nanomaterials Article
Effects of a Transverse Field in Two Mixed-Spin Ising Bilayer Films Takahito Kaneyoshi Graduate School of Information and Science, Nagoya University, 1-510, Kurosawadai, Midoriku, Nagoya 458-0003, Japan; [email protected]; Tel.: +81-528-766-607 Received: 16 June 2017; Accepted: 23 August 2017; Published: 4 September 2017
Abstract: The magnetic properties (phase diagrams and magnetizations) of two mixed-spin Ising bilayer films with a transverse field are investigated by the use of the effective field theory with correlations. The systems consist of two magnetic atoms where spin-1/2 atoms are directed to the z-direction and only spin-1 atoms are canted from the z-direction by applying a transverse field. We examined how magnetization sign reversal can be realized in the system, due to the effects of the transverse field on the spin-1 atoms. The compensation point phenomena are found in both systems, depending on the selections of physical parameters. However, the reentrant phenomena are found only for one of the two systems. Keywords: phase diagrams; magnetizations; compensation point; reentrant phenomena; mixed-spin Ising bilayer films
1. Introduction The theoretical research of various mixed-spin Ising (or transverse Ising) systems on honeycomb lattice has a long history and it is still a subject of active research. They have been examined by using various theoretical frameworks of mean-field theory (MFT), effective-field theory (EFT), the Green function method and the Monte Carlo simulation (MC). The studies of these systems are related to the experimental researches of molecular-based magnetic materials and nano-graphene films. In particular, a kind of molecular-based ferrimagnets, namely AFeII FeIII (C2 O4 )3 (A = N(n − Cn H2n+1 )4 , n = 3–5), have been the subject under extensive theoretical discussions of mixed-spin Ising (or transverse Ising) systems, where spins of FeII and FeIII atoms are respectively taken as 2 and 5/2 and they are coupled by a negative exchange interaction in each layer and two positive interactions between layers (see the references in the recent theoretical works [1–4]). For nano-graphene films, on the other hand, the magnetic properties of the ferromagnetic and antiferromagnetic two (or three) layers of honeycomb lattices described by the spin-1/2 Ising (or transverse Ising) models have been recently investigated by the use of the EFT [5–7]. The EFT [8,9] corresponds to the Zernike approximation [10] and it is believed to give more exact results than those of the MFT, since it automatically includes some correlations between a certain spin and the near-neighbor spins. Furthermore, the results obtained from the EFT have also the same topology as those obtained from the MC. In fact, the magnetic properties (magnetization, internal energy and so on) of a system are the same for the EFT and the MC, while the results obtained from the MC are smaller than those obtained from the EFT (for instance, see the works [11–13]). The aim of this work is to present another aspect of theoretical explanation for the appearance of negative magnetization in some molecular-based ferrimagnetic materials [14,15] on the basis of two layers of honeycomb lattices described by the mixed-spin (spin-1/2 and spin-1) Ising model, where the spin direction of spin-1/2 atoms is directed to the z-(or easy) direction and the spin direction of spin-1 atoms is canted from the z-direction, due to the presence of a transverse field. In [14],
Nanomaterials 2017, 7, 256; doi:10.3390/nano7090256
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for instance, A in the above-described molecular-based ferrimagnets has been replaced by A = (XR4 ) with X = N, P and R = n-propyl, n-butyl, phenyl, instead of A = N(n − Cn H2n+1 )4 , (n = 3–5). In these systems, only the system with A = NBun 4 has exhibited a negative magnetization at low-temperature. In this work, the two model systems are examined by the use of the EFT [16], in order to clarify whether such a negative magnetization can be realized within the present two model systems. In [16], the magnetizations of a mixed-spin system consisting of spin-1/2 and spin-1 atoms with different transverse fields have been examined. The temperature dependences of total magnetization in the system have shown typical ferrimagnetic behaviors usually observed in crystalline ferrimagnetic alloys Ap B1-p , due to a strong negative A-B exchange interaction, where A and B are different magnetic atoms with different spins. The physical reason for this comes from the different spin canting of spin-1/2 and spin-1 atoms. In Section 2, the two systems are proposed. At first sight, they are very similar. Even within the theoretical formulations of the EFT, the basic equations obtaining the magnetic properties (phase diagram and magnetizations) are rather different between the two systems (system A and system B). In Section 3, the magnetic properties of system A are discussed by solving the formulations numerically. In Section 4, the magnetic properties of system B are obtained numerically. The last section is devoted to the conclusion. 2. Models and Formulation We consider the two systems consisting of two layers, as depicted in Figure 1. The structure of each system is the honeycomb lattice. The white and black circles on each figure represent respectively the magnetic atoms with spin-1/2 and the magnetic atoms with spin-1 and transverse field Ω. The spins (white and black circles) on each layer are coupled by a nearest-neighbor exchange interaction-J (J > 0.0). In Figure 1A, each Ising spin on the upper layer is coupled to the corresponding same spin on the lower layer with an exchange interaction-JR (JR > 0.0). In Figure 1B, on the other hand, two exchange interactions (JA > 0.0 and JB > 0.0) exist between the two layers, depending on where the pair is located. JA is the interlayer coupling between the spin-1 atoms and JB is the interlayer coupling between the spin-1/2 atoms. The Hamiltonian of Figure 1A (or system A) is given by H = J Σ σi Z S j Z + JR Σ σi Z S j Z − Ω Σ S j X (ij)
(1a)
j
(ij)
where σi Z represents the spin-1/2 operator with σi Z = ±1/2 and Sj α (α = Z and X) is the spin-1 operator with Sj Z = ±1 and 0. The first and the second terms in (1a) represent the contributions from the intra-layer and inter-layer interactions. The last term shows the effect of a transverse field at each spin-1 atom. On the other hand, the Hamiltonian of Figure 1B (or system B) can be represented by H = J Σ σi Z S j Z − J A Σ σk Z Sl Z − JB Σ σm Z Sn Z − Ω Σ S j X (ij)
(kl )
(mn)
j
(1b)
where the second and third terms show the contributions from the inter-layer interactions. The total longitudinal magnetization (mT = mT Z ) per site in each system (A or B) is defined as mT = [mA + mB ]/2.0
(2)
where mA = mA Z = and mB = mB Z = . Within the EFT [8,9,16], the mA and mB in the system A (Figure 1A) are given by mA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [cosh(R/2.0) − 2.0 mB sinh(R/2.0)] F(x)|x=0
(3)
qA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [cosh(R/2.0) − 2.0 mB sinh(R/2.0)] G(x)|x=−0
(4)
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and mB = [qA {cosh(C) − 1.0} + 1.0 − mA sinh(C)]3 [qA {cosh(R) − 1.0} + 1.0 − mA sinh(R)] f (x)|x=0
(5)
where qA = < (Sj Z )2 >, C = J·D and R = JR ·D. D = ∂/∂x is the differential operator. Here, the functions F(x), G(x) and f (x) are defined by F(x) = 2.0 x sinh(e(x)β)/[e(x) {1.0 + 2.0 cosh(βe(x))}]
(6)
G (x) = [Ω2 + {Ω2 + 2.0 x2 } cosh(βe(x))]/[e(x)2 {1.0 + 2.0 cosh(βe(x))}]
(7)
f (x) = tanh(βx/2.0)/2.0
(8)
e(x) = [Ω2 + x2 ]1/2
(9)
with where β = 1.0/kB T. For the system B in Figure 1B, the mA and mB are given by mA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [qA {cosh(A) − 1.0} + 1.0 + mA sinh(A)] F(x)|x=0 qA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [qA {cosh(A) − 1.0} + 1.0 + mA sinh(A)] G(x)|x=−0
(10)
mB = [qA {cosh (C) − 1.0} + 1.0 − mA sinh(C)]3 [cosh(B/2.0) + 2.0 mB sinh(B/2.0)] f (x)|x=0
(11)
and
where A = JA ·D and B = JB ·D. The phase diagrams (or transition temperature) in the two systems can be determined by expanding the coupled equations of mA and mB in each system (A or B) linearly. The transition temperature of system A can be obtained from the relation [3.0 K1 + K2 ] [6.0 K3 + 2.0 K4 ] − 1.0 = 0.0
(12)
where the coefficients Kn (n = 1–4) are given by K1 = sinh(C) [qA {cosh(C) − 1.0} + 1.0]2 [qA {sinh(R) − 1.0} + 1.0] f (x)|x=0 K2 = sinh(R) [qA {cosh(C) − 1.0} + 1.0]3 f (x)|x=0 K3 = cosh2 (C/2.0) sinh(C/2.0) cosh(R/2.0) F(x)|x=0 K4 = cosh3 (C/2.0) sinh(R/2.0) F(x)|x=0
(13)
with qA = cosh3 (C/2.0) cosh(R/2.0) G(x)|x=0 . The transition temperature of system B can be obtained from the relation [L1 − 1.0] [2.0 L2 − 1.0] − 18.0 L3 L4 = 0.0
(14)
The coefficients Ln (n = 1–4) in (14) are given by L1 = cosh3 (C/2.0) sinh(A) F(x)|x=0 L2 = sinh(B/2.0) [qA {cosh(C) − 1.0} + 1.0]3 f (x)|x=0 L3 = cosh(C/2.0) sinh(C/2.0) [qA {cosh(A) − 1.0} + 1.0] F(x)|x=0 L4 = sinh(C) cosh(B/2.0) [qA {cosh(C) − 1.0} + 1.0]2 f (x)|x=0
(15)
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with qA = D1 /(1.0 + D1 − D2 )
(16)
