Author's Accepted Manuscript

The effects of next nearest-neighbor exchangeinteraction on the magnetic properties in theone-dimensional Ising system

Numan Şarlı

PII: S1386-9477(14)00244-6DOI: http://dx.doi.org/10.1016/j.physe.2014.06.028Reference: PHYSE11654

To appear in: Physica E

Received date: 29 May 2014Revised date: 26 June 2014Accepted date: 29 June 2014

Cite this article as: Numan Şarlı, The effects of next nearest-neighborexchange interaction on the magnetic properties in the one-dimensionalIsing system, Physica E, http://dx.doi.org/10.1016/j.physe.2014.06.028

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1

The effects of next nearest-neighbor exchange interaction on the magnetic properties in the one-dimensional Ising system

Numan �arl�1

1Institute of Science, Erciyes University, 38039 Kayseri, Turkey

ABSTRACT

In this work, we investigate the effects of next nearest-neighbor exchange interaction (J2) on the magnetic properties in the one-dimensional Ising system (1DIS) by using Kaneyoshi approach within the effective field theory for both ferromagnetic and antiferromagnetic case. It is found that the magnetic properties strongly depend on the J2 in the 1DIS. The critical temperature of the 1DIS decreases as the J2 decreases and it has almost a stable value (Tc=0.571) when the J2 approaches to zero (but not zero). The coercive field point, remanence magnetization and the area of the hysteresis loop of the 1DIS decrease and the hysteresis curves of the 1DIS exhibit paramagnetic behaviors as the J2 decreases. The susceptibility of the 1DIS has a distinct peak at Tc. The magnetizations of the 1DIS are m1=m2=MT=1 in the ferromagnetic case, but they are m1=-1, m2=1 and MT=0 in the antiferromagnetic case at T=0. Moreover, the hysteresis curves of the m1 and m2 of the AFM 1DIS exhibit elliptical hysteresis behaviors with two distinct loops similar to the glasses for small next nearest-neighbor exchange interaction values (J2=0.0001, 0.001 and 0.01) and they have two different coercive field points far away from H=0.000.

KEYWORDS

One-dimensional Ising system; Phase transition; Next nearest-neighbor; Effective field theory

������������������������������������������������������������1Corresponding author; e-mail: numansarli82@gmail.com, phone:90(352)4374938-33139, fax:90(352)4374931

2

1. Introduction One-dimensional systems have attracted a great deal of interest in theoretical and experimental physics. Because, their analitical calculations are generally much more amenable and easier than higher-dimensional ones [1, 2]. The researchers especially focus on whether the one-dimensional systems have phase transition or not. Many models have been carried out to understand this main query in the one-dimensional systems. Examples of one-dimensional models with phase transitions; the one-dimensional system of the KH2PO4 (KPD) is firstly proposed by Kittel and he found that one-dimensional system of the KPD undergoes a first-order phase transition at T�0 [3]. Chui-Week’s model with the infinite size transfer matrix which indicates the existence of phase transition in the one-dimensional systems [4]. Burkhardt’s model with a transfer operator [5]. Dauxois-Peyrard’s model proposed for DNA denatuation which has evidences for phase transition in the one-dimensional systems [6, 7] and the results of which are in good agreement with the experimental results of short chains obtained by Campa and Giansanti [8]. However, many works reported in the literature suggested that the one-dimensional models have no phase transitions. Such as, Van Hove’s theorem for homogeneous fluid-like models by Van Hove [9]. Generalized Van Hove’s theorem for lattice models by Ruelle [10] and Dyson [11], and the most famous argument is given by Landau and Lifshitz which reinforces that there is no phase transition in one-dimension for T�0 [12]. On the other hand, effective field theory is a very successfull method for investigations of the magnetic properties of the nanostructures. Such as, magnetic properties of the Ising nanotube, Ising nanowire, Ising nanoparticle and Ising thin film were firstly studied by Kaneyoshi [13-30], magnetic properties of the cubic nanowire were studied by Jiang et al [31, 32], dynamic behaviors of Ising nanowire were studied by Erta� and Kocakaplan [33], magnetic properties of the core/shell Ising nanostructures were studied by Kantar et al [34-36], magnetic properties of Ising nanotube were studied by Magoussi et al [37], magnetic properties of cylindrical transverse Ising nanowire were studied by Kocakaplan et al [38], phase transitions of honeycomb-structured ferroelectric thin film were studied by Wang and Ma [39], phase diagrams of transverse Ising nanowire were studied by Bouhou et al [40], magnetic behavior of nanowires was studied by Zaim et al [41], band structures of the magnetic properties in a mixed Ising nanotube were studied by �arl� [42], magnetic susceptibility and magnetic reversal events of Ising nanowire were studied by �arl� and Keskin [43], hysteresis behaviors of Ising nanowire were studied by Keskin et al [44], magnetic properties of Ising nanowire and core/shell nanoparticles were studied by Yüksel et al [45, 46] and phase diagrams of Ising nanowire were studied by Ak�nc� [47, 48]. However, to the best of our knowledge, the effects of next nearest-neighbor exchange interaction (J2) on the magnetic properties (magnetization, susceptibility, phase transition, hysteresis curves, critical temperature and coercive field point) in the one-dimensional Ising system (1DIS) have not been investigated yet. Therefore, the purpose of this paper is to investigate the effects of next nearest-neighbor exchange interaction (J2) on the magnetic properties in the 1DIS by using Kaneyoshi approach within the effective field theory for both ferromagnetic and antiferromagnetic case. The outline of this paper is as follows: In section 2, we give the theoretical method. In section 3, we present the theoretical results and discussion, followed by a brief summary.

