labeled ΔS ¼0.0) in Fig. 3(A). The region below (or inside) each TC curve shows the ferromagnetic phase.
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As noted in Section 2, when the value of JS is given by JS ¼0.0 (or ΔS ¼ 1.0) in (2), the system shown as Fig. 1(B) represents the system of Fig. 2, in which the atoms at the corner of each square (half black and half white circle) represent the nonmagnetic atoms. The system represents one of the transverse Ising islands with the same structural aspect as that of Fig. 1 (A). By selecting the value of ΔS as ΔS ¼- 1.0, let us examine the phase diagram (tC versus q plot) of the system described by Fig. 2. Then, in order to compare the tC results of Fig. 1(A) with those of Fig. 2, the value of h is fixed at h¼1.0 and the value of ΔS for the system of Fig. 1(A) is selected as ΔS ¼ 0.0 (or JS ¼J). The results are shown in Fig. 5, where the solid and dashed lines represent the tC curves of Fig. 2 and Fig. 1(A), respectively. They show the behaviors similar to those in Fig. 4, while the solid curve labeled r ¼6.0 takes a finite value (or tC ¼ 0.4) at q¼0.0. In Fig. 5, the dashed lines for system of Fig. 1(A) represent the same tC curves as those (or dashed lines) in Fig. 4 Fig. 4. The phase diagram (tC versus q plot) in the two systems with ΔS ¼ 0.0 and h ¼1.0, when the three values of r are selected. The solid and dashed lines represent the results of Fig. 1(B) and (A), respectively.
3.2. (B) Magnetizations
Fig. 5. The phase diagram (tC versus q plot) in the two systems with ΔS ¼0.0 and h ¼ 1.0, when the three values of r are selected. The solid lines represent the tC results in the system of Fig. 2 with ΔS ¼ 1.0 and the dashed lines are the same as the dashed lines in Fig. 4.
It is not clear from the results of Figs.3 and 4 from what value of r the frustration may be induced in the two transverse Ising nanoislands. In order to get one of the keys, the saturation values of mT (solid curve) and mX T (dashed curve) in the system with t¼0.0, h¼1.0 and q ¼1.0 are plotted in Fig. 6(A) as a function of r, selecting the fixed value of ΔS (or ΔS ¼ 0.0). For each curve of mT, it starts to increase from the value at r ¼0.0, exhibits a broad maximum, decreases from the maximum and finally reduces to zero at the critical value of r. The curves labeled A and B represent the systems of Fig. 1(A) and (B), respectively. In particular, one should notice that the mT curve labeled A reduces to zero at the same critical value rC (rC ¼ 6.335) as that in the corresponding tC curve (or solid curve labeled A) in Fig. 3(B). The same fact happens in the mT curve labeled B, where the critical value is also given by rC ¼7.183. These facts indicate that the frustration may be induced in the system with a larger value of r than the value of r for maximum in the plot of mT curve. For each curve of mX T , on the other hand, it decreases from the value at r¼0.0, exhibits the change of derivative at the critical point where the mT curve reduces to zero, and finally takes the same value for each mT curve in the region of r above the critical point. But, any information about the key could not be obtained from the variation of mX T . In Fig. 6(B), the saturation values of mT (solid curve) and mX T (dashed
Fig. 6. The variations of saturation magnetizations, namely mT (solid line) and mX T (dashed line), at T ¼ 0.0 K in the three systems with h ¼1.0 and q ¼1.0, are plotted as a function of r. (A) shows the results for the two systems of Fig. 1(A) (curve A) and 1(B) (curve B) with ΔS. ¼0.0. (B) shows the results for the system of Fig. 2 with ΔS. ¼ –1.0.
T. Kaneyoshi / Journal of Magnetism and Magnetic Materials 374 (2015) 321–326
curve) in the system of Fig. 2 with t ¼0.0, ΔS ¼ 1.0,h ¼1.0 and q ¼1.0 are plotted as a function of r. Then, the mT curve reduces to zero at rC ¼4.630.
325
In order to clarify whether the phase diagram (or the solid curves of Fig. 4) is correct or not, the temperature dependence of mT is examined by solving the coupled Eqs. (3) and (4) numerically. Fig. 7 shows the results of mT (solid line) in the system of Fig. 1(B) with ΔS ¼ 0.0 and h¼1.0, when the three typical pair values of r and q are selected from Fig. 4. The two mT curves labeled ( r¼1.0, q¼2.0) and (r¼4.0, q¼ 5.0) exhibit the normal behavior, since it takes the saturation values (mT ¼0.81 and mT ¼ 0.32) at T¼0.0 K, decrease with the increase of T and reduce to zero at the same TC as the corresponding values in Fig. 4. The mT curve labeled (r¼ 6.0, q¼ 4.0) clearly shows the reentrant phenomenon, since it starts to take a finite value at the lower transition temperature TCL and reduces to zero at the higher transition temperature TCH. These values of TCL and TCH are also equivalent to the corresponding values in Fig. 4. In the figure, the thermal variations of mX T (dashed line) are also plotted by selecting the same pair values as those of mT. Here, one should notice that the mX T curve labeled (r¼6.0, q¼4.0) exhibits the two discontinuous derivatives at the two TCL and TCH. In this way, both the results of the phase diagrams and the magnetizations in the system of Fig. 1(B) are consistent to each other. 3.3. (C) Well-separated nanoislands
Fig. 7. The temperature dependences of mT (solid line) and mX T (dashed line) in the system of Fig. 1(B) with ΔS ¼ 0.0 and h ¼1.0, when the three pair values of ( r, q ) are selected from Fig. 4.