where D1 and D2 are defined as D1 = cosh3 (C/2.0) G(x)|x=0 4 (17) of 12 D2 = cosh3 (C/2.0) cosh(A) G(x)|x=0 At this point, it is important to note that only spin-1 magnetic atoms in the two systems are At it is important to note thatthey onlyare spin-1 magnetic atoms in the twowhen systems are affected affectedthis by point, a transverse field Ω and hence canted from the z-direction, Ω takes a finite by a transverse field Ω and hence they are canted from the z-direction, when Ω takes a finite value. value. However, spin-1/2 atoms in them are always directed to the z-direction. At first sight, the two However, spin-1/2 atoms in them are always directed to the z-direction. At first sight, the two systems systems given in Figure 1 seem to be physically equivalent. As discussed above, however, the given in Figure 1 seem to of be the physically equivalent. As discussed above, the theoretical theoretical formulations two systems are rather different evenhowever, within the theoretical formulations of the two systems are rather different even within the theoretical formulation of the formulation of the EFT. Under these conditions, it may be important to know what phenomena can EFT. Under these conditions, it may be important to know what phenomena can be obtained in the be obtained in the two systems. Furthermore, it is also important to know whether the phenomena two systems. Furthermore, is also important to know whether phenomena are similarofbetween are similar between the twoitsystems or some differences may bethe found in the phenomena the two the two systems or some differences may be found in the phenomena of the two systems. As far systems. As far as we know, these problems have not been discussed. In the following, the magnetic as we know, thesediagram problems have not been discussed. In the following, the magnetic properties properties (phase and thermal variation of magnetizations) in system A (or Figure 1A) are (phase diagram and thermal variation of magnetizations) in system A (or Figure 1A) are examined in examined in Section 3. In Section 4, such magnetic properties of system B (or Figure 1B) are discussed. Section 3. In Section 4, such magnetic properties of system B (or Figure 1B) are discussed. Nanomaterials 2017, 7, 256
(A)
(B)
Figure 1. 1. Schematic spin Ising ferrimagnetic models on two (upper and Figure Schematicrepresentations representationsofoftwo twomixed mixed spin Ising ferrimagnetic models on two (upper lower) layered honeycomb lattices with spin-1/2 (white circle) and spin-1 (black circle) atoms. (A) and lower) layered honeycomb lattices with spin-1/2 (white circle) and spin-1 (black circle) atoms. Each Ising spin on upper layer is coupled to the corresponding same spin on the lower layer with an (A) Each Ising spin on upper layer is coupled to the corresponding same spin on the lower layer with exchange interaction-J R (JR > 0.0). (B) Two exchange interactions (JA > 0.0 and JB > 0.0) exist between an exchange interaction-J R (JR > 0.0). (B) Two exchange interactions (JA > 0.0 and JB > 0.0) exist between theinterlayer interlayercoupling couplingbetween betweenthe thespin-1 spinthe two layers, depending onwhere wherethe thepair pairisislocated. located.J JAisisthe the two layers, depending on A B is the interlayer coupling between the spin-1/2 atoms. 1 atoms and J atoms and J is the interlayer coupling between the spin-1/2 atoms. B
3. The Magnetic Properties of System A 3. The Magnetic Properties of System A At first, let us define the parameters, t, h and r as At first, let us define the parameters, t, h and r as t = kBT/J, h = Ω/J and r = JR/J (18) t = kB T/J, h = Ω/J and r = JR /J (18) In Figure 2, the phase diagram (TC versus h plot) in the system A is given by changing the value In Figure (TC versus h plot) lines in therepresent system Athe is given byof changing of r from r = 0,2,0the to phase r = 1.5.diagram In the figure, the dashed results the tC (tCthe = kvalue BTC/J) of r from r = 0,when 0 to r the = 1.5. In theAfigure, the dashed represent the results of the tC (tC = kB TC /J) versus h plot, system is described by thelines following Hamiltonian, versus h plot, when the system A is described by the following Hamiltonian,
(
H = J Σ σ iZ S jZ + JR Σ σ i Z S j Z − Ω ΣI σ i X + Σj S j X ( ij ) ( ij ) H = J Σ σi Z S j Z + JR Σ σi Z S j Z − Ω (ij)
(ij)
Σσi X + Σ S j X I
j
)
namely when both spin-1/2 and spin-1 atoms have the same transverse field Ω, instead of the present system with both the Hamiltonian this have stage,theone should noticefield thatΩthe different namely when spin-1/2 and (1,a). spin-1At atoms same transverse , instead of thebehavior present between thethe solid curve and(1,a). the At dashed curveone in should the system value of r makes an system with Hamiltonian this stage, noticewith thatthe the same different behavior between important contribution the appearance of negative magnetization at a low the solid curve and the to dashed curve in the system with the same value of temperature. r makes an important contribution to the appearance of negative magnetization at a low temperature.
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Figure 2. 2. The plot) for for the thesystem systemA, A,when whenthe thevalue value changed from Figure Thephase phasediagram diagram(t(tCC versus versus h plot) ofof r isr is changed from phase diagram (tC versus h plot) for the system A, when the value of r is changed from r =rFigure 0.0 totor2.= = 0.0 rThe =1.5. 1.5. r = 0.0 to r = 1.5.
Figure3,3,the thetemperature temperaturedependences dependences of magnetizations with fixed values of of InIn Figure of magnetizationsin inthe thesystem system with fixed values Figure 3, the dependences ofBmagnetizations the system with fixed values of r = 0.5Inand h = 0.0 are temperature plotted, where mT, mA and m are respectivelyin represented by solid, dashed and r =r 0.5 and h = 0.0 are plotted, where m , m and m are respectively represented by solid, dashed and T A B = 0.5 and h = 0.0 plotted, where mof T, mA and mB are respectively represented by solid, dashed and dotted curves. Theare thermal variation qA is also plotted in Figure 3 by the dot-dashed curve. As is dotted curves. The thermal variation of q is also plotted in Figure 3 by the dot-dashed curve. As dotted curves. The thermal of qAA is also plotted in Figure 3 by the dot-dashed curve.region As is is seen from the figure, mA andvariation mB take respectively a positive and negative value in the whole seen from the figure, m and m take respectively a positive and negative value in the whole region A A and mBB take respectively a positive and negative value in the whole region seen from the m of T below its figure, TC. Here, one should notice that the value of r = 0.5 is particularly selected, since the of of T below itsits TCT.CHere, one should notice the value value of of rr ==0.5 0.5isisparticularly particularlyselected, selected, since T below .the Here, should notice that since thethe distance between twoone layers are supposed to the be larger than the lattice interval in each layer [14,15] distance between the two layers are supposed to be larger than the lattice interval in each layer [14,15] distance between the two layers are supposed to larger than the lattice interval in each layer [14,15] and hence the exchange interaction J R is assumed to be weaker than that of intra-layer interaction J. and hence the exchange assumed to tobe beweaker weakerthan thanthat thatofofintra-layer intra-layer interaction and hence the exchangeinteraction interactionJJRR is assumed interaction J. J.
Figure 3. The temperature dependences of mT (solid curve), mA (dashed curve), mB (dotted curve) and The temperature dependences T (solid curve), mA (dashed curve), mB (dotted curve) and qFigure A (dot-dashed curve) in the system A with fixed value 0.5, when the value of h is given byand h = qA Figure 3. 3. The temperature dependences ofofmm curve),ofmr A= (dashed curve), mB (dotted curve) Ta (solid q A (dot-dashed curve) in the system A with a fixed value of r = 0.5, when the value of h is given by h = 0.0. (dot-dashed curve) in the system A with a fixed value of r = 0.5, when the value of h is given by h = 0.0. 0.0.
In Figure 4A,B, the thermal variations of mT in the system with r = 0.5 are shown by changing In In Figure 4A,B, thermal variations of m in the the system systemwith withr r==0.5 0.5 areshown shown changing Figure 4A,B,hthe the variations of in mTTFigure in byby changing the value of h from = 1.0thermal to h = 3.3. As shown 4A, the behavior of mare T may change from the thethe value of h from h = 1.0 to h = 3.3. As shown in Figure 4A, the behavior of m may change from value from h with = 1.0 the increase = 3.3. Asofshown in Figure behavior ofhm=T 3.1 may T change from the Q-type to of theh P-type h. In Figure 4B, 4A, the the curves labeled and h = 3.2 exhibit theQ-type Q-type to the P-type with the increase of h. In Figure 4B, the curves labeled h = 3.1 and h = 3.2 to the P-type thetheir increase of h. In Figure 4B, the labeled h = 3.1 and = 3.2labeled exhibit a compensation pointwith below transition temperatures (orcurves the N-type), although the hcurve exhibit compensation point below their transition temperatures (or thein N-type), although thebeen curve point below theirThe transition temperatures (orand the N-types N-type), although the curvehas labeled ha =compensation 3.3atakes the Q-type behavior. nomenclature of Q-, Pferrimagnetism h = 3.3 takes the Q-type behavior. The nomenclature of Q-, Pand N-types in ferrimagnetism has been labeled h = 3.3 takes the Q-type behavior. The nomenclature of Q-, Pand N-types in ferrimagnetism used (see [17]). In Figure 4B, the dashed curve represents the thermal variation of mT in the system used [17]). Figure represents the variation of mT in the has been (seeIn[17]). In4B, Figure 4B, the dashed curve represents thermal variation of system m in the with r(see =used 0.5, when the signs ofthe mA dashed and mB curve are changed from thethermal case the of (m A > 0.0, mB < 0.0) to theTcase with rwith = 0.5,r = when the signs mA and are changed from thefrom case of A > 0.0, mBA0.0) system 0.5, when theof signs of mmAB and mB are changed the(mcase of (m 0.0,tomthe 0.0) to B < case
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of A < 0.0, mBA> 0.0).4C, In Figure onhand, the other is shown thatbehavior the sameofbehavior of dashed curve in Figure can be 4B easily selecting the reasonable parameters of r and of h mT as the dashed curve 4B in Figure can obtained be easily by obtained by selecting the reasonable parameters (such as the curve labeled r = 1.5 and h = 4.0), even when m A > 0.0 and m B < 0.0. The curve clearly r and h (such as the curve labeled r = 1.5 and h = 4.0), even when mA > 0.0 and mB < 0.0. The curve exhibits a negative magnetization in the low temperature region. The curve labeled = 0.001(rand h= clearly exhibits a negative magnetization in the low temperature region. The curve (r labeled = 0.001 2.1) Figure exhibits the Q-type The results in Figure 4B,C are similar to those andin h= 2.1) in4C Figure 4C exhibits the behavior. Q-type behavior. The shown results shown in Figure 4B,C are similar to obtained in [14,15]. those obtained in [14,15].
