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The low temperature magnetic properties of the ferrimagnetic mixed spin Ising systems in the magnetic field
Physics Letters A 367 (2007) 483–488 www.elsevier.com/locate/pla
The low temperature magnetic properties of the ferrimagnetic mixed spin Ising systems in the magnetic field Cesur Ekiz Department of Physics, Adnan Menderes University, 09010 Aydın, Turkey Received 7 December 2006; received in revised form 26 February 2007; accepted 20 March 2007 Available online 24 March 2007 Communicated by R. Wu
Abstract The effects of external magnetic field on the low temperature magnetic properties of a ferrimagnetic mixed spin Ising systems, consisting of two magnetic sublattices A and B with spins SA = 2, 3/2, 1 and SB = 1/2, are considered within the scheme of exact recursion equations on the Bethe lattice. The magnetization plateaus and the low temperature magnetic phase diagram are investigated for the coordination number q = 4. The effect of single-ion anisotropy on the magnetic properties is also discussed by changing the values of exchange parameters in the system. © 2007 Elsevier B.V. All rights reserved. PACS: 75.50.Ee; 75.40.Cx; 75.40.Mg Keywords: Mixed spins; Magnetization plateau; Single-ion anisotropy
For many years, the two-sublattice mixed spin Ising models have attracted considerable attention. These systems are interesting for two main reasons: First, they have less translational symmetry than their single spin counterparts. Second, they are well adopted for the investigation of a certain simple kind of ferrimagnetism. A variety of mixed spin Ising systems consisting of two kinds of magnetic atom have been studied by means of different methods [1–8]. Many theoretical works dealt with two-dimensional mixed Ising systems consisting of spin-1/2 and spin-S ions. It can be found that most of these studies deal with the mixed Ising system without external magnetic field. On the other hand, low-dimensional magnetic systems have attracted considerable attention in recent years. The interesting theoretical results have been found, such as the Haldane conjecture [9], which affirms that the ground state of isotropic Heisenberg chains with integer spin are gapful, whereas half-integer spin ones are gapless. Related to the Haldane gaps are the magnetization plateaus, which are essentially macroscopic quantum phenomena in which the magnetization in an external magnetic field is quantized to fractions of the saturated value. A general condition of quantization of the magnetization was derived form the Lieb–Schultz–Mattis theorem for low-dimensional magnetic systems. Oshikawa, Yamanaka and Affleck [10] found a condition p(S − m) = integer, necessary for the appearance of the plateau in the magnetization curve of one-dimensional spin system. S is the magnitude of spin, m is the magnetization per site and p is the spatial period of the ground state, respectively. The magnetization plateaus have also been observed in one-dimensional S = 1/2 Heisenberg chains with diverse spatial structures such as dimerization, inhomogeneous magnetic field or a superlattice structure. The superlattice case with periodic boundary conditions was studied in Refs. [11] and [12]. These plateau structures have been observed not only in theoretical study but also have been obtained in experimental studies [13,14]. However, recently pure and mixed spin Ising systems in one dimension show the magnetic plateaus [15–17], i.e. quantization of the magnetization at the very low temperature or ground state of the system due to magnetic excitations, leading to the qualitatively same structures of the magnetization profiles of the Heisenberg and other systems. Thus, the study of this classical mixed spin
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.03.038
484
C. Ekiz / Physics Letters A 367 (2007) 483–488
