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Magnetism in two antiferromagnetic Ising nanoparticles under an applied transverse field
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
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Research articles
Magnetism in two antiferromagnetic Ising nanoparticles under an applied transverse field
T
T. Kaneyoshi Nagoya University, 1-510, Kurosawadai, Midoriku, Nagoya 458-0003, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Phase diagrams Magnetizations Size effect Ferrimagnetic behavior Reentrant phenomena Nanoparticles
The phase diagrams and magnetization curves of two antiferromagnetic Ising nanoparticles (A and B) under an applied transverse field are investigated by using the effective-field theory with correlations. The nanoparticle B is a size larger than that of nanoparticle A. The effects of size on the magnetic properties are discussed. Some characteristic features, such as the reentrant phenomena, are found in these properties of both systems, when a finite transverse field h is applied. The nanoparticle B exhibits the ferrimagnetic behaviors, such as a compensation point.
1. Introduction Nowadays, it is well-known experimentally and theoretically that the magnetic properties of a magnetic material change from the bulk ones, when its size reduces to the nanoscale level. Consequently, the classifications valid to bulk magnetic materials, such as ferromagnetic, antiferromagnetic and ferrimagnetic material, could not be applied straightforwardly to a nanoscale magnetic material. In fact, some case exhibits a compensation point below a transition temperature in an antiferromagnetic nanoscale material [1,2]. The compensation point phenomena are normally characteristic to bulk materials exhibiting ferrimagnetism, which are composed from at least two different atoms with different spins. Other cases show surface induced ferromagnetism in antiferromagnetic nanoscale magnetic materials, such as defects at the surface [3] and uncompensated surfaces in thin films [4]. Furthermore, the graphene-like nanoparticles, nano-islands and bi-layer films have been fabricated by doping of magnetic atoms on graphene sheets. The simplest graphene-like nano-system is made of six spin-1/2 magnetic atoms which constitute a nanoparticle with hexagonal structure. They are arranged to be antiferromagnetic, while the particle is paramagnetic. Theoretically, when the simplest graphene-like nanoparticle is discussed by using the mean-field theory (MFA), the system shows a finite transition temperature (TC ≠ 0.0), although it exhibits TC = 0.0, when the effective-field theory with correlations (EFT) is applied [5]. The difference comes from the following fact: The EFT includes automatically some correlations between the central spin and the nearest-neighbor spins. The MFA is considering only one atom and replaces the interactions from other neighbor atoms with a uniform field proportional to the averaged magnetization. The EFT corresponds
to the Zernike approximation [6]. In a nanoscaled system, particularly, we must pay attention to the fact that the magnetization at each site is heavily dependent on the surrounding situation of it. In many nanosystems [7–15] described by the transverse Ising model, an interesting phenomenon, namely the reentrant phenomenon which comes from the finite size effect of the system, has been found by using the EFT, although it has not been obtained by using the MFA. Also, mixed spin Ising nanoparticles have been discussed by using the Monte Carlo simulation [16,17]. Many works have been reported experimentally, in order to clarify the magnetic properties in antiferromagnetic nanoparticles (or nanosized manganites) [18–20]. These works have been directed to clarify the surface induced ferromagnetism in them. As far as we know, the ferrimagnetic properties in antiferromagnetic nanoparticles have not been reported experimentally, while the evidence of ferrimagnetism has been reported for a ferromagnetic nanoparticle [21]. Within the framework of the EFT, some works have discussed theoretically the possibility of ferrimagnetism in some antiferromagnetic nanoparticles [5,7,22], while the compensation point phenomenon characteristic to ferrimagnetism has not been obtained in these works. In the present study, via the EFT, an attempt is made for the two new-type antiferromagnetic nanoparticles, whether they exhibit the typical ferrimagnetic behaviors, especially the compensation point, as well as the reentrant phenomena due to the finite size effect The particularly interesting aim is to clarify what phenomena may be realized in the magnetic properties, due to the difference of particle sizes. The outline of this work is as follows. In Section 2, the models and formulations of the two (A and B) antiferromagnetic nanoparticles are given within the EFT. The nanoparticle B is a size larger than that of
E-mail address: [email protected]. https://doi.org/10.1016/j.jmmm.2019.166368 Received 8 June 2019; Received in revised form 22 October 2019; Accepted 27 December 2019 Available online 21 January 2020 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
T. Kaneyoshi
Fig. 1. Schematic representation of two antiferromagnetic nanoparticles A and B. The black and white circles represent the spin-1/2 magnetic atoms and they are respectively directed to up and down spin directions. The large circle at the center represents a nonmagnetic atom. In the two nanoparticles, the white atoms are connected by an exchange interaction JR (JR ≧ 0.0), which originates from the super-exchange interaction via a nonmagnetic atom. The two exchange interactions (−JS and −J, J> 0.0) act between the nearest-neighbor sites.
2
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
T. Kaneyoshi
site i. For the white circle in the nanoparticle A, it is given by Ei = −Js (Si+1Z + Si−1Z) + JR SlZ, where the site l represents the white circle at the opposite side. For the nanoparticle A, the magnetizations m1 and m2 in Fig. 1A can be easily and exactly written as
nanoparticle A. From the difference, the effects of size on the magnetic properties can be compared. In Section 3, the phase diagrams and the temperature dependences of magnetizations are presented numerically for the nanoparticle A. It exhibits some characteristic behaviors, such as the reentrant phenomenon, even when H = 0.0. In Section 4, the phase diagrams and the temperature dependences of magnetizations are presented numerically for the nanoparticle B. The nanoparticle B exhibits the compensation point phenomenon as well as other characteristic phenomena observed in the bulk ferrimagnetism. The last section is devoted to conclusion.