3

2. Theoretical method We investigate the magnetic properties of the ferromagnetic (J1>0) and the antiferromagnetic (J1<0) 1DIS shown in Fig. 1 within the effective field theory. Each site in Fig. 1 is occupied by the spin-1/2 Ising particle. By using Kaneyoshi approach [13-30], the Hamiltonian of the 1DIS is given by,

i j i i i j

z z z z z z1 2

ij ii i jJ S S J S S H S + S .

� �� � � � � �

� H (1)

Where, J1 is the exchange interaction between two nearest-neighbor magnetic atoms (m1 and m2) of the 1DIS. J2 is the exchange interaction between two next nearest-neighbor magnetic atoms (m1 and m1 or m2 and m2) of the 1DIS. Sz=±1 is the Pauli spin operator. H is the external magnetic field. The 1DIS shown in Fig.1 has two different magnetizations; m1 and m2 are the magnetization of the 1DIS. m1 and m2 are the nearest-neighbor atoms and they interact with the nearest-neighbor exchange interaction J1. m1 and m1 or m2 and m2 interact with the next nearest-neighbor exchange interaction J2. One notes the next nearest-neighbor exchange interaction (J2) should be small than the nearest-neighbor exchange interaction (J2<J1). The magnetizations of the 1DIS are given by,

2 2m cosh(J )+m sinh(J ) cosh(J )+m sinh(J ) F (x) ,1 2 2 1 2l l S-1/2 x=0

2 2m cosh(J )+m sinh(J ) cosh(J )+m sinh(J ) F (x) .2 1 1 1 2 2 2 S-1/2 x=0

� � � �� � �� �

� � � �� � � � � (2)

where, / x � � � is the differential operator and the function of FS-1/2(x) is defined by as follows for the spin-1/2 Ising particles.

� �F (x) tanh (x+H) .S-1/2 � � (3)

By using Kaneyoshi approach (KA) [13-30] for the magnetic properties of the nanostructure within the effective field theory, the magnetizations and susceptibility of the 1DIS are obtained as in Eq. (2). By using Eq. (2) and mathematical relation Eq.(4) [49], the magnetizations of the 1DIS are obtained as follows,

� �ae f x a . � � (4)

� �� �

2 31 0 1 1 2 1 3 1 Spin-1/2 x=0

2 32 0 1 2 2 2 3 2 Spin-1/2 x=0

m A +A m A m A m ... F (x) ,

m B +B m B m B m ... F (x) .

� � � �

� � � � (5)

By differentiating each side of the Eq. (5) with H, we get the �as follows,

4

� �� �� �

� �� �

11

22

21 1 2 1 3 1 1 Spin-1/2 x=0

211 12 2 13 2 2 Spin-1/2 x=0

2 30 1 1 2 1 3 1 Spin-1/2 x=0

22 1 2 2 3 2 2 Spin-1/2 x=0

211 12 1 13 1 1

mlim ,H 0 H

mlim ,H 0 HA 2A m 3A m F (x)

A 2A m 3A m F (x)

A +A m A m A m ... F (x) ,H

B 2B m 3B m F (x)

B 2B m 3B m

�� �

� ��

� �� �

� � � � �

� � � �

�� � � �

�� � � � �

� � � �

� �Spin-1/2 x=0

2 30 1 2 2 2 3 2 Spin-1/2 x=0

F (x)