Ising system can shed some light on the plateau mechanism and if it has purely quantum origin or can also depend on single-ion anisotropy. Up to now, the low temperature properties (magnetization curves, magnetic plateau and magnetic phase diagram) of the mixed spin Ising systems on the Bethe lattice have not been investigated. As far as we know, the most of the theoretical works have been restricted to the investigation of the systems with high temperature equilibrium properties, whereas the role of the single-ion anisotropy on the low temperature magnetic properties has not been examined yet. As is well known, when the temperature is low enough, the correlation and fluctuation are remarkable, which develop short-range order or long-range order under certain external conditions. Also, the single-ion anisotropy can induce new magnetic properties in the mixed spin Ising model. In the few decades, the Bethe lattice [18] has had great importance in the process of understanding a variety of phenomena, such as magnetic phase transitions, compensation phenomenon and tricritical phenomena. A Bethe lattice or Cayley tree is a connected cycle-free graph where each node is connected to q neighbors, where q is called the coordination number. It can be seen as a tree-like structure emanating from a central node, with all the nodes arranged in shells around the central one. The central node may be called the root or origin of the lattice. Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often exactly solvable. The solutions are related to the often used Bethe approximation for these systems. In this Letter, we focus on the effects of external magnetic field and single-ion anisotropy on the low temperature magnetic properties of the ferrimagnetic mixed-spin Ising system with single-ion anisotropy. We consider the ferrimagnetic mixed Ising spin-1/2 and spin-2 system with a single-ion anisotropy in the magnetic field, described by the Hamiltonian Si σ j + Δ Si2 − h Si + σj . H=J (1) ij
i
i
j
The Bethe lattice is composed of two different kinds of spins. One is occupied by spin-2 magnetic atoms at site i, while the other is occupied by spin-1/2 magnetic atoms at site j . J defines the exchange interaction between the spin at site i and its neighbor at site j . The analysis will be performed only for the simple case of the antiferromagnetic nearest-neighbor interaction. In all calculations we will take J = 1 in the ferrimagnetic case. The partition function in the ferrimagnetic case will be written as Z= exp(−βH) 2 exp β −J Si σ j − Δ Si + h Si + h σj . = (2) ij
(S,σ )
i
i
j
It is obvious that if the Bethe lattice is cut at the central site 0 with a spin S0 , the lattice splits into q identical disconnected branches. Each of these is a rooted tree at the central spin S0 . Thus the partition function for the central spin on the Bethe lattice can be written as
q Z= (3) exp β hS0 − ΔS02 gn (S0 ) , (S0 )
where S0 is the central spin value on the lattice, and gn (S0 ) is the partition function of an individual branch. Each branch can be cut on the site σ1 , which is the nearest to the central spin. Thus we can obtain the expressions for gn (S0 ) and gn−1 (σ1 ):
q−1 gn (S0 ) = (4) exp β(−J S0 σ1 + hσ1 ) gn−1 (σ1 ) σ1
and gn−1 (σ1 ) =
q−1 exp β −J σ1 S2 − ΔS22 + hS2 gn−2 (S2 ) .
(5)
S2
Let us introduce the following variables xn , yn , zn , tn (for spin-2 magnetic atoms) and wn−1 (for spin-1/2 magnetic atoms) respectively, xn =
gn (2) , gn (0)
yn =
gn (−2) , gn (0)
zn =
gn (1) , gn (0)
tn =
gn (−1) gn (0)
(6)
and wn−1 =
gn−1 (+1/2) . gn−1 (−1/2)
(7)
C. Ekiz / Physics Letters A 367 (2007) 483–488
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Since S0 can take the values ±2, ±1 and 0, one can obtain five differents gn (S0 ) for two possible values of σ1 . Thus
q−1 exp β(∓2J σ1 + hσ1 ) gn−1 (σ1 ) gn (±2) = σ1
q−1 1 1 1 1 q−1 = exp β ∓J + h + exp β ±J − h , gn−1 gn−1 − 2 2 2 2 gn (±1) =
(8)
q−1 exp β(∓J σ1 + hσ1 ) gn−1 (σ1 )
σ1
q−1 1 1 1 1 1 1 q−1 = exp β ∓ J + h + exp β ± J − h , gn−1 gn−1 − 2 2 2 2 2 2 and gn (0) =
(9)
q−1 exp{βhσ1 } gn−1 (σ1 )
σ1
q−1 1 1 1 1 q−1 = exp βh gn−1 + exp − βh gn−1 − . 2 2 2 2 On the other hand, since σ1 can take the values ±1/2, one can obtain two differents gn−1 (σ1 ) for two possible values of σ1 :
q−1 1 1 gn−1 ± exp β ∓ J S2 − ΔS22 + hS2 = gn−2 (S2 ) 2 2 S2
q−1
q−1 = exp β(∓J − 4Δ + 2h) gn−2 (2) + exp β(±J − 4Δ − 2h) gn−2 (−2)
q−1 1 1 1 q−1 + exp β ∓ J − Δ + h + exp β ± J − Δ − h gn−2 (1) gn−2 − 2 2 2
q−1 + gn−2 (0) .