and
m1 = < Si Z> = − 2m2 K1 + m1K2 + < SZi+1SZi−1SZl > K3
2. Models and formulation
im
mn
m1 = −2m2 K1 + m1K2 + (m2 )2m1K3
im
im
im
i
(1b)
K2 =
1 [F(2.0JS + JR) − F(2.0JS − JR) + 2.0F(]R)] 4
K3 =
1 [F(2.0JS + JR) − F(2.0JS − JR) − 2.0F(]R)] 4
m2 = −2.0m 4 P4 − 2.0m1P4 − 2.0(m1)2m 4 P5 − 2.0(m 4 )2m1P5 m1 = −2.0m2 P6 − m3P6 + m1P7 − (m2)2m3P8 + (m2)2m1P9
for the nanoparticle B, where Si (α = z or x) is the Pauli operator with SiZ = ± 1.0 at a site i. The last terms in HA and HB represent the effect of a magnetic field H applied perpendicularly to the spin direction z. The first and second terms in HA represent respectively the perimeterand super- exchange interactions. The last term represents the effect of a magnetic field H applied perpendicularly to the spin direction z. The first and third terms in HB represent respectively the perimeter- and super- exchange interactions. The second term in HB shows the contribution from the interior exchange interaction. The exchange interaction JS is given by
+ 2.0m1m2 m3P9,
(7)
where coefficients are given by B = F (2.0 JS) P1 = 1 [F (2.0 JS + J) + F(2.0 JS − J) ] 4
P2 = 1 [F (2.0 JS + J) − F(2.0 JS − J) + 2.0F (]) ] 4
P3 = 1 [F (2.0 JS + J) − F(2.0 JS − J) − 2.0F (]) ] 4
P4 = 1 [F (4.0 J) + 2.0F (2.0 J) ]
(2)
P5 =
8 1 8 1 8
[F (4.0 J) − 2.0F (2.0 ]) ]
P6 = [F (3.0 JS + JR) + F(3.0 JS − JR) + F (J + ]R) + F(J − JR) ] P7 = [F (3.0 JS + JR) − F(3.0 JS − JR) + 3.0 {F (J + ]R) − F (J − JR)} ] P8 = [F (3.0 JS + JR) + F(3.0 JS − JR) − 3.0 {F (J + ]R) + F (J − JR)} ] P9 = 1 [F (3.0 JS + JR) − F(3.0 JS − JR) − F (J + ]R) + F(J − JR) ]
in order to clarify the effect of perimeter on the magnetic properties. The present systems are a quantum system, when H ≠ 0.0, so that they could not be solved exactly. The starting equation for the EFT is the approximated relation derived in [23]. Here, one should notice that the approximated relation exactly reduces to the Callen identity [24], when H = 0.0. Applying the differential operator technique [25,26] to the relation, the longitudinal magnetization at a site i is given by
< SiZ > = < exp(Ei∇) > F(x)|x = 0
1 [F(2.0JS + JR) + F(2.0JS − JR)] 4
m5 = −m 4 B m 4 = −m5P1 − m3P1 − m2 P2 − m5m3m2 P3 m3 = −2.0m 4 P1 − m1P2−(m 4) 2m1P3
α
JS = J(1.0 + Δ)
K1 =
For the nanoparticle B, following to the same procedure as that of nanoparticle A, within the EFT, the magnetizations m1, m2, m3, m4 and m5 are given by
for the nanoparticle A and
HB = JS ∑ SZi SZm+J ∑ SZi SZm − JR ∑ SZi SZm+H ∑ SiX
(6)
The coefficients B and Kn (n = 1–3) in (5) and (6) are given by B = F (2.0 JS)
(1a)
l
(6.a)
Here, the EFT is nothing but the fact that the correlation function < Si+1Z Si-1Z SlZ > is decoupled to < Si Z Z Z > < S > < S > . Within the EFT, equation (6.a) reduces to i−1 l +1
We study two antiferromagnetic two-dimensional Ising nanoparticles, as depicted in Fig. 1. The black and white circles in the two (A and B) nanoparticles represent the magnetic atoms with spin- 1/2. They are respectively directed to opposite directions, since they are coupled by a negative exchange interaction − JS and – J (J > 0.0), where JS is the exchange interaction at the perimeter and J is the exchange interaction in the inside of nanoparticle B. In the nanoparticle A, the white atoms are also connected by an exchange interaction JR (JR ≧ 0.0), which originates from the super-exchange interaction via a nonmagnetic atom (gray atom in the center). The size of nanoparticle B is a size larger than that of nanoparticle A, where a nonmagnetic atom also exists in the center and there exists a super-exchange interaction JR between the white atoms. The Hamiltonians of two nanoparticles are described by Z HA = JS ∑ Si ZSm − JR ∑ SZm SZn − H∑ SlX
(5)
m2 = −m1B
8
Let us here define the total magnetization mT per site for these two nanoparticles A and B; For the nanoparticle A, it is given by
(3)
where ∇ = ∂ /∂ x is the differential operator. The function F(x) is defined by
mT =
x F(x) = tanh(β e(x)) e (x )
and for the nanoparticle B, it can be represented as
(4)
mT =
with
e(x) = [x2 + H2]1/2 ,
1 [m1 + m2] 2
1 [m1 + m2 + m3 + 2.0m 4 + m5] 6
(8)
(9)
In the following sections, the phase diagram and the thermal variations of magnetizations in the nanoparticles A and B are examined. For the aim, let us here introduce the reduced parameters as follows;
where β = 1/kBT. Ei in (3) represents the local field at the site i, such as Ei = −Js (Si+1Z + Si−1Z) for the black circle at a corner in the nanoparticle A, where i + 1 and i − 1 denote the nearest-neighbor spins at a
t= kB. T/J, r= JR /J and h= H/J 3
(10)
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
T. Kaneyoshi
Fig. 3. The solid line shows the variation of tC as a function of r in the nanoparticle A with Δ = 0.0 and h = 0.5. The dashed lines represent the variations of magnetizations m1 and m2 in the nanoparticle A with Δ = 0.0, h = 0.5 and t = 0.01, when the value of r is changed.
Fig. 2. The phase diagram (tC versus r plot) in the nanoparticle A with Δ = 0.0, when the number of r is changed.