(A)
(B)
(C) Figure Figure4. 4. The Thetemperature temperaturedependences dependencesof ofmmTTininthe thesystem systemA; A;the thevalue valueofofr risisfixed fixedatatrr==0.5 0.5(A,B). (A,B). The value of h is changed from h = 1.0 to h = 2.5 in (A) and from h = 3.1 to h = 3.3 in (B). The The value of h is changed from h = 1.0 to h = 2.5 in (A) and from h = 3.1 to h = 3.3 in (B). The dashed dashed curve curverepresents represents the the result result of of the the system system with with rr == 0.5 0.5 and and hh == 3.1, 3.1, when when the the signs signs of of m mAA and and m mBB are are changed from the case of (m A > 0.0, mB < 0.0) to the case of (mA < 0.0, mB > 0.0). The thermal variations changed from the case of (mA > 0.0, mB < 0.0) to the case of (mA < 0.0, mB > 0.0). The thermal variations of ofm mTTare aregiven, given,when whenthe thetwo twoset setvalues values (r (r and and h) h) are are selected selected as as (r (r == 0.001 0.001 and and hh ==2.1) 2.1)and and (r (r ==1.5 1.5and and hh==4.0) (C). 4.0) (C).
4. The Magnetic Properties of System B 4. The Magnetic Properties of System B From the structural aspect of the present two systems, system B seems to be more realistic than From the structural aspect of the present two systems, system B seems to be more realistic than system A [14,15], although the spin structure is more complicated than that of system A. In fact, for system A [14,15], although the spin structure is more complicated than that of system A. In fact, for the the inter-layer interaction, there exist two exchange interactions JA and JB in system B, while there inter-layer interaction, there exist two exchange interactions JA and JB in system B, while there exists exists only one exchange interaction JR in system A. Accordingly, let us define the following only one exchange interaction JR in system A. Accordingly, let us define the following parameters t, h, parameters t, h, r and s as r and s as t =t k=BkT/J, /J, r = JAA/J/Jand ands s= =JBJ/JB /J (19) BT/J,hh==Ω Ω/J, (19) Figure 5 shows the phase diagrams (tC versus h plot) of the system B, when the value of r (or s) is fixed at r = 0.5 in Figure 5A (or s = 0.5 in Figure 5B) and the value of s (or r) is changed from s = 0.0
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Nanomaterials 7, 256 the phase diagrams (tC versus h plot) of the system B, when the value of r 7(or of 12 Figure2017, 5 shows s) is fixed at r = 0.5 in Figure 5A (or s = 0.5 in Figure 5B) and the value of s (or r) is changed from = 2.0 from = 0.0 rto=r0.0 = 2.0). is seen from the behavior of tC curves in Figure sto=s0.0 to s(or = 2.0 (orr from to r As = 2.0). As is seenthese fromfigures, these figures, the behavior of tC curves in 5A is completely different from that in Figure 5B as well as the results of Figure 2 for system A. In Figure 5A is completely different from that in Figure 5B as well as the results of Figure 2 for system Figure 5A, the phenomenon has been obtained for the labeled r = 1.5r = (or1.5 bold A. In Figure 5A,reentrant the reentrant phenomenon has been obtained forcurve the curve labeled (or solid bold curve). However, the same phenomenon has not been obtained in Figure 5B. In order to clarify these solid curve). However, the same phenomenon has not been obtained in Figure 5B. In order to clarify differences between Figure 5A and 5B,the thehChCversus versuss s(or (orr)r)plot plothas hasbeen beenexamined examined Figure these differences between Figures 5AFigure and 5B, inin Figure 6. 6. The value of h C can be obtained from the results of Figure 5 as a critical value of h at which the t The value of hC can be obtained from the results of Figure 5 as a critical value of h at which the tCC curve reduces reduces to to zero. zero. Figure shows the = 0.5. curve Figure 6A 6A shows the hhCC versus versus ss plot, plot, when when the the value value of of rr is is fixed fixed at at rr = 0.5. Figure 6B 6B shows shows the the hhCC versus plot, when when the the value value of of ss is is fixed fixed at at ss == 0.5. 0.5. One One can can easily easily find find clear clear Figure versus rr plot, differences between betweenFigures Figure 6A 6A and and6B. Figure 6B. Asinshown Figure 7A,flat only the flat of hC in differences As shown Figurein 7A, only the region of hregion C in Figure 6A Figure 6A between h C = 1.0 and h C = 2.0 may exhibit the reentrant phenomenon, although, for the flat between hC = 1.0 and hC = 2.0 may exhibit the reentrant phenomenon, although, for the flat region of of h C > 2.0 of Figure 6A, the phenomenon has not been found, as shown in Figure 7B. In Figure hregion > 2.0 of Figure 6A, the phenomenon has not been found, as shown in Figure 7B. In Figure 6B, on the C 6B, onhand, the other hand,of the of hCsimply increases simply with theofincrease of r. A phenomenon other the value hCvalue increases with the increase r. A phenomenon similar to similar that of to that 7B of Figure 7B can be alsofor obtained A,value whenofthe of hCas is aplotted as aoffunction of Figure can be also obtained systemfor A, system when the hCvalue is plotted function r by using rthe byresults using the results of Figure 2. of Figure 2.
(A)
(B)
Figure 5. 5. The The phase phase diagrams diagrams (t (tC versus h plot) for the system B; The value of r is fixed at r = 0.5 and Figure C versus h plot) for the system B; The value of r is fixed at r = 0.5 and the value of s is changed from s = 0.0 to s 2.0(A). (A).The Thevalue valueofofs is s isfixed fixed = 0.5 and value the value of s is changed from s = 0.0 to s ==2.0 atat s =s 0.5 and thethe value of of r isr is changed from r = 0.0 to r = 2.0 (B). changed from r = 0.0 to r = 2.0 (B).
In Figure 8, the thermal variations of mT in the system with fixed values of r = 0.5 and s = 1.5 are In Figure 8, the thermal variations of m in the system with fixed values of r = 0.5 and s = 1.5 given by changing the value of h from h = T0.0 to h = 4.26, in order to clarify whether the above are given by changing the value of h from h = 0.0 to h = 4.26, in order to clarify whether the above predictions of phase diagrams are correct. In Figure 5A, the reentrant phenomenon has been obtained predictions of phase diagrams are correct. In Figure 5A, the reentrant phenomenon has been obtained in the region of 4.22 < h < 4.36 for the system with r = 0.5 and s = 1.5. In Figure 8E, the thermal variations in the region of 4.22 < h < 4.36 for the system with r = 0.5 and s = 1.5. In Figure 8E, the thermal of mT (solid curve), mA (dashed curve) and mB (dotted curve) are plotted by selecting the value of h = variations of mT (solid curve), mA (dashed curve) and mB (dotted curve) are plotted by selecting the 4.26 from the region of 4.22 < h < 4.36. They clearly exhibit the reentrant phenomenon, as predicted in value of h = 4.26 from the region of 4.22 < h < 4.36. They clearly exhibit the reentrant phenomenon, the phase diagrams. At this place, one should notice that the reentrant phenomenon is not a peculiar as predicted in the phase diagrams. At this place, one should notice that the reentrant phenomenon is behavior and it has often been found for some finite-size magnetic systems described by some not a peculiar behavior and it has often been found for some finite-size magnetic systems described by transverse Ising models, as discussed in some recent works [18–21]. As is seen from Figure 8A to some transverse Ising models, as discussed in some recent works [18–21]. As is seen from Figure 8A Figure 8D, the thermal variations of mT in the system may change a positive magnetization to a to Figure 8D, the thermal variations of mT in the system may change a positive magnetization to negative magnetization with the increase of h and may exhibit some characteristic ferrimagnetic a negative magnetization with the increase of h and may exhibit some characteristic ferrimagnetic behaviors, such as Q-, P-, N- and M-types, although some of them could not be classified by the behaviors, such as Q-, P-, N- and M-types, although some of them could not be classified by the nomenclature of ferrimagnetism [17]. In fact, the thermal variations of mT for the curves labeled h = nomenclature of ferrimagnetism [17]. In fact, the thermal variations of mT for the curves labeled h = 3.0 3.0 and h = 3.1 in Figure 8C are novel types. They start to decrease from the saturation magnetization and h = 3.1 in Figure 8C are novel types. They start to decrease from the saturation magnetization at at t = 0.0, exhibit a broad minimum, a broad maximum with the increase of t and then reduce to zero at their transition temperatures. In relation to the results of Figure 7B, the thermal variations of mT in the system with fixed value of r = 0.5 and s = 3.0 are plotted in Figure 9, selecting the three values of h. The curve labeled h = 3.0 shows the L-type behavior. The curve labeled h = 3.1 exhibits the N-type behavior. The curve labeled
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t = 0.0, exhibit a broad minimum, a broad maximum with the increase of t and then reduce to zero at their transition temperatures. In relation to the results of Figure 7B, the thermal variations of mT in the system with fixed value of r = 0.5 and s =7, 3.0 Nanomaterials 2017, 256 are plotted in Figure 9, selecting the three values of h. The curve labeled h88=of of3.0 12 Nanomaterials 2017, 7, 256 12 shows the L-type behavior. The curve labeled h = 3.1 exhibits the N-type behavior. The curve labeled 3.2 represents the M-type M-type behavior. behavior. The The thermal thermal variations variations of of m mTTT,,, corresponding corresponding to the results of 3.2 represents represents the corresponding to to the the results results of hh === 3.2 m T depicted Figure 4C for the system A with r = 0.5, are depicted in Figure 10, selecting the two depicted in Figure 4C for the system A with r = 0.5, are depicted in Figure 10, selecting the two sets mTT depicted in Figure 4C for the system A with r = 0.5, are depicted in Figure 10, selecting the two sets of ss and and insystem the system system B with with 0.5. results The results results shown in Figures Figures and 10 aresimilar also similar similar to of s and h in the B with r = 0.5. shown in Figures 9 and9910 are10 also to those sets of hh in the B rr ==The 0.5. The shown in and are also to those obtained in [14,15]. obtained in [14,15]. those obtained in [14,15]. Finally, the thermal thermal variations variationsof ofm mTTT in with fixed fixed values values of of sss = 0.5 and and rr === 1.5 1.5 are given Finally, the in the the system system with 1.5 are are given given Finally, the thermal variations of m with fixed values of == 0.5 in Figure Figure 11, 11, changing changing the the value value of 3.0 to to hh ===4.0. 4.0. These results can be compared with those of hh from from hh = 4.0. These Theseresults results can can be be compared compared with with those those in = 3.0 of Figure 8B,C for the system with = 0.5 and s = 1.5. r = 0.5 = 1.5. of Figure 8B,C for the system with r = 0.5 and s = 1.5.