In this part, let us examine the temperature dependence of α total magnetizations MT per site, when the system is consisted of
Fig. 8. The temperature dependences of MT (solid line) and MX T (dashed line) in the system consisted of two well-separated nanoislands with equal probability, namely the systems Fig. 1(A) and Fig. 2, when the values of q and h are fixed at q ¼1.0 and h ¼ 1.0 and the three values of r are selected from Fig. 5. (A) is for the case of r¼ 1.0. (B) is for the case of r¼ 4.0. (C) is for the case of r ¼6.0. In each case, the curves labeled A (ΔS ¼ 0.0) and B(ΔS ¼- 1.0) represent the mT curves for the systems of Fig. 1(A) and Fig. 2, respectively.
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two well-separated nanoislands with equal probability, namely the two nanoislands of Figs. 1(A) and 2. We write that the total α magnetizations per site in Fig. 1(A) is mT (A) and the total α α magnetizations per site in Fig. 2 is mT (B). Then, the MT is defined by M αT ¼
½mαT ðAÞ þ mαT ðBÞ ; 2 α
ð12Þ α
where mT (B) is given by (8) and mT (A) is defined by (10) in [16]. By selecting some typical cases from the results of Fig. 5, let us show the remarkable features of MT ¼MZT in the system. Fig. 8 shows such results for the system with fixed values of h¼1.0 and q ¼1.0, selecting the three values of r from Fig. 5. In Fig. 8(A), the result of MT in the system with r ¼1.0 is given, where the mT curves labeled A(ΔS ¼ 0.0) and B(ΔS ¼ 1.0) exhibit the normal behavior, decrease with the increase of T and reduce to zero at the same TC as the corresponding values in Fig. 5. Because of the difference of TC values in the two nanoislands, the discontinuity of derivative in the MT curve happens at the TC value of mT(B). The MX T (dashed) curve decreases simply from the saturation value at t¼0.0 and exhibits the two discontinuities of derivative at the two TC values of mT(A) and mT(B). In Fig. 8(B), on the other hand, all solid curves which are obtained by fixing the parameter r at r ¼4.0 express an interesting feature. At first, they may increase from the saturation value, exhibit a broad maximum and then reduce to zero at the same TC as that predicted In Fig. 5. The increase from the saturation values at T ¼0.0 K indicates that the frustration induced by r ¼ 4.0 is released by the thermal agitation. This fact is also expected from the results of Fig. 6(A) and (B), since the value of r ¼4.0 lies in the right-hand side of the maximums of Fig. 6(A) and (B). The MX T curve may show a shallow and broad minimum just in the region of the broad maximum of MT. Fig. 8(C) shows the results of the system with r ¼6.0, where the mT(B) curve labeled B(ΔS ¼ –1.0) clearly shows the reentrant phenomenon. The mT(B) curve starts to take a finite value at the lower transition temperature TCL and reduces to zero at the higher transition temperature TCH. These values of TCL and TCH are also equivalent to the corresponding values in Fig. 5. On the other hand, the mT(A) curve labeled A(ΔS ¼0.0) increases from the saturation value, exhibits a broad maximum and then reduces to zero at the same TC as that predicted In Fig. 5. Because of these characteristic behaviors of mT(A) and mT(B), the thermal variation of MT may exhibit the remarkable features, while the MX T does not express such a characteristic feature. 4. Conclusion In this work, we have, via the EFT, investigated the phase diagram and magnetizations in the two transverse Ising nanoislands with finite transverse fields. We have found many characteristic phenomena in them, which could not be obtained from the systems with zero transverse fields (ΩS ¼ Ω¼0.0), when comparing Fig. 3(A) with
other figures (or see [16]). In fact, as shown in Fig. 3(B), a new type of frustration has been obtained in the present systems as well as in [16], when a uniform transverse field (ΩS ¼Ωa0.0) is applied and a large value of r is selected. Fig. 6 confirms that the frustration can be induced by the large value of r. Such a frustration has also been obtained in the systems with different transverse fields (ΩS aΩ), as shown in Figs. 4 and 5. Here, one should notice that this type of frustration is generally found in the other nanosystems, such as ultrathin transverse Ising thin films [7–12]. As shown in Fig. 8, the nanosystem consisted of well-separated nanoislands may exhibit a lot of unexpected behaviors in the temperature dependence of total magnetization MT. Such a behavior may be observed by using the anomalous Hall effect, since the Hall resistance is proportional to the perpendicular component of magnetization [2]. Finally, the study of transverse Ising nanoislands is just at a starting point theoretically and experimentally, so that we do not have so many knowledge about it. That is to say, the study is far from the satisfactory situation. Thus, the study of transverse Ising nanoislands may open a new field in the research of magnetism. References [1] J.S. Smart, Effective Field Theories of Magnetism, W.B.Saunders Com, Philadelphia. U.S.A., 1966. [2] D Chiba, S. Fukami, K. Shimamura, N. Ishikwata, K. Kobayashi, T. Ono, Electric control of ferromagnetic phase transition in cobalt at room temperature, Nat. Mater. 