3. Nanoparticle A At first, it is important to note the following facts; When JR = 0.0, the coefficients K2 and K3 in (6,a) or (6) reduce to K2 = K3 = 0.0. Accordingly, the coupled Eqs. (5) and (6) can be exactly represented by m1 m2 [1.0 − B2 ] = 0.0, which solution indicates that the nanoparticle A with JR = 0.0 must be paramagnetic (TC = 0.0 or m1 = m2 = 0.0 for T > 0.0), just like some cases discussed for other nanoparticles [5]. When JR ≠ 0.0, the transition temperature TC (or phase diagram) can be obtained from
2.0 B K1 + K2 = 1.0
(11)
Let us examine the phase diagram (tC versus r plot) of the nanoparticle A with Δ = 0.0, changing the value of h. The results are plotted in Fig. 2. As noted above, the tC curves labeled h = 0.0 (dashed line) and h = 0.1 rapidly reduce to zero, when the value of r approaches to zero. But, the curves labeled as h ≥ 0.5 exhibit the region of tC = 0.0 in the vicinity of r = 0.0, when the value of r decreases to the value of r = 0.0. Furthermore, they exhibit the reentrant phenomena for the large values of r. In order to clarify these facts, the tC curve and the variations of saturation magnetizations (m1 and m2 at t = 0.01) for the nanoparticle A with fixed values of Δ = 0.0 and h = 0.5 are given in Fig. 3, changing the value of r. In Fig. 3, the region of tC = 0.0 in the vicinity of r = 0.0 can be found in the region of 0.06 > r ≥ 0.0 and the reentrant phenomena can be found in the region of 5.989 > r > 4.185. Both phenomena come from the competition between the strength of r and the value of an applied transverse field; The applied transverse field tends to destroy of order in the z-direction and the value of r tends to the strength of order in the z-direction. The phenomena also come from the following facts: One is that the system is paramagnetic, when r = 0.0. The other is that the system is the finite size. Accordingly, similar phenomena have also been found in other nanosized magnetic systems [5,7-15]. In Fig. 3, the saturation magnetizations m1 and m2 at t = 0.01 are also plotted and one should notice that they take finite values in the same region (4.185 > r > 0.05) as that of tC. In Fig. 4, the tC curves in the system with h = 0.0 are plotted as a function of r , selecting three values of Δ. The curve labeled Δ = 0.0 is the same as that (dashed curve labeled h = 0.0) in Fig. 2, where the two transition temperatures exist in the region of Δ > 4.362. As is seen from the results, the curve labeled Δ = – 0.5 exhibits the reentrant phenomena in the region of 12.39 > r > 2.181, while the curve labeled Δ = 0.5 has a form similar to that of Δ = 0.0. The reentrant
Fig. 4. The phase diagram (tC versus r plot) in the nanoparticle A with h = 0.0, when the number ofΔ is changed.
phenomena of this case come from the competition of exchange interactions between JS and JR, which also originate from the finite size of the present system In order to clarify whether the predictions of the phase diagrams are correct or not, let us investigate the thermal variations of magnetizations by solving the coupled equations (5) and (6) numerically. In Fig. 5, they are given by selecting three typical values of r from the dashed curves (labeled h = 0.0) in Fig. 2 or the curve labeled Δ = 0.0 in Fig. 4. In Fig. 5(A), the thermal variations of mT are given. From them, we could not distinguish which curve represents the reentrant phenomena. In Fig. 5(B) and in Fig. 7(B), therefore, the thermal variations of m1 and m2 for the three cases are given. The reentrant phenomena are clearly found in the thermal variations of m1 and m2 for the system with r = 5.0. In Fig. 6, the thermal variations of mT in the system with Δ = 0.0 and h = 0.5 are plotted by selecting the three typical values of r from Fig. 3. The curve labeled r = 5.0, as expected, exhibits the reentrant phenomenon. The curves labeled r = 2.0 and r = 4.0 show the behavior a little different from those labeled r = 1.0 and r = 2.0 in Fig. 5(A). It comes from the different values of applied transverse field, namely 4
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
T. Kaneyoshi
Fig. 7. (A) The thermal variations of mT in the nanoparticle A with h = 0.0 and Δ = −0.5, when the three values of r are selected from Fig. 4. (B) The solid curves represent the thermal variations of magnetizations mT, m1 and m2 in the nanoparticle A with △ = −0.5, h = 0.0 and r = 2.0. The dashed curves show the thermal variations of mT, m1 and m2 in the nanoparticle A with Δ = 0.0, h = 0.0 and r = 2.0.
Fig. 5. The thermal variations of magnetizations in the nanoparticle A with h = 0.0 and Δ = 0.0, when the value of r is changed. (A) The variations of mT. (B) The variations of m1 and m2 for the two systems (thin solid curves for r = 1.0 and bold solid curves for r = 5.0).
h = 0.5 in Fig. 6 and h = 0.0 in Fig. 5(A); m1 > |m2| at t = 0.0 in Fig. 6 and m1 = |m2| at t = 0.0 in Fig. 5(A), In Fig. 7(A), the thermal variations of mT in the system with Δ = – 0.5 and h = 0.0 are plotted by selecting the three typical values of r from Fig. 4. The curve labeled r = 9.0, as expected, exhibits the reentrant phenomenon. At first sight, the mT curves labeled r = 2.0 and r = 6.0 show the behavior similar to those labeled r = 1.0 and r = 2.0 in Fig. 5(A). As shown in Fig. 7(B), the thermal variations of m1 and m2 in the present system with r = 2.0 exhibit the frustration at a low temperature, since the saturation magnetizations are depressed from m1 = |m2| = 1.0. 4. Nanoparticle B The phase diagram in the particle B can be obtained by linearizing the coupled equations (7). The relation is given by
( 1.0 − P9)[{ 1.0 − B P1 − 2.0 P3 P2 }{1.0 − B P1 − 2.0 (P1)2} − 4.0 P3(P1)2P2]
Fig. 6. The thermal variations of mT in the nanoparticle A with h = 0.5 and Δ = 0.0, when the three values of r are selected from Fig. 3.
− P8 ( 1.0 − B P1) [8.0 P3P1P2 + P2 { 1.0 − B P1 − 2.0 P3P2} + 4.0 P3 {1.0 − BP1 − 2.0(P1)2}] = 0.0
(12)
Just like Fig. 2, let us show the phase diagram (tC versus r plot) of 5
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
T. Kaneyoshi
Fig. 8. The phase diagram (tC versus r plot) in the nanoparticle B with Δ = 0.0, when the value of h is changed.