(10)
(11)
Thus we can obtain a set of five recursion relations from which the sublattice magnetizations can be found. The recursion relations are found by substituting Eqs. (8)–(11) into Eqs. (6)–(7) and are given in Appendix A. The values of xn , yn , zn , tn and wn−1 have no direct physical meaning, but one can express the magnetizations and other thermodynamic quantities of interest in terms of xn , yn , zn , tn and wn−1 . Thus we can say that in the thermodynamic limit (n → ∞) the above variables determine the states of the system. The sublattice magnetizations of the mixed spin-1/2 and spin-2 Ising ferrimagnetic system are expressed by
q −1 mA = ZA S0 exp{βhS0 } gn (S0 ) =
S0 q q q q β(2h−4Δ) xn − 2eβh(−2h−4Δ) yn + eβ(h−Δ) zn − eβ(−h−Δ) tn 2e , q q q q eβ(2h−4Δ) xn + eβ(−2h−4Δ) yn + eβ(h−Δ) zn + eβ(−h−Δ) tn + 1
(12)
and mB = ZB−1
σ1
σ1 e
βhσ1
1 1 q
q 1 e 2 βh wn−1 − e− 2 βh gn−1 (σ1 ) = , 2 e 12 βh w q + e− 12 βh n−1
(13)
where ZA and ZB are the partition functions of the sublattices A and B, respectively. Now, we can examine the effects of single-ion anisotropy and magnetic field on the low temperature magnetic properties of the mixed spin-1/2 and spin-S (S = 1, 3/2 and 2) Ising spin system. Fig. 1(a) and (b) present the magnetic field dependence of sublattice magnetizations, mA and mB , for given T = 0.01, 0.8 and 2 (unit by J ) by selecting zero single-ion anisotropy. At the low temperature (T = 0.01), near the ground state, as the magnetic field changes from negative to positive values, the magnetization jumps from −1 to 1 at the saturated magnetic field hs (Fig. 1(a)). Also, when Δ 0, it is absolutely believed that there is no plateau in the present system. On the other hand, for high temperatures such as 0.8 and 2, the sublattice magnetization mA saturates more slowly with rising temperatures. Fig. 1(b) shows the sublattice magnetization mB in the absence of single-ion anisotropy. From this figure, we can see that the magnetization mB jumps from 0.5 to −0.5 with increasing magnetic field. Thus, for Δ 0, the system is isotropic and gapless. In Fig. 2(a)–(b), the single-ion anisotropy dependences of sublattice magnetizations, mA and mB , are given at fixed temperature. At T = 0.01, the sublattice magnetizations change to its new values from previous values, seen in Fig. 2(a) and (b). The low temperature just makes the transition smoother and quicker.
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C. Ekiz / Physics Letters A 367 (2007) 483–488
(a)
(b)
Fig. 1. The sublattice magnetizations, mA and mB , as a function of the magnetic field for the ferrimagnetic mixed spin-1 and spin-1/2 Ising system at different temperatures when Δ = 0.
(a)
(b)
Fig. 2. The sublattice magnetizations mA and mB , as a function of the single-ion anisotropy at h = 0.5.
(a)
(b)
Fig. 3. The sublattice magnetizations mA and mB , as a function of the magnetic field in the existence of magnetic plateau at T = 0.01. The solid and dotted lines denote the central spin value on the Bethe lattice S = 1 and 3/2, respectively.
In order to investigate the effect of single-ion anisotropy on the magnetic properties, the field dependences of sublattice magnetizations of the system are shown for the central spin S = 1 and 3/2, respectively, seen in Fig. 3. From this figure, we can see clearly an intermediate plateau at the critical magnetic field hc for Δ = 1.5, seen in Fig. 3(a). Three step-like plateaus (mA = −1, 0 and 1) are shown by the three parallel lines in the case of spin S = 1 and 3/2. The width of the plateau decreases as the temperature is raised, and the plateau disappears at higher temperatures. For the central spin S = 1, we find that for Δ < Δc1 = 0.5 the system is critical and the magnetization plateau does not exist in the magnetization process. In the range of Δc1 < Δ < Δc2 = 2.0, the system is gapful and the magnetization plateau appears in the magnetization process. However, since the sublattice B does not include single-ion anisotropy in the Hamiltonian, the magnetization versus magnetic field has not magnetic plateau in the magnetization process. In Fig. 3(b), the interesting behavior of sublattice magnetization mB at the critical magnetic field value is observed in the
C. Ekiz / Physics Letters A 367 (2007) 483–488
Fig. 4. Magnetization mA plotted as a function of the magnetic field h for Δ = 1.5. hc1 and hc2 are the critical magnetic fields and hs is the saturation magnetic field.
487
Fig. 5. The low temperature magnetic phase diagram of mixed ferrimagnetic spin-1/2 and spin-1 Ising system in the plane of magnetic field h versus single-ion anisotropy Δ.