10 (2011) 853. [3] C.M. Schneider, P. Bressler, P. Schuster, J. Kirschner, J.J. de Miguel, R. Miranda, Curie temperature of ultrathin films of fcc cobalt epitaxially grown on atomically flat Cu(100) surfaces, Phys. Rev. Lett. 64 (1990) 1059–1062. [4] R.N. Bhowrnik, Evidence of ferromagnetic La0.67Ca0.33MnO3 nanoparticle, J. Magn. Magn. Mater. 323 (2011) 311–315. [5] T.K. Yamada, L. Gerhard, R.J.H. Wesselink, A. Ernst, W. Wulfhekel, Electric field control of Fe nano magnets, J, Magn. Soc. Jpn 36 (2012) 100–103. [6] M. Loving, F. Jimenez-Villacorta, B. Kaeswurm, D.A. Arena, C.H. Marrows, L.H. Lewis, Structural evidence for stabilized ferromagnetism in epitaxial FeRh nanoislands, J. Phys. D; Appl. Phys 46 (2013) 162002–162008. [7] T. Kaneyoshi, Phase diagrams in nanoscaled Ising thin films with diluted surfaces; effects of interlayer coupling at the surfaces, Physica B 408 (2013) 126–133. [8] T. Kaneyoshi, Phase diagrams in a ultra-thin transverse Ising film with bond or site dilution at surfaces, Physica B 414 (2013) 72–77. [9] T. Kaneyoshi, Phase diagrams in an ultra-thin spin-1 transverse Ising film with bond or site dilution at surfaces, Phase Transit 87 (2014) 111–125. [10] T. Kaneyoshi, Thickness dependence in nano-scaled transverse Ising films with bond or site dilution at surfaces, Physica E 53 (2013) 14–21. [11] T. Kaneyoshi, Characteristic behaviors in a ultrathin Ising film with site- (or bond-) dilution at the surfaces, Physica B 436 (2014) 208–214. [12] T. Kaneyoshi, Unexpected magnetic properties in a ultra-thin transverse Ising film with bond or site dilution at surfaces, Physica E 59 (2014) 50–55. [13] R. Honmura, T. Kaneyoshi, Contribution to the new type of effective-field theory of the Ising model, J. Phys. C 12 (1979) 3979–3992. [14] T. Kaneyoshi, Differential operator technique in the Ising spin systems, Act. Phys. Pol. A 83 (1993) 703–738. [15] F. Zernike, The propagation of order in co-operative phenomena, The AB Case, Physica, 7, 1940565–585. [16] T. Kaneyoshi, Frustration in a transverse Ising nanoisland; effects of interlayer coupling, Phase Transit 87 (2014) 603–612. [17] S. Rusponi, T Cren, N Weiss, M Epple, P Buluschek, L Claude, H Brune, The remarkable difference between surface and step atoms in the magnetic anisotropy of two-dimensional nanostructures, Nat. Matter 2 (2003) 546–551.
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Remarkable features of magnetic properties in transverse Ising nanoislands T. Kaneyoshi 1 1–510, Kurosawadai, Midoriku, Nagoya, 458-0003, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 3 February 2014 Available online 27 August 2014
The phase diagram and magnetizations in two transverse Ising nanoislands with the same structures which are consisted of the two layers are examined by using the effective-field theory with correlations. The effects of interlayer coupling and two transverse fields at the center and the perimeter atoms on them are examined. We find remarkable features in them which come from the frustration induced by the interlayer coupling and two transverse fields. & 2014 Elsevier B.V. All rights reserved.
Keywords: Nanoisland Phase diagrams Magnetizations Transverse Ising model
1. Introduction Nowadays, magnetic nanomaterials, such as nanoparticles, nanoislands, nanofilms, nanowires, nanotubes and so on, have attracted considerable attention experimentally and theoretically, related with their technological and biomedical applications. The magnetic properties of nanomaterials are due to the distinct contributions of inside, perimeter and surface atoms. The remarkably different contributions of surface and perimeter spins govern the nanomagnetism of a material and points to the new way of finding important possible technological applications. From the academic point of view, the magnetism of a single-domain nanoparticle (or nanoisland) and a nanoscaled thin film can be addressed to the research. The thickness dependence of transition temperature, the temperature dependence of magnetization, the critical exponent β and the Neel hyperbola for paramagnetic susceptibility [1] in these materials have been obtained by the use of some elaborate experimental techniques [2–4], in addition to hysteresis loops which are important for the manufacture of magnetic recording media. From the experimental point of view, at the present time, it is almost impossible to study the morphology of the very same particle in conjunction of magnetism. The experimental topography of a nanoparticle displays a discontinuous (or island-like) architecture composed of well-separated and nonuniform particles with an average island diameter [5,6]. As far as we know, the contribution to the magnetic properties from such islands has not been taken into account theoretically. In a series of recent works [7–12], we have examined the phase diagrams and the magnetizations of nanoscaled Ising (or transverse
E-mail address: [email protected] 1 Prof. Emeritus at Nagoya University, Japan. http://dx.doi.org/10.1016/j.jmmm.2014.08.062 0304-8853/& 2014 Elsevier B.V. All rights reserved.