(A) (A)
(B) (B)
Figure 6. 6. The variations variations of hC in the the system B; B; the variation of hC in the the system with with rr == 0.5 0.5 and and tt == 0.02 0.02 Figure Figure 6. The The variations of hCC in in the system system B; the variation of hCC in in the system system 0.5 = 0.02 C in the system is given, when the value of s is changed from s = 0.0 to s = 3.5 (A). The variation of h is is given, given, when when the the value value of s is changed from s = 0.0 to s = 3.5 (A). The variation of hCC in in the the system system with s = 0.5 and t = 0.01 is given, when the value of r is changed from r = 0.0 to r = 2.5 (B). with s == 0.5 0.5 and and tt ==0.01 0.01isisgiven, given,when when the the value value of of rr is is changed changed from from rr ==0.0 0.0to torr==2.5 2.5(B). (B).
(A) (A)
(B) (B)
Figure 7. 7. The phase phase diagrams (t (tC versus hh plot) plot) for the the system B B with rr == 0.5; 0.5; the value value of ss is is changed Figure Figure 7. The The phase diagrams diagrams (tCC versus versus h plot) for for the system system B with with r = 0.5; the the value of of s is changed changed from s = 1.0 to s = 2.0 (A). Three values of s are selected as s = 2.3, s = 2.8 and s = 3.3 (B). from s = 1.0 to s = 2.0 (A). Three values of s are selected as s = 2.3, s = 2.8 and s = 3.3 (B). from s = 1.0 to s = 2.0 (A). Three values of s are selected as s = 2.3, s = 2.8 and s = 3.3 (B).
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(A)
(B)
(C)
(D)
(E) Figure 8. The temperature dependences of mT in the system B with r = 0.5 and s = 1.5; the thermal Figure 8. The temperature dependences of mT in the system B with r = 0.5 and s = 1.5; the thermal variations of mT (solid curve), mA (dashed curve), mB (dotted curve) and qA (dot-dashed curve) are variations of mT (solid curve), mA (dashed curve), mB (dotted curve) and qA (dot-dashed curve) are depicted, when the value of h is given by h = 0.0 (A). The thermal variations of mT are given, changing depicted, when the value of h is given by h = 0.0 (A). The thermal variations of mT are given, changing the value of h from h = 2.0 to h = 3.1 (B). The thermal variations of mT are given by selecting the three the value of h from h = 2.0 to h = 3.1 (B). The thermal variations of mT are given by selecting the three values of h (C,D). The reentrant phenomena of magnetizations are presented by selecting the value of values of h (C,D). The reentrant phenomena of magnetizations are presented by selecting the value of h h as h = 4.26 (E). as h = 4.26 (E).
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Figure 9. The temperature dependencesofofmm in the the system system BBwith 0.50.5 and s = s3.0, when the three Figure 9. The temperature dependences withr =r = and = 3.0, when the three TTin values of h are selected. values of h are selected. Figure 9. The temperature dependences of mT in the system B with r = 0.5 and s = 3.0, when the three values of h are selected.
Figure 10. The thermal variations ofofmm withrr==0.5 0.5are are given, when set values Figure 10. The thermal variations inthe the system system with given, when the the twotwo set values T Tin (sh) and are selected = 0.2 and and (s = = 1.0 and (s and areh) selected as (sasvariations =(s0.2 and h h=m=2.5) and (s 1.0 and =3.1). 3.1). Figure 10. The thermal of T 2.5) in the system with rh=h=0.5 are given, when the two set values (s and h) are selected as (s = 0.2 and h = 2.5) and (s = 1.0 and h = 3.1).
(A) (A)
(B) (B)
Figure 11. The temperature dependences of mT in the system B with s = 0.5 and r = 1.5; the value of h is changed from h = 3.0 to h = 3.2 (A) and from h = 3.7 to h = 4.0 (B).
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5. Conclusions In this work, within the theoretical framework of the effective-field theory with correlations (EFT), we have investigated the phase diagrams and the magnetizations in the two mixed spin Ising systems consisting of two layers with honeycomb lattice. As shown in Figure 1, at first sight, their magnetic properties seem to be similar to each other, since they have the same intra-layer spin structure. As shown in Figures 5A, 6A, 7A and 8E, the big differences between the two systems can be seen in the appearance of the reentrant phenomenon and the characteristic behavior of hC in system B. In particular, one should also notice that such characteristic features have been obtained in system B, only when the value of JA is fixed and the value of JB is changed. In fact, as shown in Figure 5B, such characteristic phenomena have not been obtained, when the value of JB is fixed and the value of JA is changed. From the numerical examinations of mT given in Sections 3 and 4, all types of behaviors normally obtained for the thermal variation of mT in bulk ferrimagntic materials, such as the Q-, P-, N-, L- and M-types, can be found in the present two systems. As shown in Figure 8C, furthermore, novel types have been obtained for the thermal variation of mT in system B. In the present work, we have examined the magnetic properties of the two systems consisting of spin-1/2 and spin-1 atoms. Realistically, such systems should be constructed from spin-2 and spin-5/2 atoms, in order to compare with experimental data in molecular-based ferriagnetic materials [14,15]. The numerical results obtained from such systems are probably similar to the present ones obtained in Sections 3 and 4. Conflicts of Interest: The author declares no conflict of interest.