B +B m B m B m ... F (x) ,H�

� � � ��

(6)

by arranging Eq. (6), we obtain the 1� and 2� as follows,

1 1 1 2 2 0 Spin-1/2 x=0

2 1 1 2 2 0 Spin-1/2 x=0

a a a F (x) ,H

b b b F (x) ,H

�� � � � � �

��

� � � � � ��

(7)

The coefficients a1, a2 and a3, are given by,

� �� �� �� �� �

21 1 2 1 3 1 Spin-1/2 x=0

22 11 12 2 13 2 Spin-1/2 x=0

2 30 0 1 1 2 1 3 1

21 1 2 2 3 2 Spin-1/2 x=0

22 11 12 1 13 1 Spin-1/2 x=0

0 0 1 1 2 1

a A 2A m 3A m ... F (x) ,

a A 2A m 3A m ... F (x) ,

a A +A m A m A m ... ,

b B 2B m 3B m ... F (x) ,

b B 2B m 3B m ... F (x) ,

b B +B m B m

� � � �

� � � �

� � � �

� � � �

� � � �

� �� �2 33 1B m ... .� �

(8)

Where, the coefficients A0-A3, A11-A13, B0-B3, B11-B13, a0-a2 and b0-b2 are very long expressions. Therefore, they will not be given here. By solving Eq. (7) numerically and inserting the values into Eq.(9), we obtain the TM and T� of the 1DIS follows,

� �� �

1M m m ,T 1 221

T 1 22.

� �

� � � �� (9)

5

3. Theoretical results and discussion We firstly considered the temperature dependence of the magnetizations and susceptibility of the 1DIS. In our calculations, the nearest-neighbor exchange interaction is J1=1. In order to find the effecs of the next nearest-neighbor exchange interaction on the magnetizations and susceptibility of the 1DIS, they are obtained for different next nearest-neighbor exchange interaction values which are small than the nearest-neighbor exchange interaction value (J2=0.0001, 0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9<J1=1). One notes that it should be J2<J1. Fig.2 shows the temperature dependence of the magnetizations and susceptibility of the ferromagnetic (0<J1) 1DIS (m1, m2 and MT). The magnetizations of the FM spin-1/2 1DIS are shown in Fig.2(a) for different next nearest-neighbor exchange interaction values. The second-order phase transition from ferromagnetic phase to paramagnetic phase occurs at Tc=0.571, 0.632, 0.803, 1.322, 1.618, 1.841, 2.063, 2.264, 2.444, 2.624, 2.782 and 2.951J/kB for J2=0.0001, 0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9, respectively. The magnetizations are m1=m2=MT=1 at T=0. The critical temperature of the 1DIS increases as the next nearest-neighbor exchange interaction increases. In Fig.2(b), the susceptibilities of the FM spin-1/2 1DIS have distinct peak at Tc. Fig.3 shows the temperature dependence of the magnetizations and susceptibility of the antiferromagnetic (0>J1) 1DIS (m1, m2 and MT). The magnetizations of the AFM spin-1/2 1DIS are shown in Fig.3(a) for different J2 similar to the FM case. The magnetizations are m1=-1, m2=1 and MT=0 at T=0. Due to the fact that m1<0, the susceptibility of the m1 ( 1� ) emerges in the negative direction in Fig.3(b). The total magnetization and susceptibility of the AFM 1DIS are zero. Fig.4 shows the applied field dependence of the magnetizations of the ferromagnetic (0<J1) 1DIS (m1, m2 and MT) for different next nearest-neighbor exchange interaction values at a constant temperature (T=1). The coercive field points of the FM 1DIS are at Hc=0.000, 0.000, 0.000, ±0.061, ±0.192, ±0.340, ±0.488, ±0.636, ±0.776, ±0.920, ±1.047 and ±1.175 for J2=0.0001, 0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9, respectively. The coercive field point of the 1DIS increases as the next nearest-neighbor exchange interaction increases. The hysteresis curves of the FM 1DIS exhibit paramagnetic behaviors for small next nearest-neighbor exchange interaction values (J2=0.0001, 0.001 and 0.01). Moreover, the remanent magnetizations of the FM 1DIS decreases as the next nearest-neighbor exchange interaction decreases. Our theoretical hysteresis results of FM 1DIS are in agreement with the results of one-dimensional polymeric [MnIII(salen)N3] and [MnIII(salen)Ag(CN)2] complexes by Panja et al [50]. Fig.5 shows the applied field dependence of the magnetizations of the antiferromagnetic (0>J1) 1DIS (m1, m2 and MT) for different next nearest-neighbor exchange interaction values at a constant temperature (T=1) similar to the FM case. In Fig.5(a), the hysteresis curves of the m1 and m2 of the AFM 1DIS are shown. The coercive field points of m1 and m2 of the AFM 1DIS are at Hc=±1.453, ±1.595, ±1.709, ±1.796, ±1.858, ±1.904, ±1.935, ±1.958 and ±1.972 for J2=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9, respectively. One notes that the hysteresis curves of the m1 and m2 of the AFM 1DIS exhibit elliptical hysteresis behaviors with two distinct loops similar to the glasses for small next nearest-neighbor exchange interaction values (J2=0.0001, 0.001 and 0.01) and they have two different coercive field points far away from H=0.000. The coercive field points are at Hc1=±0.433 and Hc2=±1.264 for J2=0.01, Hc1=±0.571 and Hc2=±1.222 for J2=0.001 and 0.0001. In Fig.5(b), the hysteresis curves of the MT of the AFM 1DIS are shown. The hysteresis curves of the total AFM 1DIS exhibit paramagnetic hysteresis behaviors and they have fluctuations at ±Hc. The fluctuations