Ising) thin films consisting of a few magnetic layers by the use of the effective-field theory with correlations (EFT) [13,14]. The EFT corresponds to the Zernike approximation (ZA) [15] and it is believed to give more exact results than those of the mean field approximation. In them, when the value of r (r¼Jr /J) is taken as a very small or a very large value, some interesting phenomena have been found in the magnetic properties of the systems with thickness L (from L¼2 to L¼10), such as the appearance of a broad maximum in the variation of transition temperature (TC) as a function of r for the site dilution, while such a phenomenon has not been obtained for the bond dilution. Here, Jr is the interlayer coupling between the surface and the next inner layer, when the value of the inner layer coupling is fixed at J. In particular, a new type of frustration has been obtained in the systems [10–12], when a uniform transverse field is applied and a large value of r is selected. Furthermore, such a frustration has been also found in the transverse Ising nanoisland consisting of two layers with eighteen atoms in total [16]. The aim of this work is, within the theoretical framework of the EFT, to investigate the effects of interlayer coupling and two transverse fields (ΩS and Ω) on the magnetic properties (phase diagram, magnetizations) in two nanoscaled transverse Ising islands with the same structural aspect as that in Ref. [16], since the effects of a uniformly applied transverse field (ΩS ¼Ω) have been examined in Ref. [16]. Here, ΩS is the transverse field at the perimeter spin and Ω is the transverse field at the central (inside) spin.
2. Model and formulation We consider the 3D nanoisland (or nanoparticle), as depicted in Fig. 1, in which they are consisted of the surface shell and the core. The core (white circles) is surrounded by the surface shell (black
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The surface exchange interaction JS is often defined as JS ¼ J ð1 þ ΔS Þ;
ð2Þ
in order to clarify the effect of surface shell on physical properties in the system. In particular, when the value of JS is given by JS ¼0.0 (or ΔS ¼ 1.0) in (3), the system shown as Fig. 1(B) represents the system as shown in Fig. 2, in which the atoms at the corner of each square (half black and half white circle) represent the nonmagnetic atoms. The system represents one of the transverse Ising islands with the same structural aspect as that in Fig. 1(A). Within the framework of the EFT [13,14], we can obtain the longitudinal magnetizations mS ( mS ¼ o σZi 4 ) at the surface shell and the longitudinal magnetization mC ( mC ¼ o σZm 4 ) in the center for the system shown in Fig. 1(B), as coupled equations: 2 mS ¼ coshðAÞ þmS sinhðAÞ coshðCÞ þmC sinhðCÞ ð3Þ coshðBÞ þmS sinhðBÞ f S ðxÞjx ¼ 0 4 mC ¼ cosh ðCÞ þ mS sinh ðCÞ ½ cosh ðBÞ þ mC sinh ðBÞfðxÞjx ¼ 0
ð4Þ
where A, B and C are defined by A ¼ JS D, B ¼ Jr D and C ¼J D. D¼ d/dx is the differential operator. Here, the functions fS (x) and f (x) are defined by f S ðxÞ ¼ ðx=yS Þ tanh ðβ yS Þ and fðxÞ ¼ ðx=yÞ tanh ðβ yÞ
ð5Þ
with yS ¼ ðx2 þ ΩS Þ1=2 2
and Fig. 1. Schematic representation of two transverse Ising nanoislands. In Fig. (A), the above and below layers are consisted of 9 spin-1/2 atoms, where 8 black circles and 1 white circle represent the same magnetic atoms. In Fig. (B), the above and below layers are consisted of 5 spin-1/2 atoms, where 4 black circles and 1 white circle represent the same magnetic atoms. The lines connecting the black and white circles in each figure represent the three nearest-neighbor exchange interactions (JS, J and Jr).