References 1. 2. 3.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Shi, X.; Qi, Y. Existence of a dynamic compensation temperature of the mixed spin-1 and spin-3/2 Ising model within the effective-field theory. Physica B 2016, 459, 117. [CrossRef] Bi, J.L.; Wang, W.; Li, Q. Monte Carlo study of a ferrimagnetic mixed-spin (2,5/2) system with the nearest and next-nearest neighbors exchange couplings. Supperlattices Microstruct. 2017, 107, 104–117. [CrossRef] Wang, W.; Lv, D.; Zhang, F.; Bi, J.I.; Chen, J.N. Monte Carlo simulation of magnetic properties of a mixed spin-2 and spin-5/2 ferrimagnetic Ising system in a longitudinal magnetic field. J. Magn. Magn. Mater. 2015, 358, 16–26. [CrossRef] Ertas, M.; Keskin, M.; Deviren, D. Dynamic magnetic properties in the kinetic mixed spin-2 and spin-5/2 Ising model under a time-dependent magnetic field. Phys. A 2012, 391, 1038–1047. [CrossRef] Sali, N.; Akbudak, S.; Elliatioghu, M.R. The peak effect (PE) region of the antiferromagnetic two layer Ising nanographene. Phys. B 2014, 452, 18–22. Santos, J.P.; Barreto, F.C.S. An effective-field theory of trilayer Ising nanostructure: Thermodynamic and magnetic properties. J. Magn. Magn. Mater. 2017, 439, 114–119. [CrossRef] Jiang, W.; Yang, Y.Y.; Guo, A.B. Study on magnetic properties of a nano-graphene bilayer. Carbon 2015, 95, 190–198. [CrossRef] Honmura, R.; Kaneyoshi, T. Contribution to the new type of effective-field theory of the Ising model. J. Phys. C 1979, 12, 3979. [CrossRef] Kaneyoshi, T. Differential operator technique in the Ising spin systems. Acta Phys. Pol. A 1993, 83, 703. [CrossRef] Zernike, F. The propagation of order in co-operative phenomena: Part 1. The AB case. Physica 1940, 7, 565–585. [CrossRef] Liu, J.W.; Xin, H.Z.; Chen, L.S.; Zhang, Y.C. Effective-field and Monte Carlo studies of mixed spin-2 and spin-1/2 Ising diamond chain. Chin. Phys. B 2013, 22, 027501. [CrossRef] Magoussi, H.; Zaim, A.; Kerouad, M. Magnetic properties of a nanoscaled ferromagnetic thin film; Monte Carlo and effective field treatments. Superlattices Microstruct. 2016, 89, 188–203. [CrossRef] Zhang, Q.; Wei, G.; Xin, Z.; Liang, Y. Effective-field theory and Monte Carlo study of a layered mixed spin-1 and spin-2 Ising system on honeycomb lattice. J. Magn. Magn. Mater. 2004, 280, 14–22. [CrossRef]
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Mathoniere, C.; Carling, S.G.; Yusheng, D.; Day, P. Molecular-based Mixed valency Ferrimagnets (XR4 ) FeII FeIII (C2 O4 )3 (X = N, P; R = n-propyl, n-butyl, phenyl); Anomalous Negative Magnetization in the Tera-n-Butylammonium Derivative. J. Chem. Soc. Commun. Issue 1994, 13, 1551–1552. [CrossRef] Bhattacharjee, A.; Feyerherm, R.; Steiner, M. Study of the Negative Magnetization Phenomenon in NBu4 [FeII FeIII (ox)3 ]. J. Phys. Soc. Jpn. 1999, 68, 1679–1683. [CrossRef] Kaneyoshi, T.; Sarmento, E.F.; Fittipaldi, I.P. A Compensation Temperature Induced by Transverse Fields in a Mixed Ising Ferrimagnetic System. Jpn. J. Appl. Phys. 1988, 27, L690. [CrossRef] Herpin, A. Theory of Magnetism; Press University of France: Paris, France, 1968. Kaneyoshi, T. Ferrimagnetism in two antiferromagnetic transverse Ising thin films. Ph. Trans. 2016, 90, 131–142. [CrossRef] Kaneyoshi, T. Temperature-Induced Magnetization Reversal in a Transverse Ising Antiferromagnetic Thin Film. J. Supercond. Novel Magn. 2017, 30, 157–164. [CrossRef] Kaneyoshi, T. Unconventional phenomena in a transverse Ising quadrangular nanowire. J. Phys. Chem. Solids 2017, 106, 105–111. [CrossRef] Kaneyoshi, T. Unconventional Phenomena in Transverse Ising Nanowires with Simple Core-Shell Structure. J. Supercond. Novel Magn. 2017, 30, 1867–1876. [CrossRef] © 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Effects of a Transverse Field in Two Mixed-Spin Ising Bilayer Films Takahito Kaneyoshi Graduate School of Information and Science, Nagoya University, 1-510, Kurosawadai, Midoriku, Nagoya 458-0003, Japan; [email protected]; Tel.: +81-528-766-607 Received: 16 June 2017; Accepted: 23 August 2017; Published: 4 September 2017
Abstract: The magnetic properties (phase diagrams and magnetizations) of two mixed-spin Ising bilayer films with a transverse field are investigated by the use of the effective field theory with correlations. The systems consist of two magnetic atoms where spin-1/2 atoms are directed to the z-direction and only spin-1 atoms are canted from the z-direction by applying a transverse field. We examined how magnetization sign reversal can be realized in the system, due to the effects of the transverse field on the spin-1 atoms. The compensation point phenomena are found in both systems, depending on the selections of physical parameters. However, the reentrant phenomena are found only for one of the two systems. Keywords: phase diagrams; magnetizations; compensation point; reentrant phenomena; mixed-spin Ising bilayer films
1. Introduction The theoretical research of various mixed-spin Ising (or transverse Ising) systems on honeycomb lattice has a long history and it is still a subject of active research. They have been examined by using various theoretical frameworks of mean-field theory (MFT), effective-field theory (EFT), the Green function method and the Monte Carlo simulation (MC). The studies of these systems are related to the experimental researches of molecular-based magnetic materials and nano-graphene films. In particular, a kind of molecular-based ferrimagnets, namely AFeII FeIII (C2 O4 )3 (A = N(n − Cn H2n+1 )4 , n = 3–5), have been the subject under extensive theoretical discussions of mixed-spin Ising (or transverse Ising) systems, where spins of FeII and FeIII atoms are respectively taken as 2 and 5/2 and they are coupled by a negative exchange interaction in each layer and two positive interactions between layers (see the references in the recent theoretical works [1–4]). For nano-graphene films, on the other hand, the magnetic properties of the ferromagnetic and antiferromagnetic two (or three) layers of honeycomb lattices described by the spin-1/2 Ising (or transverse Ising) models have been recently investigated by the use of the EFT [5–7]. The EFT [8,9] corresponds to the Zernike approximation [10] and it is believed to give more exact results than those of the MFT, since it automatically includes some correlations between a certain spin and the near-neighbor spins. Furthermore, the results obtained from the EFT have also the same topology as those obtained from the MC. In fact, the magnetic properties (magnetization, internal energy and so on) of a system are the same for the EFT and the MC, while the results obtained from the MC are smaller than those obtained from the EFT (for instance, see the works [11–13]). The aim of this work is to present another aspect of theoretical explanation for the appearance of negative magnetization in some molecular-based ferrimagnetic materials [14,15] on the basis of two layers of honeycomb lattices described by the mixed-spin (spin-1/2 and spin-1) Ising model, where the spin direction of spin-1/2 atoms is directed to the z-(or easy) direction and the spin direction of spin-1 atoms is canted from the z-direction, due to the presence of a transverse field. In [14],
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for instance, A in the above-described molecular-based ferrimagnets has been replaced by A = (XR4 ) with X = N, P and R = n-propyl, n-butyl, phenyl, instead of A = N(n − Cn H2n+1 )4 , (n = 3–5). In these systems, only the system with A = NBun 4 has exhibited a negative magnetization at low-temperature. In this work, the two model systems are examined by the use of the EFT [16], in order to clarify whether such a negative magnetization can be realized within the present two model systems. In [16], the magnetizations of a mixed-spin system consisting of spin-1/2 and spin-1 atoms with different transverse fields have been examined. The temperature dependences of total magnetization in the system have shown typical ferrimagnetic behaviors usually observed in crystalline ferrimagnetic alloys Ap B1-p , due to a strong negative A-B exchange interaction, where A and B are different magnetic atoms with different spins. The physical reason for this comes from the different spin canting of spin-1/2 and spin-1 atoms. In Section 2, the two systems are proposed. At first sight, they are very similar. Even within the theoretical formulations of the EFT, the basic equations obtaining the magnetic properties (phase diagram and magnetizations) are rather different between the two systems (system A and system B). In Section 3, the magnetic properties of system A are discussed by solving the formulations numerically. In Section 4, the magnetic properties of system B are obtained numerically. The last section is devoted to the conclusion. 2. Models and Formulation We consider the two systems consisting of two layers, as depicted in Figure 1. The structure of each system is the honeycomb lattice. The white and black circles on each figure represent respectively the magnetic atoms with spin-1/2 and the magnetic atoms with spin-1 and transverse field Ω. The spins (white and black circles) on each layer are coupled by a nearest-neighbor exchange interaction-J (J > 0.0). In Figure 1A, each Ising spin on the upper layer is coupled to the corresponding same spin on the lower layer with an exchange interaction-JR (JR > 0.0). In Figure 1B, on the other hand, two exchange interactions (JA > 0.0 and JB > 0.0) exist between the two layers, depending on where the pair is located. JA is the interlayer coupling between the spin-1 atoms and JB is the interlayer coupling between the spin-1/2 atoms. The Hamiltonian of Figure 1A (or system A) is given by H = J Σ σi Z S j Z + JR Σ σi Z S j Z − Ω Σ S j X (ij)
(1a)
j
(ij)
where σi Z represents the spin-1/2 operator with σi Z = ±1/2 and Sj α (α = Z and X) is the spin-1 operator with Sj Z = ±1 and 0. The first and the second terms in (1a) represent the contributions from the intra-layer and inter-layer interactions. The last term shows the effect of a transverse field at each spin-1 atom. On the other hand, the Hamiltonian of Figure 1B (or system B) can be represented by H = J Σ σi Z S j Z − J A Σ σk Z Sl Z − JB Σ σm Z Sn Z − Ω Σ S j X (ij)
(kl )
(mn)
j
(1b)
where the second and third terms show the contributions from the inter-layer interactions. The total longitudinal magnetization (mT = mT Z ) per site in each system (A or B) is defined as mT = [mA + mB ]/2.0
(2)