Fig. 10. The thermal variations of magnetizations m1, m2, m3, m4 and m5 in the nanoparticle B with h = 0.5 , Δ = 0.0 and r = 10.44.
Fig. 9. The variations of magnetizations in the nanoparticles B (solid curves) and A (dashed curves) with h = 0.0, Δ = 0.0 and t = 0.01, when the value of r is changed.
But, another type of reentrant phenomena exists in Fig. 9, which result (bold solid curve) is labeled as h = 0.5. In the curve, the three transition temperatures can be obtained in the region of 10.455 > r > 10.401. In Fig. 10, the thermal variations of magnetizations in the nanoparticle B with r = 10.44, Δ = 0.0 and h = 0.5 are plotted, in order to confirm whether the three transition temperatures are correctly realized or not. They clearly show the reentrant phenomena. In Fig. 11, the phase diagrams (tC versus h plot) of the nanoparticle B with three selected values of Δ are given by changing the value of r. In Fig. 11(A), the results of Δ = 0.0 are depicted. As discussed in Fig. 8 and Fig. 10, the curve labeled r = 10.44 exhibits the reentrant phenomena in the region of 0.5024 > h > 0.4923. The results of Fig. 10 have been obtained for the system with Δ = 0.0, h = 0.5 and r = 10.44. The region exhibiting the reentrant phenomenon in the curve labeled r = 11.0 becomes wider than that of the curve labeled r = 10.44, while such a behavior has not been obtained for the curve labeled r = 10.0. In Fig. 11(B), the results of Δ = −0.5 are depicted. Comparing with those depicted for Δ = 0.0, the reentrant phenomena can be more easily found in the system with Δ = −0.5. As shown in Fig. 11(C), on the other hand, it become more difficult to find such phenomena in the system with Δ = 0.5. Thus. the phenomena are closely related to the competition between the three values of h, r and Δ. In Fig. 12, the thermal variations of mT in the nanoparticle B with fixed values of Δ = 0.0 and h = 0.5 are plotted by selecting the six typical values of r from Fig. 8. In Fig. 12(A), the three curves are plotted; The curves labeled r = 4.0 and r = 5.0 exhibit a compensation point below their transition temperatures, which may be classified as the N-type in the nomenclature of ferrimagnetism [27]. The curve labeled r = 2.0 shows the behavior being similar to those labeled r = 2.0 and r = 4.0 in Fig. 6 for the nanoparticle A, which may be classified as the P-type in the nomenclature of ferrimagnetism [27]. In Fig. 12(B), the three curves are also plotted; The curve labeled r = 9.5 exhibits a compensation point below their transition temperatures. The curve labeled r = 10.44 represents the thermal variation of mT, where the thermal variations of m1, m2, m3, m4 and m5 in the system have already been given in Fig. 10. It clearly shows the reentrant phenomena. The dotted curve labeled r = 11.0 exhibits the standard behavior, which may be classified as the Q-type in the nomenclature of ferrimagnetism [28].
the nanoparticle B with Δ = 0.0, changing the value of h. The results are given in Fig. 8. As is seen from the results, the tC values at r = 0.0 take finite values, where the value of h = 0.0 is the largest (tC = 2.199) and with the increase of h it decreases and reduces to zero at h = 1.91 (see also the curve labeled r = 0.0 in Fig. 11(A). One should notice that the similar behavior has been normally obtained in the systems described by the transverse Ising model [23]). This fact is clearly different from the behavior of nanoparticle A. Furthermore, the result labeled h = 0.0 (dashed curve in Fig. 8) does not exhibit any possibility of two transition temperatures in contrast to the results of Fig. 2. The difference can be understood more clearly by plotting the behavior of saturation magnetizations (at t = 0.01) for the two nanoparticles with Δ = 0.0 and h = 0.0 as a function of r. The results are given as Fig. 9; The dashed curves (m1 and m2) represent the results of nanoparticle A, which reduce to zero at r = 4.37. In Fig. 2, one should also notice that the possibility of two transition temperatures has been found in the region of r > 4.37 (or for the dashed curve labeled h = 0.0). In Fig. 9, however, such a behavior has not been found for the nanoparticle B. When comparing with the results of Fig. 2 and those of Fig. 8, the behaviors of tC curves with h ≠ 0.0 are drastically different. One may think that the nanoparticle B does not exhibit the reentrant phenomena. 6
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
T. Kaneyoshi
Fig. 12. The thermal variations of mT in the nanoparticle B with h = 0.5 and △ = 0.0; (A) for the three values of r (r = 2.0, r = 4.0 and r = 5.0). (B) for the three values of r (r = 9.5, r = 10.44 and r = 11.0).
As shown in Figs. 2–4, 8 and 11, the reentrant phenomena have been observed in both systems, although the reentrant phenomena of nanoparticle B are clearly different from those of nanoparticle A. The similar reentrant phenomena observed for the nanoparticle A have also been found in other nanosized magnetic systems [7–15], while the reentrant phenomena obtained for the nanoparticle B have not been reported before. These phenomena also come from the finite size effect and the competition between exchange interactions JS, JR and an applied transverse field. As shown in Fig. 12(A) and (B), the ferrimagnetic behaviors, namely the existence of a compensation point, have been found in the nanoparticle B. Such ferrimagnetic behaviors have also been found in the nanoparticle B, even when the parameters JR, JS and H have been selected as other appropriate values different from those in Fig. 12, although such results have not been given in this work. Thus, we can have a possibility to find some ferrimagnetic behaviors, even in a nanosized nanoparticle with an antiferromagnetic spin configuration, when a transverse field is applied to the system. At last, as discussed in [27], one should notice that such a possibility extremely depends on the shapes of nanoparticles.
Fig. 11. The phase diagrams (tC versus h plot) in the nanoparticle B with the three typical values ofΔ, when the value of r is changed; (A) Δ = 0.0, (B) Δ = −0.5 and (C) Δ = 0.5.
5. Conclusion Declaration of Competing Interest
In this work, within the theoretical framework of the EFT, we have investigated the phase diagrams and magnetization curves of two antiferromagnetic Ising nanoparticles under an applied transverse field.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to 7
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
T. Kaneyoshi
influence the work reported in this paper.