circles). Each site on the figure is occupied by a Ising spin. Each spin is connected to the nearest neighbor spins with an exchange interaction. The surface spins are coupled to the center spin with an exchange interaction J. The atoms on the surface shell are connected by the exchange interaction JS. Each spin on the upper layer is connected to the corresponding spin on the lower layer with an exchange interaction Jr. The choice corresponds to the experimental facts [5,17], as discussed in [16]. In this work, let us formulate the system depicted as Fig. 1(B), since the phase diagram and the magnetizations of the system shown in Fig. 1 (A) has been formulated in [16] and the numerical results of them have been examined for the case of ΩS ¼ Ω. The Hamiltonian of the system is given by
y ¼ ðx2 þ Ω Þ1=2 ; 2
ð6Þ
where β ¼1 / kBT and T is a temperature. X Furthermore, the transverse magnetizations mX S ¼ o σi 4 and X X mC ¼ o σm 4 are also given by the same equations as those of Eqs. (3,4), only replacing the functions fS(x) and f(x) in Eqs. (3,4) by the new functions hS(x) and h(x), respectively. The new functions hS(x) and h(x) are defined by hS ðxÞ ¼ ðΩS =yS Þ tanh ðβ yS Þ and hðxÞ ¼ ðΩ=yÞ tanh ðβ yÞ
ð7Þ
H ¼ J S ∑σ i Z σ j Z J∑σ m Z σ n Z J r ∑σ i Z σ m Z ðmnÞ
ðijÞ
ΩS Σσ i X ΩΣσ m X ; ðiÞ
ðmÞ
ðimÞ
ð1Þ
α
where σi (α ¼z, x) is the Pauli spin operator with σZi ¼ 71. ΩS and Ω represent the transverse fields at the surface shell and in the core, respectively, since the transverse fields may be different at the surface and in the core. The first (ij) and second (mn) terms in the Hamiltonian (1) represent the contributions from the surface shell and the core, respectively. The third term shows the contribution from the interlayer interaction.
Fig. 2. Schematic representation of a transverse Ising nanoisland. The above and below layers are consisted of 9 spin-1/2 atoms, where 8 black circles and 1 white circle represent the same magnetic atoms and the half black and half white atoms represent a nonmagnetic atom. The lines connecting the black and white circles represent the two nearest-neighbor exchange interactions (J and Jr).
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Let us here define the total longitudinal and transverse magnetizations (mT ¼ mZT and mX T ) per site as α 4mS þ mαc ; ð8Þ mαT ¼ 5 where α ¼z or x. By expanding the right-hand sides of coupled Eqs. (3,4), the transition temperature (or phase diagram) can be determined from ½1 ð2K 1 þ K 2 Þ½1 K 3 4K 4 K 5 ¼ 0;
ð9Þ
where the coefficients Kn (n ¼1–6) are given by K1 ¼ coshðAÞ sinhðAÞ coshðBÞ coshðCÞ f S ðxÞjx ¼ 0 2
K2 ¼ cosh ðAÞ sinhðBÞ coshðCÞ f S ðxÞjx ¼ 0 4
K3 ¼ cosh ðCÞ sinhðBÞ fðxÞjx ¼ 0 2
K4 ¼ cosh ðAÞ coshðBÞ sinhðCÞ fsðxÞjx ¼ 0 3
K5 ¼ cosh ðCÞ sinhðCÞ coshðBÞ fðxÞjx ¼ 0;
ð10Þ
The coefficients Kn and the coupled Eqs. (3) and (4) can be easily calculated by applying a mathematical relation exp(a D)F (x)¼ F(xþa). For the following discussions, let us here define the four parameters t, r h and q as t¼
kB T J Ω ΩS ; r ¼ r ; h ¼ and q ¼ J J J Ω
ð11Þ
3. Numerical results In this section, let us examine the magnetic properties (phase diagram and magnetization) in the nanoisland (or Fig. 1(B)) by solving the Eqs. (3), (4) and (9) given in Section 2 numerically. In particular, we are interested in the behaviors of them, when the parameter q is selected as qa 1.0. The reasons are based on the previous work [16], in which the case of q ¼1.0 has been examined. In order to compare the results of Fig. 1(B) with those of Fig. 1(A) in [16], one should notice that the exchange interaction JS (or ΔS) in Fig. 1(B) is assumed to be the same value as that in Fig. 1(A), for convincing.
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3.1. (A) Phase diagrams At first, let us show the two typical results of phase diagrams for the two systems (Fig. 1(A) and (B)) with q ¼1.0 (or ΩS ¼Ω) in Fig. 3. Fig. 3(A) shows the phase diagram (tC ¼kBTC/J versus r plots) in the two systems with ΩS ¼ Ω¼0.0, when the three values of ΔS are selected. The solid lines express the results for the system depicted as Fig. 1(B) and the dashed lines represent the previous results in [16] for the system depicted as Fig. 1(A). For each value of ΔS, the curves at first exhibit the rather rapid increase and then change to the slow variation, when the value of r is increased. Next, let us examine whether the unconventional phase diagram (or Fig. 5 in [16]) in the nanoisland of Fig. 1(A) can be found for the system of Fig. 1(B) with fixed values of ΔS ¼0.0 and q ¼1.0, when the finite transverse field (or h ¼1.0) is applied. Fig. 3(B) shows such results for the two nanoislands, in which the dashed lines are equivalent to the corresponding results with h¼0.0 in Fig. 3(A), namely the sold and dashed curves labeled ΔS ¼0.0. The solid curve labeled A at first increases with following to the dashed line labeled A, shows the existence of two transition temperatures (or the reentrant phenomenon) in the region between r ¼12.39 and r ¼6.335, and reduces to zero at the critical value of rC (rC ¼6.335), as shown in [16]. The same behaviors as those labeled A can be also found for the system of Fig. 1(B). They are represented as the solid and dashed curves labeled B. Then, the critical value rC for the solid curve labeled B is given by rC ¼7.183. In [16], the phase diagrams have been obtained for the case of q¼ 1.0. In this work, let us examine another case (or tC versus q plot) for the two nanoislands, changing the value of r. The results for the two systems with fixed values of ΔS ¼0.0 and h¼ 1.0 are depicted in Fig. 4, when the three values of r are selected. The solid and dashed lines in the figure represent the results of Fig. 1(B) and Fig. 1(A), respectively. In the figure, the tC curve labeled r ¼4.0 is larger than the tC curve labeled r ¼ 1.0 and the tC curve labeled r ¼6.0 exhibits the reentrant behavior, for each system. In the Figs. 3 and 4, one should notice that the tc results of Fig. 1 (B) are always larger than those of Fig. 1(A). The reason is that the system of Fig. 1(B) is more closely packed than that of Fig. 1(A), even though the exchange interaction JS in the two systems is selected as the same value. In fact, the behavior of mS1 in Fig. 1(A), which is loosely connected as a whole to mS2, gives a lower tC value than that of Fig. 1(B).