6

of the total AFM 1DIS decrease as the next nearest-neighbor exchange interaction decreases similar to the coercive field points. Fig.6 shows the effects of the next nearest-neighbor exchange interaction between two next nearest-neighbor magnetic atoms (m1 and m1 or m2 and m2) on the critical temperature of the 1DIS. The critical temperature of the 1DIS increases as the next nearest-neighbor exchange interaction increases (but not high than nearest-neighbor exchange interaction, J2<J1). The critical temperature of the 1DIS has almost a constant value (Tc=0.517) as the next nearest-neighbor exchange interaction approaches to zero (but not zero, J2�0). Therefore, we suggest that the critical temperature (Tc) of the 1DIS strongly depends on the next nearest-neighbor exchange interaction (J2) in the 1DIS. In summary, we have investigated the effects of next nearest-neighbor exchange interaction (J2) on the magnetic properties (magnetization, susceptibility, phase transition, hysteresis curves, critical temperature and coercive field point) in the one-dimensional Ising system (1DIS) by using Kaneyoshi approach within the effective field theory. It is found that; a) the critical temperature of the 1DIS decreases as the next nearest-neighbor exchange interaction decreases and it has almost stable value for small next nearest-neighbor exchange interaction, b) the coercive field point, remanent magnetization and the area of the hysteresis loop of the 1DIS decreases as the J2 decreases, c) the hysteresis curves of the 1DIS exhibit paramagnetic hysteresis behavior for small J2 in the ferromagnetic case, d) the hysteresis curves of the m1 and m2 exhibit elliptical hysteresis behaviors with two distinct loops similar to the glasses for small J2 in the antiferromagnetic case and they have two different coercive field points far away from H=0.000, e) the magnetic properties strongly depend on the next nearest-neighbor exchange interaction in the 1DIS and f) If J2=0, 1DIS has phase transition but its susceptibility can’t be calculated.

.

7

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T. Dauxois, N. Theodorakopoulos, and M. Peyrard, J. Stat. Phys. 107 (2002) 869. [8] L. van Hove, Physica 16 (1950) 137 (reprinted in ref. 9, p. 28). [9] E. H. Lieb and D. C. Mattis, Mathematical Physics in One Dimension (Academic Press,

London, 1966). [10] D. Ruelle, Statistical Mechanics: Rigorous Results (Addison–Wesley, Reading, 1989). [11] F. J. Dyson, Comm. Math. Phys. 12 (1969) 91. [12] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon, New York,