case of spin value σ = 1/2. This behavior is possible since the single-ion anisotropy is necessary for a plateau mechanism. However this behavior is not observed at the higher spin values such as S = 3/2. Thus, the single-ion anisotropy is indispensable to appear the plateaus in the mixed Ising system, which are validated both in the cases of pure spin-1 and other spin-S values. Fig. 4 represents the magnetic field dependence of sublattice magnetization mA for mixed spin-2 and spin-1/2 Ising ferrimagnetic system by selecting T = 0.01, 1.0 and 2.0, respectively and single-ion anisotropy Δ = 1.5. In order to study low temperature properties, we use the mixed spin ferrimagnetic system under an external magnetic field at sufficiently low temperatures, near the ground state. For sufficiently low temperatures such as T = 0.01 near the ground state, the four step-like plateaus (mB = −2, −1, 0 and 2) are observed by the four parallel solid lines. The existence of the magnetic plateau indicates that the mixed spin ferrimagnetic Ising system has a gap mechanism due to the single-ion anisotropy. The transitions between magnetic plateaus take place at critical magnetic field values called as hc1 and hc2 . The last plateau corresponds to the saturation magnetization of the system called as hs . In the system, the positive single-ion anisotropy is an important condition for the appearance of the plateau at m = −1 and 0. On the other hand, we again have no obtained a magnetization plateau for negative and zero single-ion anisotropies. For the above parameters, as the temperature is increased, the magnetic plateaus disappear in the high temperatures. Finally, Fig. 5 shows the low temperature magnetic phase diagram for sublattice magnetization mA on the magnetic field h versus single-ion anisotropy plane. The lower critical values correspond to the critical magnetic fields for the plateau existence and the upper ones to the saturation fields. It suggests that the magnetic plateau in the mixed spin-1/2 and spin-1 Ising system appears for 0.5 < Δ/J < 2.0 and the length of the plateau becomes longer as the magnitude of single-ion anisotropy larger until it disappears when Δ > 2.0. In conclusion, in this Letter the low temperature magnetization plateaus and magnetic plateau phase diagram were studied because the effect of the single-ion anisotropy on the magnetic properties of mixed spin Ising system is unknown. With the iteration technique based on the exact recursion relations on the Bethe lattice, we determined boundaries between the plateau and the noplateau regions. The Bethe lattice study showed that the anisotropic ferrimagnetic mixed spin-1 and spin-1/2 Ising system has the magnetization plateau for Δ > 0.5. With the consideration of field and anisotropy at the same time, the magnetization curves reach saturation more slowly in the high temperature region. Unfortunately, we have not compared our theoretical results with experimental results of real ferrimagnetic materials since there are no enough experimental studies for these systems in the literature. However, it can be understood from the present theoretical results that the magnetic properties of the mixed spin ferrimagnetic systems are the similar. Appendix A The explicit formulations of the recursion relations in the ferrimagnetic case are given as SA = 3/2 case: 3
xn =
1
3
1
1
1
eβ(− 4 J + 2 h) wn−1 + eβ( 4 J − 2 h) 1
q−1
1
q−1
3
9
eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h)
yn =
,
3
1
q−1
1
1
q−1
3
1
1
1
eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h) eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h)
1
zn =
,
1
wn−1 =
3
9
3
q−1
3
9
3
q−1
1
1
1
1
1
1
1
1
1
eβ(− 4 J − 4 Δ+ 2 h) xn−2 + eβ( 4 J − 4 Δ− 2 h) yn−2 + eβ(− 4 J − 4 Δ+ 2 h) zn−2 + eβ( 4 J − 4 Δ− 2 h) 3
9
3
q−1
1
1
1
q−1
eβ( 4 J − 4 Δ+ 2 h) xn−2 + eβ(− 4 J − 4 Δ− 2 h) yn−2 + eβ( 4 J − 4 Δ+ 2 h) zn−2 + eβ(− 4 J − 4 Δ− 2 h) q−1
q−1
1
q−1
1
1
1
1
eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h)
and 3
1
eβ(− 4 J + 2 h) wn−1 + eβ( 4 J − 2 h)
.
q−1
,
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C. Ekiz / Physics Letters A 367 (2007) 483–488
SA = 1 case: 1
xn =
1
1
1
1
1
eβ(− 2 J + 2 h) wn−1 + eβ( 2 J − 2 h) 1
1
q−1
eβ( 2 J + 2 h) wn−1 + eβ(− 2 J − 2 h) q−1
,
yn =
1
1
q−1
1
1
q−1
1
1
1
1
eβ( 2 J + 2 h) wn−1 + eβ(− 2 J − 2 h) eβ( 2 J + 2 h) wn−1 + eβ(− 2 J − 2 h)
and 1
wn−1 =
1
q−1
1
q−1
eβ(− 2 J −Δ+h) xn−2 + eβ( 2 J −Δ−h) yn−2 + 1 1
q−1
eβ( 2 J −Δ+h) xn−2 + eβ(− 2 J −Δ−h) yn−2 + 1 q−1
.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
S.L. Yan, C.Z. Yang, Phys. Rev. B 57 (1998) 3512. X. Zhang, X.M. Kong, Physica A 369 (2006) 589. C. Ekiz, R. Erdem, Phys. Lett. A 352 (2006) 291. S.L. Yan, C.Z. Yang, Z. Phys. B 103 (1997) 93. S.L. Yan, C.Z. Yang, Phys. Status Solidi B 208 (1998) 151. G.Z. Wei, Y.-Q. Liang, Q. Zhang, Z.H. Xin, J. Magn. Magn. Mater. 271 (2004) 246. W. Jiang, G.Z. Wei, Z.H. Xin, Physica A 23 (2001) 455. T. Kaneyoshi, J. Appl. Phys. 64 (1988) 254. F.D.M. Haldane, Phys. Lett. A 93 (1983) 464. M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 78 (1997) 1984. J. Silva-Valencia, E. Miranda, Phys. Rev. B 65 (2002) 24443. J. Silva-Valencia, R. Franco, Physica B 384 (2006) 2005. Y. Narumi, et al., Physica B 246 (1998) 5090. Y. Narumi, et al., J. Magn. Magn. Mater. 272 (2004) 878. X.Y. Chen, Q. Jiang, W.Z. Shen, C.G. Zhong, J. Magn. Magn. Mater. 262 (2003) 258. H.H. Fu, K.L. Yao, Z.L. Liu, J. Magn. Magn. Mater. 305 (2006) 253. X.Y. Chen, Q. Jiang, Y.Z. Wu, Solid. State Commun. 121 (2002) 641. R.J. Baxter, Exacly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.