Fig. 3. (A) The phase diagrams (tC versus r plot) in the two systems (or Figs.1(A) and (B)) with zero transverse fields (ΩS ¼ Ω ¼0.0), when the three values of ΔS are selected. The solid and dashed lines represent the results of Fig. 1(B) and (A), respectively. (B) The phase diagrams (tC versus r plot) in the two systems with fixed values of ΔS ¼0.0 and q¼ 1.0, where the curves labeled A and B show the results of Fig. 1(B) and (A). The dashed line labeled h ¼ 0.0 is respectively equivalent to the corresponding one (or the curve labeled ΔS ¼0.0) in Fig. 3(A). The region below (or inside) each TC curve shows the ferromagnetic phase.
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As noted in Section 2, when the value of JS is given by JS ¼0.0 (or ΔS ¼ 1.0) in (2), the system shown as Fig. 1(B) represents the system of Fig. 2, in which the atoms at the corner of each square (half black and half white circle) represent the nonmagnetic atoms. The system represents one of the transverse Ising islands with the same structural aspect as that of Fig. 1 (A). By selecting the value of ΔS as ΔS ¼- 1.0, let us examine the phase diagram (tC versus q plot) of the system described by Fig. 2. Then, in order to compare the tC results of Fig. 1(A) with those of Fig. 2, the value of h is fixed at h¼1.0 and the value of ΔS for the system of Fig. 1(A) is selected as ΔS ¼ 0.0 (or JS ¼J). The results are shown in Fig. 5, where the solid and dashed lines represent the tC curves of Fig. 2 and Fig. 1(A), respectively. They show the behaviors similar to those in Fig. 4, while the solid curve labeled r ¼6.0 takes a finite value (or tC ¼ 0.4) at q¼0.0. In Fig. 5, the dashed lines for system of Fig. 1(A) represent the same tC curves as those (or dashed lines) in Fig. 4 Fig. 4. The phase diagram (tC versus q plot) in the two systems with ΔS ¼ 0.0 and h ¼1.0, when the three values of r are selected. The solid and dashed lines represent the results of Fig. 1(B) and (A), respectively.
3.2. (B) Magnetizations
Fig. 5. The phase diagram (tC versus q plot) in the two systems with ΔS ¼0.0 and h ¼ 1.0, when the three values of r are selected. The solid lines represent the tC results in the system of Fig. 2 with ΔS ¼ 1.0 and the dashed lines are the same as the dashed lines in Fig. 4.
It is not clear from the results of Figs.3 and 4 from what value of r the frustration may be induced in the two transverse Ising nanoislands. In order to get one of the keys, the saturation values of mT (solid curve) and mX T (dashed curve) in the system with t¼0.0, h¼1.0 and q ¼1.0 are plotted in Fig. 6(A) as a function of r, selecting the fixed value of ΔS (or ΔS ¼ 0.0). For each curve of mT, it starts to increase from the value at r ¼0.0, exhibits a broad maximum, decreases from the maximum and finally reduces to zero at the critical value of r. The curves labeled A and B represent the systems of Fig. 1(A) and (B), respectively. In particular, one should notice that the mT curve labeled A reduces to zero at the same critical value rC (rC ¼ 6.335) as that in the corresponding tC curve (or solid curve labeled A) in Fig. 3(B). The same fact happens in the mT curve labeled B, where the critical value is also given by rC ¼7.183. These facts indicate that the frustration may be induced in the system with a larger value of r than the value of r for maximum in the plot of mT curve. For each curve of mX T , on the other hand, it decreases from the value at r¼0.0, exhibits the change of derivative at the critical point where the mT curve reduces to zero, and finally takes the same value for each mT curve in the region of r above the critical point. But, any information about the key could not be obtained from the variation of mX T . In Fig. 6(B), the saturation values of mT (solid curve) and mX T (dashed
Fig. 6. The variations of saturation magnetizations, namely mT (solid line) and mX T (dashed line), at T ¼ 0.0 K in the three systems with h ¼1.0 and q ¼1.0, are plotted as a function of r. (A) shows the results for the two systems of Fig. 1(A) (curve A) and 1(B) (curve B) with ΔS. ¼0.0. (B) shows the results for the system of Fig. 2 with ΔS. ¼ –1.0.