where mA = mA Z = and mB = mB Z = . Within the EFT [8,9,16], the mA and mB in the system A (Figure 1A) are given by mA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [cosh(R/2.0) − 2.0 mB sinh(R/2.0)] F(x)|x=0
(3)
qA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [cosh(R/2.0) − 2.0 mB sinh(R/2.0)] G(x)|x=−0
(4)
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and mB = [qA {cosh(C) − 1.0} + 1.0 − mA sinh(C)]3 [qA {cosh(R) − 1.0} + 1.0 − mA sinh(R)] f (x)|x=0
(5)
where qA = < (Sj Z )2 >, C = J·D and R = JR ·D. D = ∂/∂x is the differential operator. Here, the functions F(x), G(x) and f (x) are defined by F(x) = 2.0 x sinh(e(x)β)/[e(x) {1.0 + 2.0 cosh(βe(x))}]
(6)
G (x) = [Ω2 + {Ω2 + 2.0 x2 } cosh(βe(x))]/[e(x)2 {1.0 + 2.0 cosh(βe(x))}]
(7)
f (x) = tanh(βx/2.0)/2.0
(8)
e(x) = [Ω2 + x2 ]1/2
(9)
with where β = 1.0/kB T. For the system B in Figure 1B, the mA and mB are given by mA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [qA {cosh(A) − 1.0} + 1.0 + mA sinh(A)] F(x)|x=0 qA = [cosh(C/2.0) − 2.0 mB sinh(C/2.0)]3 [qA {cosh(A) − 1.0} + 1.0 + mA sinh(A)] G(x)|x=−0
(10)
mB = [qA {cosh (C) − 1.0} + 1.0 − mA sinh(C)]3 [cosh(B/2.0) + 2.0 mB sinh(B/2.0)] f (x)|x=0
(11)
and
where A = JA ·D and B = JB ·D. The phase diagrams (or transition temperature) in the two systems can be determined by expanding the coupled equations of mA and mB in each system (A or B) linearly. The transition temperature of system A can be obtained from the relation [3.0 K1 + K2 ] [6.0 K3 + 2.0 K4 ] − 1.0 = 0.0
(12)
where the coefficients Kn (n = 1–4) are given by K1 = sinh(C) [qA {cosh(C) − 1.0} + 1.0]2 [qA {sinh(R) − 1.0} + 1.0] f (x)|x=0 K2 = sinh(R) [qA {cosh(C) − 1.0} + 1.0]3 f (x)|x=0 K3 = cosh2 (C/2.0) sinh(C/2.0) cosh(R/2.0) F(x)|x=0 K4 = cosh3 (C/2.0) sinh(R/2.0) F(x)|x=0
(13)
with qA = cosh3 (C/2.0) cosh(R/2.0) G(x)|x=0 . The transition temperature of system B can be obtained from the relation [L1 − 1.0] [2.0 L2 − 1.0] − 18.0 L3 L4 = 0.0
(14)
The coefficients Ln (n = 1–4) in (14) are given by L1 = cosh3 (C/2.0) sinh(A) F(x)|x=0 L2 = sinh(B/2.0) [qA {cosh(C) − 1.0} + 1.0]3 f (x)|x=0 L3 = cosh(C/2.0) sinh(C/2.0) [qA {cosh(A) − 1.0} + 1.0] F(x)|x=0 L4 = sinh(C) cosh(B/2.0) [qA {cosh(C) − 1.0} + 1.0]2 f (x)|x=0
(15)
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with qA = D1 /(1.0 + D1 − D2 )
(16)
where D1 and D2 are defined as D1 = cosh3 (C/2.0) G(x)|x=0 4 (17) of 12 D2 = cosh3 (C/2.0) cosh(A) G(x)|x=0 At this point, it is important to note that only spin-1 magnetic atoms in the two systems are At it is important to note thatthey onlyare spin-1 magnetic atoms in the twowhen systems are affected affectedthis by point, a transverse field Ω and hence canted from the z-direction, Ω takes a finite by a transverse field Ω and hence they are canted from the z-direction, when Ω takes a finite value. value. However, spin-1/2 atoms in them are always directed to the z-direction. At first sight, the two However, spin-1/2 atoms in them are always directed to the z-direction. At first sight, the two systems systems given in Figure 1 seem to be physically equivalent. As discussed above, however, the given in Figure 1 seem to of be the physically equivalent. As discussed above, the theoretical theoretical formulations two systems are rather different evenhowever, within the theoretical formulations of the two systems are rather different even within the theoretical formulation of the formulation of the EFT. Under these conditions, it may be important to know what phenomena can EFT. Under these conditions, it may be important to know what phenomena can be obtained in the be obtained in the two systems. Furthermore, it is also important to know whether the phenomena two systems. Furthermore, is also important to know whether phenomena are similarofbetween are similar between the twoitsystems or some differences may bethe found in the phenomena the two the two systems or some differences may be found in the phenomena of the two systems. As far systems. As far as we know, these problems have not been discussed. In the following, the magnetic as we know, thesediagram problems have not been discussed. In the following, the magnetic properties properties (phase and thermal variation of magnetizations) in system A (or Figure 1A) are (phase diagram and thermal variation of magnetizations) in system A (or Figure 1A) are examined in examined in Section 3. In Section 4, such magnetic properties of system B (or Figure 1B) are discussed. Section 3. In Section 4, such magnetic properties of system B (or Figure 1B) are discussed. Nanomaterials 2017, 7, 256
(A)
(B)
Figure 1. 1. Schematic spin Ising ferrimagnetic models on two (upper and Figure Schematicrepresentations representationsofoftwo twomixed mixed spin Ising ferrimagnetic models on two (upper lower) layered honeycomb lattices with spin-1/2 (white circle) and spin-1 (black circle) atoms. (A) and lower) layered honeycomb lattices with spin-1/2 (white circle) and spin-1 (black circle) atoms. Each Ising spin on upper layer is coupled to the corresponding same spin on the lower layer with an (A) Each Ising spin on upper layer is coupled to the corresponding same spin on the lower layer with exchange interaction-J R (JR > 0.0). (B) Two exchange interactions (JA > 0.0 and JB > 0.0) exist between an exchange interaction-J R (JR > 0.0). (B) Two exchange interactions (JA > 0.0 and JB > 0.0) exist between theinterlayer interlayercoupling couplingbetween betweenthe thespin-1 spinthe two layers, depending onwhere wherethe thepair pairisislocated. located.J JAisisthe the two layers, depending on A B is the interlayer coupling between the spin-1/2 atoms. 1 atoms and J atoms and J is the interlayer coupling between the spin-1/2 atoms. B
3. The Magnetic Properties of System A 3. The Magnetic Properties of System A At first, let us define the parameters, t, h and r as At first, let us define the parameters, t, h and r as t = kBT/J, h = Ω/J and r = JR/J (18) t = kB T/J, h = Ω/J and r = JR /J (18) In Figure 2, the phase diagram (TC versus h plot) in the system A is given by changing the value In Figure (TC versus h plot) lines in therepresent system Athe is given byof changing of r from r = 0,2,0the to phase r = 1.5.diagram In the figure, the dashed results the tC (tCthe = kvalue BTC/J) of r from r = 0,when 0 to r the = 1.5. In theAfigure, the dashed represent the results of the tC (tC = kB TC /J) versus h plot, system is described by thelines following Hamiltonian, versus h plot, when the system A is described by the following Hamiltonian,
(
H = J Σ σ iZ S jZ + JR Σ σ i Z S j Z − Ω ΣI σ i X + Σj S j X ( ij ) ( ij ) H = J Σ σi Z S j Z + JR Σ σi Z S j Z − Ω (ij)
(ij)
Σσi X + Σ S j X I
j
)
namely when both spin-1/2 and spin-1 atoms have the same transverse field Ω, instead of the present system with both the Hamiltonian this have stage,theone should noticefield thatΩthe different namely when spin-1/2 and (1,a). spin-1At atoms same transverse , instead of thebehavior present between thethe solid curve and(1,a). the At dashed curveone in should the system value of r makes an system with Hamiltonian this stage, noticewith thatthe the same different behavior between important contribution the appearance of negative magnetization at a low the solid curve and the to dashed curve in the system with the same value of temperature. r makes an important contribution to the appearance of negative magnetization at a low temperature.
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Figure 2. 2. The plot) for for the thesystem systemA, A,when whenthe thevalue value changed from Figure Thephase phasediagram diagram(t(tCC versus versus h plot) ofof r isr is changed from phase diagram (tC versus h plot) for the system A, when the value of r is changed from r =rFigure 0.0 totor2.= = 0.0 rThe =1.5. 1.5. r = 0.0 to r = 1.5.
Figure3,3,the thetemperature temperaturedependences dependences of magnetizations with fixed values of of InIn Figure of magnetizationsin inthe thesystem system with fixed values Figure 3, the dependences ofBmagnetizations the system with fixed values of r = 0.5Inand h = 0.0 are temperature plotted, where mT, mA and m are respectivelyin represented by solid, dashed and r =r 0.5 and h = 0.0 are plotted, where m , m and m are respectively represented by solid, dashed and T A B = 0.5 and h = 0.0 plotted, where mof T, mA and mB are respectively represented by solid, dashed and dotted curves. Theare thermal variation qA is also plotted in Figure 3 by the dot-dashed curve. As is dotted curves. The thermal variation of q is also plotted in Figure 3 by the dot-dashed curve. As dotted curves. The thermal of qAA is also plotted in Figure 3 by the dot-dashed curve.region As is is seen from the figure, mA andvariation mB take respectively a positive and negative value in the whole seen from the figure, m and m take respectively a positive and negative value in the whole region A A and mBB take respectively a positive and negative value in the whole region seen from the m of T below its figure, TC. Here, one should notice that the value of r = 0.5 is particularly selected, since the of of T below itsits TCT.CHere, one should notice the value value of of rr ==0.5 0.5isisparticularly particularlyselected, selected, since T below .the Here, should notice that since thethe distance between twoone layers are supposed to the be larger than the lattice interval in each layer [14,15] distance between the two layers are supposed to be larger than the lattice interval in each layer [14,15] distance between the two layers are supposed to larger than the lattice interval in each layer [14,15] and hence the exchange interaction J R is assumed to be weaker than that of intra-layer interaction J. and hence the exchange assumed to tobe beweaker weakerthan thanthat thatofofintra-layer intra-layer interaction and hence the exchangeinteraction interactionJJRR is assumed interaction J. J.
Figure 3. The temperature dependences of mT (solid curve), mA (dashed curve), mB (dotted curve) and The temperature dependences T (solid curve), mA (dashed curve), mB (dotted curve) and qFigure A (dot-dashed curve) in the system A with fixed value 0.5, when the value of h is given byand h = qA Figure 3. 3. The temperature dependences ofofmm curve),ofmr A= (dashed curve), mB (dotted curve) Ta (solid q A (dot-dashed curve) in the system A with a fixed value of r = 0.5, when the value of h is given by h = 0.0. (dot-dashed curve) in the system A with a fixed value of r = 0.5, when the value of h is given by h = 0.0. 0.0.