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Research articles
Magnetism in two antiferromagnetic Ising nanoparticles under an applied transverse field
T
T. Kaneyoshi Nagoya University, 1-510, Kurosawadai, Midoriku, Nagoya 458-0003, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Phase diagrams Magnetizations Size effect Ferrimagnetic behavior Reentrant phenomena Nanoparticles
The phase diagrams and magnetization curves of two antiferromagnetic Ising nanoparticles (A and B) under an applied transverse field are investigated by using the effective-field theory with correlations. The nanoparticle B is a size larger than that of nanoparticle A. The effects of size on the magnetic properties are discussed. Some characteristic features, such as the reentrant phenomena, are found in these properties of both systems, when a finite transverse field h is applied. The nanoparticle B exhibits the ferrimagnetic behaviors, such as a compensation point.
1. Introduction Nowadays, it is well-known experimentally and theoretically that the magnetic properties of a magnetic material change from the bulk ones, when its size reduces to the nanoscale level. Consequently, the classifications valid to bulk magnetic materials, such as ferromagnetic, antiferromagnetic and ferrimagnetic material, could not be applied straightforwardly to a nanoscale magnetic material. In fact, some case exhibits a compensation point below a transition temperature in an antiferromagnetic nanoscale material [1,2]. The compensation point phenomena are normally characteristic to bulk materials exhibiting ferrimagnetism, which are composed from at least two different atoms with different spins. Other cases show surface induced ferromagnetism in antiferromagnetic nanoscale magnetic materials, such as defects at the surface [3] and uncompensated surfaces in thin films [4]. Furthermore, the graphene-like nanoparticles, nano-islands and bi-layer films have been fabricated by doping of magnetic atoms on graphene sheets. The simplest graphene-like nano-system is made of six spin-1/2 magnetic atoms which constitute a nanoparticle with hexagonal structure. They are arranged to be antiferromagnetic, while the particle is paramagnetic. Theoretically, when the simplest graphene-like nanoparticle is discussed by using the mean-field theory (MFA), the system shows a finite transition temperature (TC ≠ 0.0), although it exhibits TC = 0.0, when the effective-field theory with correlations (EFT) is applied [5]. The difference comes from the following fact: The EFT includes automatically some correlations between the central spin and the nearest-neighbor spins. The MFA is considering only one atom and replaces the interactions from other neighbor atoms with a uniform field proportional to the averaged magnetization. The EFT corresponds
to the Zernike approximation [6]. In a nanoscaled system, particularly, we must pay attention to the fact that the magnetization at each site is heavily dependent on the surrounding situation of it. In many nanosystems [7–15] described by the transverse Ising model, an interesting phenomenon, namely the reentrant phenomenon which comes from the finite size effect of the system, has been found by using the EFT, although it has not been obtained by using the MFA. Also, mixed spin Ising nanoparticles have been discussed by using the Monte Carlo simulation [16,17]. Many works have been reported experimentally, in order to clarify the magnetic properties in antiferromagnetic nanoparticles (or nanosized manganites) [18–20]. These works have been directed to clarify the surface induced ferromagnetism in them. As far as we know, the ferrimagnetic properties in antiferromagnetic nanoparticles have not been reported experimentally, while the evidence of ferrimagnetism has been reported for a ferromagnetic nanoparticle [21]. Within the framework of the EFT, some works have discussed theoretically the possibility of ferrimagnetism in some antiferromagnetic nanoparticles [5,7,22], while the compensation point phenomenon characteristic to ferrimagnetism has not been obtained in these works. In the present study, via the EFT, an attempt is made for the two new-type antiferromagnetic nanoparticles, whether they exhibit the typical ferrimagnetic behaviors, especially the compensation point, as well as the reentrant phenomena due to the finite size effect The particularly interesting aim is to clarify what phenomena may be realized in the magnetic properties, due to the difference of particle sizes. The outline of this work is as follows. In Section 2, the models and formulations of the two (A and B) antiferromagnetic nanoparticles are given within the EFT. The nanoparticle B is a size larger than that of
E-mail address: [email protected]. https://doi.org/10.1016/j.jmmm.2019.166368 Received 8 June 2019; Received in revised form 22 October 2019; Accepted 27 December 2019 Available online 21 January 2020 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
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Fig. 1. Schematic representation of two antiferromagnetic nanoparticles A and B. The black and white circles represent the spin-1/2 magnetic atoms and they are respectively directed to up and down spin directions. The large circle at the center represents a nonmagnetic atom. In the two nanoparticles, the white atoms are connected by an exchange interaction JR (JR ≧ 0.0), which originates from the super-exchange interaction via a nonmagnetic atom. The two exchange interactions (−JS and −J, J> 0.0) act between the nearest-neighbor sites.
2
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site i. For the white circle in the nanoparticle A, it is given by Ei = −Js (Si+1Z + Si−1Z) + JR SlZ, where the site l represents the white circle at the opposite side. For the nanoparticle A, the magnetizations m1 and m2 in Fig. 1A can be easily and exactly written as
nanoparticle A. From the difference, the effects of size on the magnetic properties can be compared. In Section 3, the phase diagrams and the temperature dependences of magnetizations are presented numerically for the nanoparticle A. It exhibits some characteristic behaviors, such as the reentrant phenomenon, even when H = 0.0. In Section 4, the phase diagrams and the temperature dependences of magnetizations are presented numerically for the nanoparticle B. The nanoparticle B exhibits the compensation point phenomenon as well as other characteristic phenomena observed in the bulk ferrimagnetism. The last section is devoted to conclusion.