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curve) in the system of Fig. 2 with t ¼0.0, ΔS ¼ 1.0,h ¼1.0 and q ¼1.0 are plotted as a function of r. Then, the mT curve reduces to zero at rC ¼4.630.
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In order to clarify whether the phase diagram (or the solid curves of Fig. 4) is correct or not, the temperature dependence of mT is examined by solving the coupled Eqs. (3) and (4) numerically. Fig. 7 shows the results of mT (solid line) in the system of Fig. 1(B) with ΔS ¼ 0.0 and h¼1.0, when the three typical pair values of r and q are selected from Fig. 4. The two mT curves labeled ( r¼1.0, q¼2.0) and (r¼4.0, q¼ 5.0) exhibit the normal behavior, since it takes the saturation values (mT ¼0.81 and mT ¼ 0.32) at T¼0.0 K, decrease with the increase of T and reduce to zero at the same TC as the corresponding values in Fig. 4. The mT curve labeled (r¼ 6.0, q¼ 4.0) clearly shows the reentrant phenomenon, since it starts to take a finite value at the lower transition temperature TCL and reduces to zero at the higher transition temperature TCH. These values of TCL and TCH are also equivalent to the corresponding values in Fig. 4. In the figure, the thermal variations of mX T (dashed line) are also plotted by selecting the same pair values as those of mT. Here, one should notice that the mX T curve labeled (r¼6.0, q¼4.0) exhibits the two discontinuous derivatives at the two TCL and TCH. In this way, both the results of the phase diagrams and the magnetizations in the system of Fig. 1(B) are consistent to each other. 3.3. (C) Well-separated nanoislands
Fig. 7. The temperature dependences of mT (solid line) and mX T (dashed line) in the system of Fig. 1(B) with ΔS ¼ 0.0 and h ¼1.0, when the three pair values of ( r, q ) are selected from Fig. 4.
In this part, let us examine the temperature dependence of α total magnetizations MT per site, when the system is consisted of
Fig. 8. The temperature dependences of MT (solid line) and MX T (dashed line) in the system consisted of two well-separated nanoislands with equal probability, namely the systems Fig. 1(A) and Fig. 2, when the values of q and h are fixed at q ¼1.0 and h ¼ 1.0 and the three values of r are selected from Fig. 5. (A) is for the case of r¼ 1.0. (B) is for the case of r¼ 4.0. (C) is for the case of r ¼6.0. In each case, the curves labeled A (ΔS ¼ 0.0) and B(ΔS ¼- 1.0) represent the mT curves for the systems of Fig. 1(A) and Fig. 2, respectively.
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two well-separated nanoislands with equal probability, namely the two nanoislands of Figs. 1(A) and 2. We write that the total α magnetizations per site in Fig. 1(A) is mT (A) and the total α α magnetizations per site in Fig. 2 is mT (B). Then, the MT is defined by M αT ¼
½mαT ðAÞ þ mαT ðBÞ ; 2 α
ð12Þ α
where mT (B) is given by (8) and mT (A) is defined by (10) in [16]. By selecting some typical cases from the results of Fig. 5, let us show the remarkable features of MT ¼MZT in the system. Fig. 8 shows such results for the system with fixed values of h¼1.0 and q ¼1.0, selecting the three values of r from Fig. 5. In Fig. 8(A), the result of MT in the system with r ¼1.0 is given, where the mT curves labeled A(ΔS ¼ 0.0) and B(ΔS ¼ 1.0) exhibit the normal behavior, decrease with the increase of T and reduce to zero at the same TC as the corresponding values in Fig. 5. Because of the difference of TC values in the two nanoislands, the discontinuity of derivative in the MT curve happens at the TC value of mT(B). The MX T (dashed) curve decreases simply from the saturation value at t¼0.0 and exhibits the two discontinuities of derivative at the two TC values of mT(A) and mT(B). In Fig. 8(B), on the other hand, all solid curves which are obtained by fixing the parameter r at r ¼4.0 express an interesting feature. At first, they may increase from the saturation value, exhibit a broad maximum and then reduce to zero at the same TC as that predicted In Fig. 5. The increase from the saturation values at T ¼0.0 K indicates that the frustration induced by r ¼ 4.0 is released by the thermal agitation. This fact is also expected from the results of Fig. 6(A) and (B), since the value of r ¼4.0 lies in the right-hand side of the maximums of Fig. 6(A) and (B). The MX T curve may show a shallow and broad minimum just in the region of the broad maximum of MT. Fig. 8(C) shows the results of the system with r ¼6.0, where the mT(B) curve labeled B(ΔS ¼ –1.0) clearly shows the reentrant phenomenon. The mT(B) curve starts to take a finite value at the lower transition temperature TCL and reduces to zero at the higher transition temperature TCH. These values of TCL and TCH are also equivalent to the corresponding values in Fig. 5. On the other hand, the mT(A) curve labeled A(ΔS ¼0.0) increases from the saturation value, exhibits a broad maximum and then reduces to zero at the same TC as that predicted In Fig. 5. Because of these characteristic behaviors of mT(A) and mT(B), the thermal variation of MT may exhibit the remarkable features, while the MX T does not express such a characteristic feature. 