In Figure 4A,B, the thermal variations of mT in the system with r = 0.5 are shown by changing In In Figure 4A,B, thermal variations of m in the the system systemwith withr r==0.5 0.5 areshown shown changing Figure 4A,B,hthe the variations of in mTTFigure in byby changing the value of h from = 1.0thermal to h = 3.3. As shown 4A, the behavior of mare T may change from the thethe value of h from h = 1.0 to h = 3.3. As shown in Figure 4A, the behavior of m may change from value from h with = 1.0 the increase = 3.3. Asofshown in Figure behavior ofhm=T 3.1 may T change from the Q-type to of theh P-type h. In Figure 4B, 4A, the the curves labeled and h = 3.2 exhibit theQ-type Q-type to the P-type with the increase of h. In Figure 4B, the curves labeled h = 3.1 and h = 3.2 to the P-type thetheir increase of h. In Figure 4B, the labeled h = 3.1 and = 3.2labeled exhibit a compensation pointwith below transition temperatures (orcurves the N-type), although the hcurve exhibit compensation point below their transition temperatures (or thein N-type), although thebeen curve point below theirThe transition temperatures (orand the N-types N-type), although the curvehas labeled ha =compensation 3.3atakes the Q-type behavior. nomenclature of Q-, Pferrimagnetism h = 3.3 takes the Q-type behavior. The nomenclature of Q-, Pand N-types in ferrimagnetism has been labeled h = 3.3 takes the Q-type behavior. The nomenclature of Q-, Pand N-types in ferrimagnetism used (see [17]). In Figure 4B, the dashed curve represents the thermal variation of mT in the system used [17]). Figure represents the variation of mT in the has been (seeIn[17]). In4B, Figure 4B, the dashed curve represents thermal variation of system m in the with r(see =used 0.5, when the signs ofthe mA dashed and mB curve are changed from thethermal case the of (m A > 0.0, mB < 0.0) to theTcase with rwith = 0.5,r = when the signs mA and are changed from thefrom case of A > 0.0, mBA0.0) system 0.5, when theof signs of mmAB and mB are changed the(mcase of (m 0.0,tomthe 0.0) to B < case
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of A < 0.0, mBA> 0.0).4C, In Figure onhand, the other is shown thatbehavior the sameofbehavior of dashed curve in Figure can be 4B easily selecting the reasonable parameters of r and of h mT as the dashed curve 4B in Figure can obtained be easily by obtained by selecting the reasonable parameters (such as the curve labeled r = 1.5 and h = 4.0), even when m A > 0.0 and m B < 0.0. The curve clearly r and h (such as the curve labeled r = 1.5 and h = 4.0), even when mA > 0.0 and mB < 0.0. The curve exhibits a negative magnetization in the low temperature region. The curve labeled = 0.001(rand h= clearly exhibits a negative magnetization in the low temperature region. The curve (r labeled = 0.001 2.1) Figure exhibits the Q-type The results in Figure 4B,C are similar to those andin h= 2.1) in4C Figure 4C exhibits the behavior. Q-type behavior. The shown results shown in Figure 4B,C are similar to obtained in [14,15]. those obtained in [14,15].
(A)
(B)
(C) Figure Figure4. 4. The Thetemperature temperaturedependences dependencesof ofmmTTininthe thesystem systemA; A;the thevalue valueofofr risisfixed fixedatatrr==0.5 0.5(A,B). (A,B). The value of h is changed from h = 1.0 to h = 2.5 in (A) and from h = 3.1 to h = 3.3 in (B). The The value of h is changed from h = 1.0 to h = 2.5 in (A) and from h = 3.1 to h = 3.3 in (B). The dashed dashed curve curverepresents represents the the result result of of the the system system with with rr == 0.5 0.5 and and hh == 3.1, 3.1, when when the the signs signs of of m mAA and and m mBB are are changed from the case of (m A > 0.0, mB < 0.0) to the case of (mA < 0.0, mB > 0.0). The thermal variations changed from the case of (mA > 0.0, mB < 0.0) to the case of (mA < 0.0, mB > 0.0). The thermal variations of ofm mTTare aregiven, given,when whenthe thetwo twoset setvalues values (r (r and and h) h) are are selected selected as as (r (r == 0.001 0.001 and and hh ==2.1) 2.1)and and (r (r ==1.5 1.5and and hh==4.0) (C). 4.0) (C).
4. The Magnetic Properties of System B 4. The Magnetic Properties of System B From the structural aspect of the present two systems, system B seems to be more realistic than From the structural aspect of the present two systems, system B seems to be more realistic than system A [14,15], although the spin structure is more complicated than that of system A. In fact, for system A [14,15], although the spin structure is more complicated than that of system A. In fact, for the the inter-layer interaction, there exist two exchange interactions JA and JB in system B, while there inter-layer interaction, there exist two exchange interactions JA and JB in system B, while there exists exists only one exchange interaction JR in system A. Accordingly, let us define the following only one exchange interaction JR in system A. Accordingly, let us define the following parameters t, h, parameters t, h, r and s as r and s as t =t k=BkT/J, /J, r = JAA/J/Jand ands s= =JBJ/JB /J (19) BT/J,hh==Ω Ω/J, (19) Figure 5 shows the phase diagrams (tC versus h plot) of the system B, when the value of r (or s) is fixed at r = 0.5 in Figure 5A (or s = 0.5 in Figure 5B) and the value of s (or r) is changed from s = 0.0
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Nanomaterials 7, 256 the phase diagrams (tC versus h plot) of the system B, when the value of r 7(or of 12 Figure2017, 5 shows s) is fixed at r = 0.5 in Figure 5A (or s = 0.5 in Figure 5B) and the value of s (or r) is changed from = 2.0 from = 0.0 rto=r0.0 = 2.0). is seen from the behavior of tC curves in Figure sto=s0.0 to s(or = 2.0 (orr from to r As = 2.0). As is seenthese fromfigures, these figures, the behavior of tC curves in 5A is completely different from that in Figure 5B as well as the results of Figure 2 for system A. In Figure 5A is completely different from that in Figure 5B as well as the results of Figure 2 for system Figure 5A, the phenomenon has been obtained for the labeled r = 1.5r = (or1.5 bold A. In Figure 5A,reentrant the reentrant phenomenon has been obtained forcurve the curve labeled (or solid bold curve). However, the same phenomenon has not been obtained in Figure 5B. In order to clarify these solid curve). However, the same phenomenon has not been obtained in Figure 5B. In order to clarify differences between Figure 5A and 5B,the thehChCversus versuss s(or (orr)r)plot plothas hasbeen beenexamined examined Figure these differences between Figures 5AFigure and 5B, inin Figure 6. 6. The value of h C can be obtained from the results of Figure 5 as a critical value of h at which the t The value of hC can be obtained from the results of Figure 5 as a critical value of h at which the tCC curve reduces reduces to to zero. zero. Figure shows the = 0.5. curve Figure 6A 6A shows the hhCC versus versus ss plot, plot, when when the the value value of of rr is is fixed fixed at at rr = 0.5. Figure 6B 6B shows shows the the hhCC versus plot, when when the the value value of of ss is is fixed fixed at at ss == 0.5. 0.5. One One can can easily easily find find clear clear Figure versus rr plot, differences between betweenFigures Figure 6A 6A and and6B. Figure 6B. Asinshown Figure 7A,flat only the flat of hC in differences As shown Figurein 7A, only the region of hregion C in Figure 6A Figure 6A between h C = 1.0 and h C = 2.0 may exhibit the reentrant phenomenon, although, for the flat between hC = 1.0 and hC = 2.0 may exhibit the reentrant phenomenon, although, for the flat region of of h C > 2.0 of Figure 6A, the phenomenon has not been found, as shown in Figure 7B. In Figure hregion > 2.0 of Figure 6A, the phenomenon has not been found, as shown in Figure 7B. In Figure 6B, on the C 6B, onhand, the other hand,of the of hCsimply increases simply with theofincrease of r. A phenomenon other the value hCvalue increases with the increase r. A phenomenon similar to similar that of to that 7B of Figure 7B can be alsofor obtained A,value whenofthe of hCas is aplotted as aoffunction of Figure can be also obtained systemfor A, system when the hCvalue is plotted function r by using rthe byresults using the results of Figure 2. of Figure 2.
(A)
(B)
Figure 5. 5. The The phase phase diagrams diagrams (t (tC versus h plot) for the system B; The value of r is fixed at r = 0.5 and Figure C versus h plot) for the system B; The value of r is fixed at r = 0.5 and the value of s is changed from s = 0.0 to s 2.0(A). (A).The Thevalue valueofofs is s isfixed fixed = 0.5 and value the value of s is changed from s = 0.0 to s ==2.0 atat s =s 0.5 and thethe value of of r isr is changed from r = 0.0 to r = 2.0 (B). changed from r = 0.0 to r = 2.0 (B).