1980). [13] T. Kaneyoshi, J. Magn. Magn. Mater. 321 (2009) 3430. [14] T. Kaneyoshi, J. Magn. Magn. Mater.321 (2009) 3630. [15] T. Kaneyoshi, J. Magn. Magn. Mater. 322 (2010) 3410. [16] T. Kaneyoshi, J. Magn. Magn. Mater. 322 (2010) 3014. [17] T. Kaneyoshi, J. Magn. Magn. Mater. 323 (2011) 2483. [18] T. Kaneyoshi, J. Magn. Magn. Mater. 323 (2011) 1145. [19] T. Kaneyoshi, J. Magn. Magn. Mater. 336 (2013) 8. [20] T. Kaneyoshi, J. Magn. Magn. Mater. 339 (2013) 151. [21] T. Kaneyoshi, Phys. Status Solidi (b) 246 (2009) 2359. [22] T. Kaneyoshi, Phys. Status Solidi (b) 248 (2011) 250. [23] T. Kaneyoshi, Solid State Commun. 151 (2011) 1528. [24] T. Kaneyoshi, Solid State Commun. 152 (2012) 883. [25] T. Kaneyoshi, Phys. Lett. A 376 (2012) 2352. [26] T. Kaneyoshi, Physica A 391 (2012) 3616. [27] T. Kaneyoshi, Physica A 392 (2013) 2406. [28] T. Kaneyoshi, Physica B 407 (2012) 4358. [29] T. Kaneyoshi, Physica B 414 (2013) 72. [30] T. Kaneyoshi, Physica B 436 (2014) 208. [31] W. Jiang, X. X. Li, L. M. Liu, J. N. Chen and F. Zhang, J. Magn. Magn. Mater. 353

(2014) 90. [32] W. Jiang, X. X. Li and L. M. Liu, Physica E 53 (2013) 29. [33] M. Erta� and Y. Kocakaplan, Phys. Lett. A 378 (2014) 845. [34] E. Kantar and Y. Kocakaplan, Solid State Commun. 177 (2014) 1. [35] E. Kantar and M. Keskin, J. Magn. Magn. Mater. 349 (2014) 165. [36] E. Kantar, B. Deviren and M. Keskin, Eur. Phys. J. B 86 (2013) 40080. [37] H. Magoussi, A. Zaim and M. Kerouad, Chinese Phys. B 22 (2013) 116401. [38] Y. Kocakaplan, E. Kantar and M. Keskin, Eur. Phys. J. B 86 (2013) 40659. [39] C. D. Wang and R. G. Ma, Physica A 392 (2013) 3570. [40] S. Bouhou, I. Essaoudi, A. Ainane, M. Saber, R. Ahuja and F. Dujardin, J. Magn. Magn.

Mater. 336 (2013) 75. [41] A. Zaim, M. Kerouad and M. Boughrara, J. Magn. Magn. Mater. 331 (2013) 37.

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[42] N. �arl�, Physica B 411 (2013) 12. [43] N. �arl� and M. Keskin, Solid State Commun. 152 (2012) 354. [44] M. Keskin, N. �arl� and B. Deviren, Solid State Commun. 151 (2011) 1025. [45] Y. Yüksel, Ü. Ak�nc� and H. Polat, Phys. Status Solidi (b) 250 (2013) 196. [46] Y. Yüksel, Ü. Ak�nc� and H. Polat, Physica A 392 (2013) 2347. [47] Ü. Ak�nc�, J. Magn. Magn. Mater. 324 (2012) 4237. [48] Ü. Ak�nc�, J. Magn. Magn. Mater. 324 (2012) 3951. [49] T. Kaneyoshi, Phys. Rev. B 24 (1981) 2693. [50] A. Panja, N. Shaikh, P. Vojtisek, S. Gao, P. Banerjee, New J. Chem. 26 (2002) 1025.

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Figure captions

Fig.1. (Color online) Schematic representation of the one-dimensional Ising system (1DIS). Fig.2. (Color online) The temperature dependence of the magnetizations (a) and susceptibility

(b) of the FM 1DIS. Fig.3. (Color online) The temperature dependence of the magnetizations (a) and susceptibility

(b) of the AFM 1DIS. Fig.4. (Color online) The applied field dependence of the magnetizations of the FM 1DIS. Fig.5. (Color online) The applied field dependence of the magnetizations AFM m1 and m2 (a)

and the total AFM 1DIS (b). Fig.6. The effects of next nearest-neighbor on the critical temperature of the 1DIS.

10

Highlights

� Magnetic properties of the one-dimensional Ising system (1DIS) were studied by EFT.

� Effect of next nearest-neighbor exchange interaction (J2) in the 1DIS was studied.

� Magnetic properties of FM and AFM 1DIS strongly depend on the J2.

� AFM 1DIS has two hysteresis loops similar to the glasses for the small J2.

� If J2=0, 1DIS has a phase transition but its susceptibility can’t be calculated.

Fig.1. (Color online) Schematic representation of the one-dimensional Ising system (1DIS).