and
m1 = < Si Z> = − 2m2 K1 + m1K2 + < SZi+1SZi−1SZl > K3
2. Models and formulation
im
mn
m1 = −2m2 K1 + m1K2 + (m2 )2m1K3
im
im
im
i
(1b)
K2 =
1 [F(2.0JS + JR) − F(2.0JS − JR) + 2.0F(]R)] 4
K3 =
1 [F(2.0JS + JR) − F(2.0JS − JR) − 2.0F(]R)] 4
m2 = −2.0m 4 P4 − 2.0m1P4 − 2.0(m1)2m 4 P5 − 2.0(m 4 )2m1P5 m1 = −2.0m2 P6 − m3P6 + m1P7 − (m2)2m3P8 + (m2)2m1P9
for the nanoparticle B, where Si (α = z or x) is the Pauli operator with SiZ = ± 1.0 at a site i. The last terms in HA and HB represent the effect of a magnetic field H applied perpendicularly to the spin direction z. The first and second terms in HA represent respectively the perimeterand super- exchange interactions. The last term represents the effect of a magnetic field H applied perpendicularly to the spin direction z. The first and third terms in HB represent respectively the perimeter- and super- exchange interactions. The second term in HB shows the contribution from the interior exchange interaction. The exchange interaction JS is given by
+ 2.0m1m2 m3P9,
(7)
where coefficients are given by B = F (2.0 JS) P1 = 1 [F (2.0 JS + J) + F(2.0 JS − J) ] 4
P2 = 1 [F (2.0 JS + J) − F(2.0 JS − J) + 2.0F (]) ] 4
P3 = 1 [F (2.0 JS + J) − F(2.0 JS − J) − 2.0F (]) ] 4
P4 = 1 [F (4.0 J) + 2.0F (2.0 J) ]
(2)
P5 =
8 1 8 1 8
[F (4.0 J) − 2.0F (2.0 ]) ]
P6 = [F (3.0 JS + JR) + F(3.0 JS − JR) + F (J + ]R) + F(J − JR) ] P7 = [F (3.0 JS + JR) − F(3.0 JS − JR) + 3.0 {F (J + ]R) − F (J − JR)} ] P8 = [F (3.0 JS + JR) + F(3.0 JS − JR) − 3.0 {F (J + ]R) + F (J − JR)} ] P9 = 1 [F (3.0 JS + JR) − F(3.0 JS − JR) − F (J + ]R) + F(J − JR) ]
in order to clarify the effect of perimeter on the magnetic properties. The present systems are a quantum system, when H ≠ 0.0, so that they could not be solved exactly. The starting equation for the EFT is the approximated relation derived in [23]. Here, one should notice that the approximated relation exactly reduces to the Callen identity [24], when H = 0.0. Applying the differential operator technique [25,26] to the relation, the longitudinal magnetization at a site i is given by
< SiZ > = < exp(Ei∇) > F(x)|x = 0
1 [F(2.0JS + JR) + F(2.0JS − JR)] 4
m5 = −m 4 B m 4 = −m5P1 − m3P1 − m2 P2 − m5m3m2 P3 m3 = −2.0m 4 P1 − m1P2−(m 4) 2m1P3
α
JS = J(1.0 + Δ)
K1 =
For the nanoparticle B, following to the same procedure as that of nanoparticle A, within the EFT, the magnetizations m1, m2, m3, m4 and m5 are given by
for the nanoparticle A and
HB = JS ∑ SZi SZm+J ∑ SZi SZm − JR ∑ SZi SZm+H ∑ SiX
(6)
The coefficients B and Kn (n = 1–3) in (5) and (6) are given by B = F (2.0 JS)
(1a)
l
(6.a)
Here, the EFT is nothing but the fact that the correlation function < Si+1Z Si-1Z SlZ > is decoupled to < Si Z Z Z > < S > < S > . Within the EFT, equation (6.a) reduces to i−1 l +1
We study two antiferromagnetic two-dimensional Ising nanoparticles, as depicted in Fig. 1. The black and white circles in the two (A and B) nanoparticles represent the magnetic atoms with spin- 1/2. They are respectively directed to opposite directions, since they are coupled by a negative exchange interaction − JS and – J (J > 0.0), where JS is the exchange interaction at the perimeter and J is the exchange interaction in the inside of nanoparticle B. In the nanoparticle A, the white atoms are also connected by an exchange interaction JR (JR ≧ 0.0), which originates from the super-exchange interaction via a nonmagnetic atom (gray atom in the center). The size of nanoparticle B is a size larger than that of nanoparticle A, where a nonmagnetic atom also exists in the center and there exists a super-exchange interaction JR between the white atoms. The Hamiltonians of two nanoparticles are described by Z HA = JS ∑ Si ZSm − JR ∑ SZm SZn − H∑ SlX
(5)
m2 = −m1B
8
Let us here define the total magnetization mT per site for these two nanoparticles A and B; For the nanoparticle A, it is given by
(3)
where ∇ = ∂ /∂ x is the differential operator. The function F(x) is defined by
mT =
x F(x) = tanh(β e(x)) e (x )
and for the nanoparticle B, it can be represented as
(4)
mT =
with
e(x) = [x2 + H2]1/2 ,
1 [m1 + m2] 2
1 [m1 + m2 + m3 + 2.0m 4 + m5] 6
(8)
(9)
In the following sections, the phase diagram and the thermal variations of magnetizations in the nanoparticles A and B are examined. For the aim, let us here introduce the reduced parameters as follows;
where β = 1/kBT. Ei in (3) represents the local field at the site i, such as Ei = −Js (Si+1Z + Si−1Z) for the black circle at a corner in the nanoparticle A, where i + 1 and i − 1 denote the nearest-neighbor spins at a
t= kB. T/J, r= JR /J and h= H/J 3
(10)
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Fig. 3. The solid line shows the variation of tC as a function of r in the nanoparticle A with Δ = 0.0 and h = 0.5. The dashed lines represent the variations of magnetizations m1 and m2 in the nanoparticle A with Δ = 0.0, h = 0.5 and t = 0.01, when the value of r is changed.
Fig. 2. The phase diagram (tC versus r plot) in the nanoparticle A with Δ = 0.0, when the number of r is changed.