The low temperature magnetic properties of the ferrimagnetic mixed spin Ising systems in the magnetic field Cesur Ekiz Department of Physics, Adnan Menderes University, 09010 Aydın, Turkey Received 7 December 2006; received in revised form 26 February 2007; accepted 20 March 2007 Available online 24 March 2007 Communicated by R. Wu
Abstract The effects of external magnetic field on the low temperature magnetic properties of a ferrimagnetic mixed spin Ising systems, consisting of two magnetic sublattices A and B with spins SA = 2, 3/2, 1 and SB = 1/2, are considered within the scheme of exact recursion equations on the Bethe lattice. The magnetization plateaus and the low temperature magnetic phase diagram are investigated for the coordination number q = 4. The effect of single-ion anisotropy on the magnetic properties is also discussed by changing the values of exchange parameters in the system. © 2007 Elsevier B.V. All rights reserved. PACS: 75.50.Ee; 75.40.Cx; 75.40.Mg Keywords: Mixed spins; Magnetization plateau; Single-ion anisotropy
For many years, the two-sublattice mixed spin Ising models have attracted considerable attention. These systems are interesting for two main reasons: First, they have less translational symmetry than their single spin counterparts. Second, they are well adopted for the investigation of a certain simple kind of ferrimagnetism. A variety of mixed spin Ising systems consisting of two kinds of magnetic atom have been studied by means of different methods [1–8]. Many theoretical works dealt with two-dimensional mixed Ising systems consisting of spin-1/2 and spin-S ions. It can be found that most of these studies deal with the mixed Ising system without external magnetic field. On the other hand, low-dimensional magnetic systems have attracted considerable attention in recent years. The interesting theoretical results have been found, such as the Haldane conjecture [9], which affirms that the ground state of isotropic Heisenberg chains with integer spin are gapful, whereas half-integer spin ones are gapless. Related to the Haldane gaps are the magnetization plateaus, which are essentially macroscopic quantum phenomena in which the magnetization in an external magnetic field is quantized to fractions of the saturated value. A general condition of quantization of the magnetization was derived form the Lieb–Schultz–Mattis theorem for low-dimensional magnetic systems. Oshikawa, Yamanaka and Affleck [10] found a condition p(S − m) = integer, necessary for the appearance of the plateau in the magnetization curve of one-dimensional spin system. S is the magnitude of spin, m is the magnetization per site and p is the spatial period of the ground state, respectively. The magnetization plateaus have also been observed in one-dimensional S = 1/2 Heisenberg chains with diverse spatial structures such as dimerization, inhomogeneous magnetic field or a superlattice structure. The superlattice case with periodic boundary conditions was studied in Refs. [11] and [12]. These plateau structures have been observed not only in theoretical study but also have been obtained in experimental studies [13,14]. However, recently pure and mixed spin Ising systems in one dimension show the magnetic plateaus [15–17], i.e. quantization of the magnetization at the very low temperature or ground state of the system due to magnetic excitations, leading to the qualitatively same structures of the magnetization profiles of the Heisenberg and other systems. Thus, the study of this classical mixed spin
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.03.038
484
C. Ekiz / Physics Letters A 367 (2007) 483–488