In Figure 8, the thermal variations of mT in the system with fixed values of r = 0.5 and s = 1.5 are In Figure 8, the thermal variations of m in the system with fixed values of r = 0.5 and s = 1.5 given by changing the value of h from h = T0.0 to h = 4.26, in order to clarify whether the above are given by changing the value of h from h = 0.0 to h = 4.26, in order to clarify whether the above predictions of phase diagrams are correct. In Figure 5A, the reentrant phenomenon has been obtained predictions of phase diagrams are correct. In Figure 5A, the reentrant phenomenon has been obtained in the region of 4.22 < h < 4.36 for the system with r = 0.5 and s = 1.5. In Figure 8E, the thermal variations in the region of 4.22 < h < 4.36 for the system with r = 0.5 and s = 1.5. In Figure 8E, the thermal of mT (solid curve), mA (dashed curve) and mB (dotted curve) are plotted by selecting the value of h = variations of mT (solid curve), mA (dashed curve) and mB (dotted curve) are plotted by selecting the 4.26 from the region of 4.22 < h < 4.36. They clearly exhibit the reentrant phenomenon, as predicted in value of h = 4.26 from the region of 4.22 < h < 4.36. They clearly exhibit the reentrant phenomenon, the phase diagrams. At this place, one should notice that the reentrant phenomenon is not a peculiar as predicted in the phase diagrams. At this place, one should notice that the reentrant phenomenon is behavior and it has often been found for some finite-size magnetic systems described by some not a peculiar behavior and it has often been found for some finite-size magnetic systems described by transverse Ising models, as discussed in some recent works [18–21]. As is seen from Figure 8A to some transverse Ising models, as discussed in some recent works [18–21]. As is seen from Figure 8A Figure 8D, the thermal variations of mT in the system may change a positive magnetization to a to Figure 8D, the thermal variations of mT in the system may change a positive magnetization to negative magnetization with the increase of h and may exhibit some characteristic ferrimagnetic a negative magnetization with the increase of h and may exhibit some characteristic ferrimagnetic behaviors, such as Q-, P-, N- and M-types, although some of them could not be classified by the behaviors, such as Q-, P-, N- and M-types, although some of them could not be classified by the nomenclature of ferrimagnetism [17]. In fact, the thermal variations of mT for the curves labeled h = nomenclature of ferrimagnetism [17]. In fact, the thermal variations of mT for the curves labeled h = 3.0 3.0 and h = 3.1 in Figure 8C are novel types. They start to decrease from the saturation magnetization and h = 3.1 in Figure 8C are novel types. They start to decrease from the saturation magnetization at at t = 0.0, exhibit a broad minimum, a broad maximum with the increase of t and then reduce to zero at their transition temperatures. In relation to the results of Figure 7B, the thermal variations of mT in the system with fixed value of r = 0.5 and s = 3.0 are plotted in Figure 9, selecting the three values of h. The curve labeled h = 3.0 shows the L-type behavior. The curve labeled h = 3.1 exhibits the N-type behavior. The curve labeled
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t = 0.0, exhibit a broad minimum, a broad maximum with the increase of t and then reduce to zero at their transition temperatures. In relation to the results of Figure 7B, the thermal variations of mT in the system with fixed value of r = 0.5 and s =7, 3.0 Nanomaterials 2017, 256 are plotted in Figure 9, selecting the three values of h. The curve labeled h88=of of3.0 12 Nanomaterials 2017, 7, 256 12 shows the L-type behavior. The curve labeled h = 3.1 exhibits the N-type behavior. The curve labeled 3.2 represents the M-type M-type behavior. behavior. The The thermal thermal variations variations of of m mTTT,,, corresponding corresponding to the results of 3.2 represents represents the corresponding to to the the results results of hh === 3.2 m T depicted Figure 4C for the system A with r = 0.5, are depicted in Figure 10, selecting the two depicted in Figure 4C for the system A with r = 0.5, are depicted in Figure 10, selecting the two sets mTT depicted in Figure 4C for the system A with r = 0.5, are depicted in Figure 10, selecting the two sets of ss and and insystem the system system B with with 0.5. results The results results shown in Figures Figures and 10 aresimilar also similar similar to of s and h in the B with r = 0.5. shown in Figures 9 and9910 are10 also to those sets of hh in the B rr ==The 0.5. The shown in and are also to those obtained in [14,15]. obtained in [14,15]. those obtained in [14,15]. Finally, the thermal thermal variations variationsof ofm mTTT in with fixed fixed values values of of sss = 0.5 and and rr === 1.5 1.5 are given Finally, the in the the system system with 1.5 are are given given Finally, the thermal variations of m with fixed values of == 0.5 in Figure Figure 11, 11, changing changing the the value value of 3.0 to to hh ===4.0. 4.0. These results can be compared with those of hh from from hh = 4.0. These Theseresults results can can be be compared compared with with those those in = 3.0 of Figure 8B,C for the system with = 0.5 and s = 1.5. r = 0.5 = 1.5. of Figure 8B,C for the system with r = 0.5 and s = 1.5.
(A) (A)
(B) (B)
Figure 6. 6. The variations variations of hC in the the system B; B; the variation of hC in the the system with with rr == 0.5 0.5 and and tt == 0.02 0.02 Figure Figure 6. The The variations of hCC in in the system system B; the variation of hCC in in the system system 0.5 = 0.02 C in the system is given, when the value of s is changed from s = 0.0 to s = 3.5 (A). The variation of h is is given, given, when when the the value value of s is changed from s = 0.0 to s = 3.5 (A). The variation of hCC in in the the system system with s = 0.5 and t = 0.01 is given, when the value of r is changed from r = 0.0 to r = 2.5 (B). with s == 0.5 0.5 and and tt ==0.01 0.01isisgiven, given,when when the the value value of of rr is is changed changed from from rr ==0.0 0.0to torr==2.5 2.5(B). (B).
(A) (A)
(B) (B)
Figure 7. 7. The phase phase diagrams (t (tC versus hh plot) plot) for the the system B B with rr == 0.5; 0.5; the value value of ss is is changed Figure Figure 7. The The phase diagrams diagrams (tCC versus versus h plot) for for the system system B with with r = 0.5; the the value of of s is changed changed from s = 1.0 to s = 2.0 (A). Three values of s are selected as s = 2.3, s = 2.8 and s = 3.3 (B). from s = 1.0 to s = 2.0 (A). Three values of s are selected as s = 2.3, s = 2.8 and s = 3.3 (B). from s = 1.0 to s = 2.0 (A). Three values of s are selected as s = 2.3, s = 2.8 and s = 3.3 (B).
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(A)
(B)
(C)
(D)
(E) Figure 8. The temperature dependences of mT in the system B with r = 0.5 and s = 1.5; the thermal Figure 8. The temperature dependences of mT in the system B with r = 0.5 and s = 1.5; the thermal variations of mT (solid curve), mA (dashed curve), mB (dotted curve) and qA (dot-dashed curve) are variations of mT (solid curve), mA (dashed curve), mB (dotted curve) and qA (dot-dashed curve) are depicted, when the value of h is given by h = 0.0 (A). The thermal variations of mT are given, changing depicted, when the value of h is given by h = 0.0 (A). The thermal variations of mT are given, changing the value of h from h = 2.0 to h = 3.1 (B). The thermal variations of mT are given by selecting the three the value of h from h = 2.0 to h = 3.1 (B). The thermal variations of mT are given by selecting the three values of h (C,D). The reentrant phenomena of magnetizations are presented by selecting the value of values of h (C,D). The reentrant phenomena of magnetizations are presented by selecting the value of h h as h = 4.26 (E). as h = 4.26 (E).
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Figure 9. The temperature dependencesofofmm in the the system system BBwith 0.50.5 and s = s3.0, when the three Figure 9. The temperature dependences withr =r = and = 3.0, when the three TTin values of h are selected. values of h are selected. Figure 9. The temperature dependences of mT in the system B with r = 0.5 and s = 3.0, when the three values of h are selected.
Figure 10. The thermal variations ofofmm withrr==0.5 0.5are are given, when set values Figure 10. The thermal variations inthe the system system with given, when the the twotwo set values T Tin (sh) and are selected = 0.2 and and (s = = 1.0 and (s and areh) selected as (sasvariations =(s0.2 and h h=m=2.5) and (s 1.0 and =3.1). 3.1). Figure 10. The thermal of T 2.5) in the system with rh=h=0.5 are given, when the two set values (s and h) are selected as (s = 0.2 and h = 2.5) and (s = 1.0 and h = 3.1).
(A) (A)
(B) (B)
Figure 11. The temperature dependences of mT in the system B with s = 0.5 and r = 1.5; the value of h is changed from h = 3.0 to h = 3.2 (A) and from h = 3.7 to h = 4.0 (B).
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5. Conclusions In this work, within the theoretical framework of the effective-field theory with correlations (EFT), we have investigated the phase diagrams and the magnetizations in the two mixed spin Ising systems consisting of two layers with honeycomb lattice. As shown in Figure 1, at first sight, their magnetic properties seem to be similar to each other, since they have the same intra-layer spin structure. As shown in Figures 5A, 6A, 7A and 8E, the big differences between the two systems can be seen in the appearance of the reentrant phenomenon and the characteristic behavior of hC in system B. In particular, one should also notice that such characteristic features have been obtained in system B, only when the value of JA is fixed and the value of JB is changed. In fact, as shown in Figure 5B, such characteristic phenomena have not been obtained, when the value of JB is fixed and the value of JA is changed. From the numerical examinations of mT given in Sections 3 and 4, all types of behaviors normally obtained for the thermal variation of mT in bulk ferrimagntic materials, such as the Q-, P-, N-, L- and M-types, can be found in the present two systems. As shown in Figure 8C, furthermore, novel types have been obtained for the thermal variation of mT in system B. In the present work, we have examined the magnetic properties of the two systems consisting of spin-1/2 and spin-1 atoms. Realistically, such systems should be constructed from spin-2 and spin-5/2 atoms, in order to compare with experimental data in molecular-based ferriagnetic materials [14,15]. The numerical results obtained from such systems are probably similar to the present ones obtained in Sections 3 and 4. Conflicts of Interest: The author declares no conflict of interest.
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