3. Nanoparticle A At first, it is important to note the following facts; When JR = 0.0, the coefficients K2 and K3 in (6,a) or (6) reduce to K2 = K3 = 0.0. Accordingly, the coupled Eqs. (5) and (6) can be exactly represented by m1 m2 [1.0 − B2 ] = 0.0, which solution indicates that the nanoparticle A with JR = 0.0 must be paramagnetic (TC = 0.0 or m1 = m2 = 0.0 for T > 0.0), just like some cases discussed for other nanoparticles [5]. When JR ≠ 0.0, the transition temperature TC (or phase diagram) can be obtained from
2.0 B K1 + K2 = 1.0
(11)
Let us examine the phase diagram (tC versus r plot) of the nanoparticle A with Δ = 0.0, changing the value of h. The results are plotted in Fig. 2. As noted above, the tC curves labeled h = 0.0 (dashed line) and h = 0.1 rapidly reduce to zero, when the value of r approaches to zero. But, the curves labeled as h ≥ 0.5 exhibit the region of tC = 0.0 in the vicinity of r = 0.0, when the value of r decreases to the value of r = 0.0. Furthermore, they exhibit the reentrant phenomena for the large values of r. In order to clarify these facts, the tC curve and the variations of saturation magnetizations (m1 and m2 at t = 0.01) for the nanoparticle A with fixed values of Δ = 0.0 and h = 0.5 are given in Fig. 3, changing the value of r. In Fig. 3, the region of tC = 0.0 in the vicinity of r = 0.0 can be found in the region of 0.06 > r ≥ 0.0 and the reentrant phenomena can be found in the region of 5.989 > r > 4.185. Both phenomena come from the competition between the strength of r and the value of an applied transverse field; The applied transverse field tends to destroy of order in the z-direction and the value of r tends to the strength of order in the z-direction. The phenomena also come from the following facts: One is that the system is paramagnetic, when r = 0.0. The other is that the system is the finite size. Accordingly, similar phenomena have also been found in other nanosized magnetic systems [5,7-15]. In Fig. 3, the saturation magnetizations m1 and m2 at t = 0.01 are also plotted and one should notice that they take finite values in the same region (4.185 > r > 0.05) as that of tC. In Fig. 4, the tC curves in the system with h = 0.0 are plotted as a function of r , selecting three values of Δ. The curve labeled Δ = 0.0 is the same as that (dashed curve labeled h = 0.0) in Fig. 2, where the two transition temperatures exist in the region of Δ > 4.362. As is seen from the results, the curve labeled Δ = – 0.5 exhibits the reentrant phenomena in the region of 12.39 > r > 2.181, while the curve labeled Δ = 0.5 has a form similar to that of Δ = 0.0. The reentrant
Fig. 4. The phase diagram (tC versus r plot) in the nanoparticle A with h = 0.0, when the number ofΔ is changed.
phenomena of this case come from the competition of exchange interactions between JS and JR, which also originate from the finite size of the present system In order to clarify whether the predictions of the phase diagrams are correct or not, let us investigate the thermal variations of magnetizations by solving the coupled equations (5) and (6) numerically. In Fig. 5, they are given by selecting three typical values of r from the dashed curves (labeled h = 0.0) in Fig. 2 or the curve labeled Δ = 0.0 in Fig. 4. In Fig. 5(A), the thermal variations of mT are given. From them, we could not distinguish which curve represents the reentrant phenomena. In Fig. 5(B) and in Fig. 7(B), therefore, the thermal variations of m1 and m2 for the three cases are given. The reentrant phenomena are clearly found in the thermal variations of m1 and m2 for the system with r = 5.0. In Fig. 6, the thermal variations of mT in the system with Δ = 0.0 and h = 0.5 are plotted by selecting the three typical values of r from Fig. 3. The curve labeled r = 5.0, as expected, exhibits the reentrant phenomenon. The curves labeled r = 2.0 and r = 4.0 show the behavior a little different from those labeled r = 1.0 and r = 2.0 in Fig. 5(A). It comes from the different values of applied transverse field, namely 4
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
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Fig. 7. (A) The thermal variations of mT in the nanoparticle A with h = 0.0 and Δ = −0.5, when the three values of r are selected from Fig. 4. (B) The solid curves represent the thermal variations of magnetizations mT, m1 and m2 in the nanoparticle A with △ = −0.5, h = 0.0 and r = 2.0. The dashed curves show the thermal variations of mT, m1 and m2 in the nanoparticle A with Δ = 0.0, h = 0.0 and r = 2.0.
Fig. 5. The thermal variations of magnetizations in the nanoparticle A with h = 0.0 and Δ = 0.0, when the value of r is changed. (A) The variations of mT. (B) The variations of m1 and m2 for the two systems (thin solid curves for r = 1.0 and bold solid curves for r = 5.0).
h = 0.5 in Fig. 6 and h = 0.0 in Fig. 5(A); m1 > |m2| at t = 0.0 in Fig. 6 and m1 = |m2| at t = 0.0 in Fig. 5(A), In Fig. 7(A), the thermal variations of mT in the system with Δ = – 0.5 and h = 0.0 are plotted by selecting the three typical values of r from Fig. 4. The curve labeled r = 9.0, as expected, exhibits the reentrant phenomenon. At first sight, the mT curves labeled r = 2.0 and r = 6.0 show the behavior similar to those labeled r = 1.0 and r = 2.0 in Fig. 5(A). As shown in Fig. 7(B), the thermal variations of m1 and m2 in the present system with r = 2.0 exhibit the frustration at a low temperature, since the saturation magnetizations are depressed from m1 = |m2| = 1.0. 4. Nanoparticle B The phase diagram in the particle B can be obtained by linearizing the coupled equations (7). The relation is given by
( 1.0 − P9)[{ 1.0 − B P1 − 2.0 P3 P2 }{1.0 − B P1 − 2.0 (P1)2} − 4.0 P3(P1)2P2]
Fig. 6. The thermal variations of mT in the nanoparticle A with h = 0.5 and Δ = 0.0, when the three values of r are selected from Fig. 3.
− P8 ( 1.0 − B P1) [8.0 P3P1P2 + P2 { 1.0 − B P1 − 2.0 P3P2} + 4.0 P3 {1.0 − BP1 − 2.0(P1)2}] = 0.0
(12)
Just like Fig. 2, let us show the phase diagram (tC versus r plot) of 5
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
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Fig. 8. The phase diagram (tC versus r plot) in the nanoparticle B with Δ = 0.0, when the value of h is changed.
Fig. 10. The thermal variations of magnetizations m1, m2, m3, m4 and m5 in the nanoparticle B with h = 0.5 , Δ = 0.0 and r = 10.44.