Ising system can shed some light on the plateau mechanism and if it has purely quantum origin or can also depend on single-ion anisotropy. Up to now, the low temperature properties (magnetization curves, magnetic plateau and magnetic phase diagram) of the mixed spin Ising systems on the Bethe lattice have not been investigated. As far as we know, the most of the theoretical works have been restricted to the investigation of the systems with high temperature equilibrium properties, whereas the role of the single-ion anisotropy on the low temperature magnetic properties has not been examined yet. As is well known, when the temperature is low enough, the correlation and fluctuation are remarkable, which develop short-range order or long-range order under certain external conditions. Also, the single-ion anisotropy can induce new magnetic properties in the mixed spin Ising model. In the few decades, the Bethe lattice [18] has had great importance in the process of understanding a variety of phenomena, such as magnetic phase transitions, compensation phenomenon and tricritical phenomena. A Bethe lattice or Cayley tree is a connected cycle-free graph where each node is connected to q neighbors, where q is called the coordination number. It can be seen as a tree-like structure emanating from a central node, with all the nodes arranged in shells around the central one. The central node may be called the root or origin of the lattice. Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often exactly solvable. The solutions are related to the often used Bethe approximation for these systems. In this Letter, we focus on the effects of external magnetic field and single-ion anisotropy on the low temperature magnetic properties of the ferrimagnetic mixed-spin Ising system with single-ion anisotropy. We consider the ferrimagnetic mixed Ising spin-1/2 and spin-2 system with a single-ion anisotropy in the magnetic field, described by the Hamiltonian Si σ j + Δ Si2 − h Si + σj . H=J (1) ij
i
i
j
The Bethe lattice is composed of two different kinds of spins. One is occupied by spin-2 magnetic atoms at site i, while the other is occupied by spin-1/2 magnetic atoms at site j . J defines the exchange interaction between the spin at site i and its neighbor at site j . The analysis will be performed only for the simple case of the antiferromagnetic nearest-neighbor interaction. In all calculations we will take J = 1 in the ferrimagnetic case. The partition function in the ferrimagnetic case will be written as Z= exp(−βH) 2 exp β −J Si σ j − Δ Si + h Si + h σj . = (2) ij
(S,σ )
i
i
j
It is obvious that if the Bethe lattice is cut at the central site 0 with a spin S0 , the lattice splits into q identical disconnected branches. Each of these is a rooted tree at the central spin S0 . Thus the partition function for the central spin on the Bethe lattice can be written as
q Z= (3) exp β hS0 − ΔS02 gn (S0 ) , (S0 )
where S0 is the central spin value on the lattice, and gn (S0 ) is the partition function of an individual branch. Each branch can be cut on the site σ1 , which is the nearest to the central spin. Thus we can obtain the expressions for gn (S0 ) and gn−1 (σ1 ):
q−1 gn (S0 ) = (4) exp β(−J S0 σ1 + hσ1 ) gn−1 (σ1 ) σ1
and gn−1 (σ1 ) =
q−1 exp β −J σ1 S2 − ΔS22 + hS2 gn−2 (S2 ) .
(5)
S2
Let us introduce the following variables xn , yn , zn , tn (for spin-2 magnetic atoms) and wn−1 (for spin-1/2 magnetic atoms) respectively, xn =
gn (2) , gn (0)
yn =
gn (−2) , gn (0)
zn =
gn (1) , gn (0)
tn =
gn (−1) gn (0)
(6)
and wn−1 =
gn−1 (+1/2) . gn−1 (−1/2)
(7)
C. Ekiz / Physics Letters A 367 (2007) 483–488
485
Since S0 can take the values ±2, ±1 and 0, one can obtain five differents gn (S0 ) for two possible values of σ1 . Thus
q−1 exp β(∓2J σ1 + hσ1 ) gn−1 (σ1 ) gn (±2) = σ1
q−1 1 1 1 1 q−1 = exp β ∓J + h + exp β ±J − h , gn−1 gn−1 − 2 2 2 2 gn (±1) =
(8)
q−1 exp β(∓J σ1 + hσ1 ) gn−1 (σ1 )
σ1
q−1 1 1 1 1 1 1 q−1 = exp β ∓ J + h + exp β ± J − h , gn−1 gn−1 − 2 2 2 2 2 2 and gn (0) =
(9)
q−1 exp{βhσ1 } gn−1 (σ1 )
σ1
q−1 1 1 1 1 q−1 = exp βh gn−1 + exp − βh gn−1 − . 2 2 2 2 On the other hand, since σ1 can take the values ±1/2, one can obtain two differents gn−1 (σ1 ) for two possible values of σ1 :
q−1 1 1 gn−1 ± exp β ∓ J S2 − ΔS22 + hS2 = gn−2 (S2 ) 2 2 S2
q−1
q−1 = exp β(∓J − 4Δ + 2h) gn−2 (2) + exp β(±J − 4Δ − 2h) gn−2 (−2)
q−1 1 1 1 q−1 + exp β ∓ J − Δ + h + exp β ± J − Δ − h gn−2 (1) gn−2 − 2 2 2
q−1 + gn−2 (0) .