Figure 1.

b )

k B T /J

0 1 2 3 4 5

Susc

eptib

ility

0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

T c = 0 .5 7 1 ... . . .J 2 = 0 .0 0 0 1T c = 0 .6 3 2 J 2 = 0 .0 0 1T c = 0 .8 0 3 J 2 = 0 .0 1T c = 1 .3 2 2 J 2 = 0 .1T c = 1 .6 1 8 J 2 = 0 .2T c = 1 .8 4 1 J 2 = 0 .3T c = 2 .0 6 3 J 2 = 0 .4T c = 2 .2 6 4 J 2 = 0 .5T c = 2 .4 4 4 J 2 = 0 .6T c = 2 .6 2 4 J 2 = 0 .7T c = 2 .7 8 2 J 2 = 0 .8T c = 2 .9 5 1 J 2 = 0 .9

a )

Mag

netiz

atio

n

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0F M 1 D IS

(m 1 = m 2 = M T )

F M 1 D IS

(� 1 =� 2 =� T )

Fig.2. (Color online) The temperature dependence of the magnetizations (a) and susceptibility (b) of the FM 1DIS.

Figure 2.

b )

k B T /J

0 1 2 3 4 5

Susc

eptib

ility

- 1 0 0

-5 0

0

5 0

1 0 0

a )

T c = 0 .5 7 1 ... . . .J 2 = 0 .0 0 0 1T c = 0 .6 3 2 J 2 = 0 .0 0 1T c = 0 .8 0 3 J 2 = 0 .0 1T c = 1 .3 2 2 J 2 = 0 .1T c = 1 .6 1 8 J 2 = 0 .2T c = 1 .8 4 1 J 2 = 0 .3T c = 2 .0 6 3 J 2 = 0 .4T c = 2 .2 6 4 J 2 = 0 .5T c = 2 .4 4 4 J 2 = 0 .6T c = 2 .6 2 4 J 2 = 0 .7T c = 2 .7 8 2 J 2 = 0 .8T c = 2 .9 5 1 J 2 = 0 .9

Mag

netiz

atio

n

- 1 .0

-0 .5

0 .0

0 .5

1 .0m 2

M T

m 1

A F M 1 D IS

(m 1 = -m 2 , M T = 0 )

A F M 1 D IS

(� 1 = -� 2 , � T = 0 )

Fig.3. (Color online) The temperature dependence of the magnetizations (a) and susceptibility (b) of the AFM 1DIS.

Figure 3.

. . . . . .J 2 = 0 .0 0 0 1 J 2 = 0 .0 0 1 J 2 = 0 .0 1 J 2 = 0 .1 J 2 = 0 .2 J 2 = 0 .3 J 2 = 0 .4 J 2 = 0 .5 J 2 = 0 .6 J 2 = 0 .7 J 2 = 0 .8 J 2 = 0 .9

H

-2 -1 0 1 2

Mag

netiz

atio

n

-1 .0

-0 .5

0 .0

0 .5

1 .0F M 1 D IS (m 1 = m 2 = m T )

Fig.4. (Color online) The applied field dependence of the magnetizations of the FM 1DIS.

Figure 4.

. . . . . .J 2 = 0 .0 0 0 1 J 2 = 0 .0 0 1 J 2 = 0 .0 1 J 2 = 0 .1 J 2 = 0 .2 J 2 = 0 .3 J 2 = 0 .4 J 2 = 0 .5 J 2 = 0 .6 J 2 = 0 .7 J 2 = 0 .8 J 2 = 0 .9

Mag

netiz

atio

n

- 1 .0

-0 .5

0 .0

0 .5

1 .0

m1=

-m2

m1=

-m2

A F M m 1 = -m 2

H

-4 -2 0 2 4

Mag

netiz

atio

n

- 1 .0

-0 .5

0 .0

0 .5

1 .0 A F M 1 D IS (M T )

a )

b )

Fig.5. (Color online) The applied field dependence of the magnetizations AFM m1 and m2 (a) and the total AFM 1DIS (b).

Figure 5.

T h e e ffe c t o f th e n e x t n e a re s t n e ig h b o r o n T c o f th e 1 D IS

J 2

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9

Tc

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

3 .5

T c o f th e 1 D IS

Fig.6. The effects of next nearest-neighbor on the critical temperature of the 1DIS.

Figure 6.

Graphical Abstract

The figure shows the effects of next nearest-neighbor on the critical temperature of the 1DIS.

�

The effect of the next nearest neighbor on Tc of the 1DIS

J2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Tc

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Tc of the 1DIS