Fig. 9. The variations of magnetizations in the nanoparticles B (solid curves) and A (dashed curves) with h = 0.0, Δ = 0.0 and t = 0.01, when the value of r is changed.
But, another type of reentrant phenomena exists in Fig. 9, which result (bold solid curve) is labeled as h = 0.5. In the curve, the three transition temperatures can be obtained in the region of 10.455 > r > 10.401. In Fig. 10, the thermal variations of magnetizations in the nanoparticle B with r = 10.44, Δ = 0.0 and h = 0.5 are plotted, in order to confirm whether the three transition temperatures are correctly realized or not. They clearly show the reentrant phenomena. In Fig. 11, the phase diagrams (tC versus h plot) of the nanoparticle B with three selected values of Δ are given by changing the value of r. In Fig. 11(A), the results of Δ = 0.0 are depicted. As discussed in Fig. 8 and Fig. 10, the curve labeled r = 10.44 exhibits the reentrant phenomena in the region of 0.5024 > h > 0.4923. The results of Fig. 10 have been obtained for the system with Δ = 0.0, h = 0.5 and r = 10.44. The region exhibiting the reentrant phenomenon in the curve labeled r = 11.0 becomes wider than that of the curve labeled r = 10.44, while such a behavior has not been obtained for the curve labeled r = 10.0. In Fig. 11(B), the results of Δ = −0.5 are depicted. Comparing with those depicted for Δ = 0.0, the reentrant phenomena can be more easily found in the system with Δ = −0.5. As shown in Fig. 11(C), on the other hand, it become more difficult to find such phenomena in the system with Δ = 0.5. Thus. the phenomena are closely related to the competition between the three values of h, r and Δ. In Fig. 12, the thermal variations of mT in the nanoparticle B with fixed values of Δ = 0.0 and h = 0.5 are plotted by selecting the six typical values of r from Fig. 8. In Fig. 12(A), the three curves are plotted; The curves labeled r = 4.0 and r = 5.0 exhibit a compensation point below their transition temperatures, which may be classified as the N-type in the nomenclature of ferrimagnetism [27]. The curve labeled r = 2.0 shows the behavior being similar to those labeled r = 2.0 and r = 4.0 in Fig. 6 for the nanoparticle A, which may be classified as the P-type in the nomenclature of ferrimagnetism [27]. In Fig. 12(B), the three curves are also plotted; The curve labeled r = 9.5 exhibits a compensation point below their transition temperatures. The curve labeled r = 10.44 represents the thermal variation of mT, where the thermal variations of m1, m2, m3, m4 and m5 in the system have already been given in Fig. 10. It clearly shows the reentrant phenomena. The dotted curve labeled r = 11.0 exhibits the standard behavior, which may be classified as the Q-type in the nomenclature of ferrimagnetism [28].
the nanoparticle B with Δ = 0.0, changing the value of h. The results are given in Fig. 8. As is seen from the results, the tC values at r = 0.0 take finite values, where the value of h = 0.0 is the largest (tC = 2.199) and with the increase of h it decreases and reduces to zero at h = 1.91 (see also the curve labeled r = 0.0 in Fig. 11(A). One should notice that the similar behavior has been normally obtained in the systems described by the transverse Ising model [23]). This fact is clearly different from the behavior of nanoparticle A. Furthermore, the result labeled h = 0.0 (dashed curve in Fig. 8) does not exhibit any possibility of two transition temperatures in contrast to the results of Fig. 2. The difference can be understood more clearly by plotting the behavior of saturation magnetizations (at t = 0.01) for the two nanoparticles with Δ = 0.0 and h = 0.0 as a function of r. The results are given as Fig. 9; The dashed curves (m1 and m2) represent the results of nanoparticle A, which reduce to zero at r = 4.37. In Fig. 2, one should also notice that the possibility of two transition temperatures has been found in the region of r > 4.37 (or for the dashed curve labeled h = 0.0). In Fig. 9, however, such a behavior has not been found for the nanoparticle B. When comparing with the results of Fig. 2 and those of Fig. 8, the behaviors of tC curves with h ≠ 0.0 are drastically different. One may think that the nanoparticle B does not exhibit the reentrant phenomena. 6
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
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Fig. 12. The thermal variations of mT in the nanoparticle B with h = 0.5 and △ = 0.0; (A) for the three values of r (r = 2.0, r = 4.0 and r = 5.0). (B) for the three values of r (r = 9.5, r = 10.44 and r = 11.0).
As shown in Figs. 2–4, 8 and 11, the reentrant phenomena have been observed in both systems, although the reentrant phenomena of nanoparticle B are clearly different from those of nanoparticle A. The similar reentrant phenomena observed for the nanoparticle A have also been found in other nanosized magnetic systems [7–15], while the reentrant phenomena obtained for the nanoparticle B have not been reported before. These phenomena also come from the finite size effect and the competition between exchange interactions JS, JR and an applied transverse field. As shown in Fig. 12(A) and (B), the ferrimagnetic behaviors, namely the existence of a compensation point, have been found in the nanoparticle B. Such ferrimagnetic behaviors have also been found in the nanoparticle B, even when the parameters JR, JS and H have been selected as other appropriate values different from those in Fig. 12, although such results have not been given in this work. Thus, we can have a possibility to find some ferrimagnetic behaviors, even in a nanosized nanoparticle with an antiferromagnetic spin configuration, when a transverse field is applied to the system. At last, as discussed in [27], one should notice that such a possibility extremely depends on the shapes of nanoparticles.
Fig. 11. The phase diagrams (tC versus h plot) in the nanoparticle B with the three typical values ofΔ, when the value of r is changed; (A) Δ = 0.0, (B) Δ = −0.5 and (C) Δ = 0.5.
5. Conclusion Declaration of Competing Interest
In this work, within the theoretical framework of the EFT, we have investigated the phase diagrams and magnetization curves of two antiferromagnetic Ising nanoparticles under an applied transverse field.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to 7
Journal of Magnetism and Magnetic Materials 502 (2020) 166368
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influence the work reported in this paper.
[13] [14] [15] [16]
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