(10)
(11)
Thus we can obtain a set of five recursion relations from which the sublattice magnetizations can be found. The recursion relations are found by substituting Eqs. (8)–(11) into Eqs. (6)–(7) and are given in Appendix A. The values of xn , yn , zn , tn and wn−1 have no direct physical meaning, but one can express the magnetizations and other thermodynamic quantities of interest in terms of xn , yn , zn , tn and wn−1 . Thus we can say that in the thermodynamic limit (n → ∞) the above variables determine the states of the system. The sublattice magnetizations of the mixed spin-1/2 and spin-2 Ising ferrimagnetic system are expressed by
q −1 mA = ZA S0 exp{βhS0 } gn (S0 ) =
S0 q q q q β(2h−4Δ) xn − 2eβh(−2h−4Δ) yn + eβ(h−Δ) zn − eβ(−h−Δ) tn 2e , q q q q eβ(2h−4Δ) xn + eβ(−2h−4Δ) yn + eβ(h−Δ) zn + eβ(−h−Δ) tn + 1
(12)
and mB = ZB−1
σ1
σ1 e
βhσ1
1 1 q
q 1 e 2 βh wn−1 − e− 2 βh gn−1 (σ1 ) = , 2 e 12 βh w q + e− 12 βh n−1
(13)
where ZA and ZB are the partition functions of the sublattices A and B, respectively. Now, we can examine the effects of single-ion anisotropy and magnetic field on the low temperature magnetic properties of the mixed spin-1/2 and spin-S (S = 1, 3/2 and 2) Ising spin system. Fig. 1(a) and (b) present the magnetic field dependence of sublattice magnetizations, mA and mB , for given T = 0.01, 0.8 and 2 (unit by J ) by selecting zero single-ion anisotropy. At the low temperature (T = 0.01), near the ground state, as the magnetic field changes from negative to positive values, the magnetization jumps from −1 to 1 at the saturated magnetic field hs (Fig. 1(a)). Also, when Δ 0, it is absolutely believed that there is no plateau in the present system. On the other hand, for high temperatures such as 0.8 and 2, the sublattice magnetization mA saturates more slowly with rising temperatures. Fig. 1(b) shows the sublattice magnetization mB in the absence of single-ion anisotropy. From this figure, we can see that the magnetization mB jumps from 0.5 to −0.5 with increasing magnetic field. Thus, for Δ 0, the system is isotropic and gapless. In Fig. 2(a)–(b), the single-ion anisotropy dependences of sublattice magnetizations, mA and mB , are given at fixed temperature. At T = 0.01, the sublattice magnetizations change to its new values from previous values, seen in Fig. 2(a) and (b). The low temperature just makes the transition smoother and quicker.
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C. Ekiz / Physics Letters A 367 (2007) 483–488
(a)
(b)
Fig. 1. The sublattice magnetizations, mA and mB , as a function of the magnetic field for the ferrimagnetic mixed spin-1 and spin-1/2 Ising system at different temperatures when Δ = 0.
(a)
(b)
Fig. 2. The sublattice magnetizations mA and mB , as a function of the single-ion anisotropy at h = 0.5.
(a)
(b)
Fig. 3. The sublattice magnetizations mA and mB , as a function of the magnetic field in the existence of magnetic plateau at T = 0.01. The solid and dotted lines denote the central spin value on the Bethe lattice S = 1 and 3/2, respectively.
In order to investigate the effect of single-ion anisotropy on the magnetic properties, the field dependences of sublattice magnetizations of the system are shown for the central spin S = 1 and 3/2, respectively, seen in Fig. 3. From this figure, we can see clearly an intermediate plateau at the critical magnetic field hc for Δ = 1.5, seen in Fig. 3(a). Three step-like plateaus (mA = −1, 0 and 1) are shown by the three parallel lines in the case of spin S = 1 and 3/2. The width of the plateau decreases as the temperature is raised, and the plateau disappears at higher temperatures. For the central spin S = 1, we find that for Δ < Δc1 = 0.5 the system is critical and the magnetization plateau does not exist in the magnetization process. In the range of Δc1 < Δ < Δc2 = 2.0, the system is gapful and the magnetization plateau appears in the magnetization process. However, since the sublattice B does not include single-ion anisotropy in the Hamiltonian, the magnetization versus magnetic field has not magnetic plateau in the magnetization process. In Fig. 3(b), the interesting behavior of sublattice magnetization mB at the critical magnetic field value is observed in the
C. Ekiz / Physics Letters A 367 (2007) 483–488
Fig. 4. Magnetization mA plotted as a function of the magnetic field h for Δ = 1.5. hc1 and hc2 are the critical magnetic fields and hs is the saturation magnetic field.