4. Conclusion In this work, we have, via the EFT, investigated the phase diagram and magnetizations in the two transverse Ising nanoislands with finite transverse fields. We have found many characteristic phenomena in them, which could not be obtained from the systems with zero transverse fields (ΩS ¼ Ω¼0.0), when comparing Fig. 3(A) with
other figures (or see [16]). In fact, as shown in Fig. 3(B), a new type of frustration has been obtained in the present systems as well as in [16], when a uniform transverse field (ΩS ¼Ωa0.0) is applied and a large value of r is selected. Fig. 6 confirms that the frustration can be induced by the large value of r. Such a frustration has also been obtained in the systems with different transverse fields (ΩS aΩ), as shown in Figs. 4 and 5. Here, one should notice that this type of frustration is generally found in the other nanosystems, such as ultrathin transverse Ising thin films [7–12]. As shown in Fig. 8, the nanosystem consisted of well-separated nanoislands may exhibit a lot of unexpected behaviors in the temperature dependence of total magnetization MT. Such a behavior may be observed by using the anomalous Hall effect, since the Hall resistance is proportional to the perpendicular component of magnetization [2]. Finally, the study of transverse Ising nanoislands is just at a starting point theoretically and experimentally, so that we do not have so many knowledge about it. That is to say, the study is far from the satisfactory situation. Thus, the study of transverse Ising nanoislands may open a new field in the research of magnetism. References [1] J.S. Smart, Effective Field Theories of Magnetism, W.B.Saunders Com, Philadelphia. U.S.A., 1966. [2] D Chiba, S. Fukami, K. Shimamura, N. Ishikwata, K. Kobayashi, T. Ono, Electric control of ferromagnetic phase transition in cobalt at room temperature, Nat. Mater. 10 (2011) 853. [3] C.M. Schneider, P. Bressler, P. Schuster, J. Kirschner, J.J. de Miguel, R. Miranda, Curie temperature of ultrathin films of fcc cobalt epitaxially grown on atomically flat Cu(100) surfaces, Phys. Rev. Lett. 64 (1990) 1059–1062. [4] R.N. Bhowrnik, Evidence of ferromagnetic La0.67Ca0.33MnO3 nanoparticle, J. Magn. Magn. Mater. 323 (2011) 311–315. [5] T.K. Yamada, L. Gerhard, R.J.H. Wesselink, A. Ernst, W. Wulfhekel, Electric field control of Fe nano magnets, J, Magn. Soc. Jpn 36 (2012) 100–103. [6] M. Loving, F. Jimenez-Villacorta, B. Kaeswurm, D.A. Arena, C.H. Marrows, L.H. Lewis, Structural evidence for stabilized ferromagnetism in epitaxial FeRh nanoislands, J. Phys. D; Appl. Phys 46 (2013) 162002–162008. [7] T. Kaneyoshi, Phase diagrams in nanoscaled Ising thin films with diluted surfaces; effects of interlayer coupling at the surfaces, Physica B 408 (2013) 126–133. [8] T. Kaneyoshi, Phase diagrams in a ultra-thin transverse Ising film with bond or site dilution at surfaces, Physica B 414 (2013) 72–77. [9] T. Kaneyoshi, Phase diagrams in an ultra-thin spin-1 transverse Ising film with bond or site dilution at surfaces, Phase Transit 87 (2014) 111–125. [10] T. Kaneyoshi, Thickness dependence in nano-scaled transverse Ising films with bond or site dilution at surfaces, Physica E 53 (2013) 14–21. [11] T. Kaneyoshi, Characteristic behaviors in a ultrathin Ising film with site- (or bond-) dilution at the surfaces, Physica B 436 (2014) 208–214. [12] T. Kaneyoshi, Unexpected magnetic properties in a ultra-thin transverse Ising film with bond or site dilution at surfaces, Physica E 59 (2014) 50–55. [13] R. Honmura, T. Kaneyoshi, Contribution to the new type of effective-field theory of the Ising model, J. Phys. C 12 (1979) 3979–3992. [14] T. Kaneyoshi, Differential operator technique in the Ising spin systems, Act. Phys. Pol. A 83 (1993) 703–738. [15] F. Zernike, The propagation of order in co-operative phenomena, The AB Case, Physica, 7, 1940565–585. [16] T. Kaneyoshi, Frustration in a transverse Ising nanoisland; effects of interlayer coupling, Phase Transit 87 (2014) 603–612. [17] S. Rusponi, T Cren, N Weiss, M Epple, P Buluschek, L Claude, H Brune, The remarkable difference between surface and step atoms in the magnetic anisotropy of two-dimensional nanostructures, Nat. Matter 2 (2003) 546–551.