487
Fig. 5. The low temperature magnetic phase diagram of mixed ferrimagnetic spin-1/2 and spin-1 Ising system in the plane of magnetic field h versus single-ion anisotropy Δ.
case of spin value σ = 1/2. This behavior is possible since the single-ion anisotropy is necessary for a plateau mechanism. However this behavior is not observed at the higher spin values such as S = 3/2. Thus, the single-ion anisotropy is indispensable to appear the plateaus in the mixed Ising system, which are validated both in the cases of pure spin-1 and other spin-S values. Fig. 4 represents the magnetic field dependence of sublattice magnetization mA for mixed spin-2 and spin-1/2 Ising ferrimagnetic system by selecting T = 0.01, 1.0 and 2.0, respectively and single-ion anisotropy Δ = 1.5. In order to study low temperature properties, we use the mixed spin ferrimagnetic system under an external magnetic field at sufficiently low temperatures, near the ground state. For sufficiently low temperatures such as T = 0.01 near the ground state, the four step-like plateaus (mB = −2, −1, 0 and 2) are observed by the four parallel solid lines. The existence of the magnetic plateau indicates that the mixed spin ferrimagnetic Ising system has a gap mechanism due to the single-ion anisotropy. The transitions between magnetic plateaus take place at critical magnetic field values called as hc1 and hc2 . The last plateau corresponds to the saturation magnetization of the system called as hs . In the system, the positive single-ion anisotropy is an important condition for the appearance of the plateau at m = −1 and 0. On the other hand, we again have no obtained a magnetization plateau for negative and zero single-ion anisotropies. For the above parameters, as the temperature is increased, the magnetic plateaus disappear in the high temperatures. Finally, Fig. 5 shows the low temperature magnetic phase diagram for sublattice magnetization mA on the magnetic field h versus single-ion anisotropy plane. The lower critical values correspond to the critical magnetic fields for the plateau existence and the upper ones to the saturation fields. It suggests that the magnetic plateau in the mixed spin-1/2 and spin-1 Ising system appears for 0.5 < Δ/J < 2.0 and the length of the plateau becomes longer as the magnitude of single-ion anisotropy larger until it disappears when Δ > 2.0. In conclusion, in this Letter the low temperature magnetization plateaus and magnetic plateau phase diagram were studied because the effect of the single-ion anisotropy on the magnetic properties of mixed spin Ising system is unknown. With the iteration technique based on the exact recursion relations on the Bethe lattice, we determined boundaries between the plateau and the noplateau regions. The Bethe lattice study showed that the anisotropic ferrimagnetic mixed spin-1 and spin-1/2 Ising system has the magnetization plateau for Δ > 0.5. With the consideration of field and anisotropy at the same time, the magnetization curves reach saturation more slowly in the high temperature region. Unfortunately, we have not compared our theoretical results with experimental results of real ferrimagnetic materials since there are no enough experimental studies for these systems in the literature. However, it can be understood from the present theoretical results that the magnetic properties of the mixed spin ferrimagnetic systems are the similar. Appendix A The explicit formulations of the recursion relations in the ferrimagnetic case are given as SA = 3/2 case: 3
xn =
1
3
1
1
1
eβ(− 4 J + 2 h) wn−1 + eβ( 4 J − 2 h) 1
q−1
1
q−1
3
9
eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h)
yn =
,
3
1
q−1
1
1
q−1
3
1
1
1
eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h) eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h)
1
zn =
,
1
wn−1 =
3
9
3
q−1
3
9
3
q−1
1
1
1
1
1
1
1
1
1
eβ(− 4 J − 4 Δ+ 2 h) xn−2 + eβ( 4 J − 4 Δ− 2 h) yn−2 + eβ(− 4 J − 4 Δ+ 2 h) zn−2 + eβ( 4 J − 4 Δ− 2 h) 3
9
3
q−1
1
1
1
q−1
eβ( 4 J − 4 Δ+ 2 h) xn−2 + eβ(− 4 J − 4 Δ− 2 h) yn−2 + eβ( 4 J − 4 Δ+ 2 h) zn−2 + eβ(− 4 J − 4 Δ− 2 h) q−1
q−1
1
q−1
1
1
1
1
eβ( 4 J + 2 h) wn−1 + eβ(− 4 J − 2 h)
and 3
1
eβ(− 4 J + 2 h) wn−1 + eβ( 4 J − 2 h)
.
q−1
,
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C. Ekiz / Physics Letters A 367 (2007) 483–488
SA = 1 case: 1
xn =
1
1
1
1
1
eβ(− 2 J + 2 h) wn−1 + eβ( 2 J − 2 h) 1
1
q−1
eβ( 2 J + 2 h) wn−1 + eβ(− 2 J − 2 h) q−1
,
yn =
1
1
q−1
1
1
q−1
1
1
1
1
eβ( 2 J + 2 h) wn−1 + eβ(− 2 J − 2 h) eβ( 2 J + 2 h) wn−1 + eβ(− 2 J − 2 h)
and 1
wn−1 =
1
q−1
1
q−1
eβ(− 2 J −Δ+h) xn−2 + eβ( 2 J −Δ−h) yn−2 + 1 1
q−1
eβ( 2 J −Δ+h) xn−2 + eβ(− 2 J −Δ−h) yn−2 + 1 q−1
.
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