The magnetic and thermodynamic properties of a spin-2 Heisenberg ferromagnetic system

Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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The magnetic and thermodynamic properties of a spin-2 Heisenberg ferromagnetic system Gülistan Mert n Department of Physics, Selçuk University, 42075 Konya, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 17 June 2014 Received in revised form 25 July 2014

The magnetic and thermodynamic properties such as the magnetization, internal energy, specific heat and susceptibility of spin-2 Heisenberg ferromagnetic system on a square lattice are studied by using Green's function technique. Without including the next nearest neighbor interaction, one doesn't observe the second-order phase transitions. We found that only when the next nearest neighbor interaction is greater than the nearest neighbor interaction, the second-order phase transitions exist for the small single-ion anisotropy values. Indeed, in the case of negative anisotropy which corresponds to first-order phase transitions, the energies have discontinuities. At the same time, the specific heat shows two peaks. & 2014 Published by Elsevier B.V.

Keywords: Ferromagnet Green's function Phase transition Internal energy Specific heat

1. Introduction Owing to the exchange interaction, some materials exhibit a spontaneous magnetic moment below a certain critical temperature which is associated with a second-order phase transition from ferromagnetic to paramagnetic. The various magnetic properties of Heisenberg ferromagnetic system can be investigated by the double-time temperature-dependence Green's function theory [1–5]; Tahir-Kheli and ter Haar have calculated the internal energy and specific heat [2,3]. Fröbrich et al. have developed a method to calculate more than one component of the magnetization for ferromagnetic films [6–9]. Junger et al. have found that the specific heat had two maxima and power laws for the position and height of the susceptibility maximum did not support the predictions of Landau theory [10,11]. Wang et al. have very recently presented a very nice formalism which gave the internal energy as the thermal average in terms of z-component of spin operator by using time derivatives of correlation function [12,13]. Indeed, there have been many interesting works dealing with the Heisenberg ferromagnet in one- and two-dimensional systems [14–19]. For some ferromagnetic low dimensional systems, Hamedoun et al. have found that there existed a phase transition from ferromagnetic to paramagnetic state at a finite temperature [20]. In this work, we will apply Green's function technique to investigate the magnetization, internal energy, specific heat, susceptibility of spin-2 Heisenberg ferromagnet on the square lattice.

n

Fax: þ 90 332 2376195. E-mail address: [email protected]

In order to obtain the internal energy, we will follow the abovementioned formalism of Wang et al. [13], but we introduce the next nearest neighbor interaction. The outline of this paper is as follows. In Section 2, we present the formalism of the Green's function method. In Section 3, the numerical results are discussed for the magnetization, internal energy, specific heat and susceptibility. Finally, Section 4 contributes to conclusions.

2. Model and formalism Let us consider the spin-2 Heisenberg ferromagnetic model on a square lattice whose lattice constant is a. Hamiltonian which includes nearest neighbor and next nearest neighbor interactions, single-ion anisotropy and an external magnetic field in the z-direction is as follows: 1 1 H ¼  J∑S i  S j  J 1 ∑S i  S i0  D∑ ðSzi Þ2  h∑ Szi ; 2 ij 2 ii' i i

ð1Þ

The first summation runs over pairs of the nearest neighbor sites and the second runs over pairs of the next nearest neighbor sites. J is the exchange coupling constant between the nearest neighboring sites and J1 between the next nearest neighboring sites. We consider only ferromagnetic cases which have positive exchange parameters (J and J1 40). D is the single-ion anisotropy parameter and the magnetic field h is applied along the z-axis. In order to study magnetic properties of the model, we z introduce the Green's function 〈〈Siþ ðtÞ; Bl ð0Þ〉〉; where Bl ¼ eηSi Si

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2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

(η is a parameter) and Siþ ðtÞ and Si ð0Þ are Heisenberg spin operators [4]. We have obtained the following equation of motion for the Green's function:

U LC ¼ 

n 1 ½Sp ð2Sp þ 1Þ〈Sz 〉 3〈ðSz Þ2 〉 þ 4〈ðSz Þ3 〉Φ1 2½Sp þ 1

þ ½  Sp  3〈Sz 〉þ 3〈ðSz Þ2 Φ2 þ 4ðJ þJ 1 Þ½Sp þ 〈ðSz Þ3 〈Sz 〉 þ D½2〈ðSz Þ4 þ 〈ðSz Þ3  2ðSp þ 1Þ〈ðSz Þ2 þ ðSp þ 1Þ〈Sz 〉 þ Sp 

ω〈〈Siþ ; Bl 〉〉 ¼ δil 〈½Siþ ; Bl 〉  J∑ 〈〈ðSzi Sjþ  Siþ Szj Þ; Bl 〉〉

þ h½Sp ðSp þ 1Þ〈Sz 〉 þ〈ðSz Þ3 g

j

 J 1 ∑〈〈ðSzi Siþ0 Siþ Szi0 Þ; Bl 〉〉

ð16Þ

where

i0

ð2Þ where δ is Dirac's delta function and 〈…〉 denotes the canonical thermal average. We have taken the value of ℏ as unit. In the equations of motion, the higher order Green's functions will appear. For the decoupling process, we perform the random phase approximation concerning the exchange interaction terms [2,3] 〈〈Szi Sjþ ; Bl 〉〉 ffi 〈Szi 〉〈〈Sjþ ; Bl 〉〉;

ð3Þ

and a generalization of the Anderson–Callen's decoupling schema concerning the single-ion anisotropy term [5] 〈〈ðSiþ Szi þ Szi Siþ Þ; Bl 〉〉 ffi τ〈〈Siþ ; Bl 〉〉;

ð4Þ

where τ ¼ f2  ½Sp  2〈ðSz Þ2 〉=S2 g〈Sz 〉:

ð5Þ

1 JðkÞ Φ1 ¼ ∑ βωðkÞ ; Nk e 1 1 ωðkÞ ; Φ2 ¼ ∑ βωðkÞ Nk e 1

þ z z þ þD〈〈ðSð17Þ i Si þ Si Si Þ; Bl 〉〉

ð18Þ

where ω(k) is given by Eq. (7) and JðkÞ ¼ J∑m eik U m þJ 1 ∑n eik U n , here the first summation is over nearest neighbors and second is over the next nearest neighbors. Specific heat C is defined as the derivative of the internal energy with respect to temperature: C¼

∂UðTÞ : ∂T

ð19Þ

Parallel susceptibility χ == is calculated as follows: χ == ¼

〈Sz 〉h  〈Sz 〉0 : h

ð20Þ

here Sp ¼S(S þ1). Then we obtain the following Green's function Gðω; kÞ ¼

3. Results and discussions

〈½S þ ; B〉 ; ω ωðkÞ

ð6Þ

where ωðkÞ ¼ fDτ þ 4ðJ þ J 1 Þ  2Jγ  4J 1 γ 1 g〈Sz 〉 þh;

ð7Þ

γ ¼ cos ðkx aÞ þ cos ðky aÞ;

ð8Þ

pffiffiffi pffiffiffi γ 1 ¼ cos ð 2kx aÞ cos ð 2ky aÞ:

ð9Þ

One has the following expressions for 〈Sz 〉and its other power thermal averages by means of spectral theorem and Callen's technique [4] 〈Sz 〉 ¼

ðS ΦÞð1 þ ΦÞ2S þ 1 þ ðS þ1 þ ΦÞΦ2S þ 1 ð1 þ ΦÞ2S þ 1  Φ2S þ 1

;

ð10Þ

〈ðSz Þ2 〉 ¼ Sp  ð1 þ 2ΦÞ〈Sz 〉;

ð11Þ

〈ðSz Þ3 〉 ¼ fð1 þ 2ΦÞ½Sp  3〈ðSz Þ2 〉 þ ½2Sp  1〈Sz 〉g=2;

ð12Þ

〈ðSz Þ4 〉 ¼ S2p  〈ðSz Þ2 〉  2ð1 þ2Φ〈ðSz Þ3 〉;

ð13Þ

where 1 1 ; Φ ¼ ∑ βωðkÞ Nk e 1

ð14Þ

here k sums over the first Brillouin Zone and N is the total site number in the system. In order to obtain internal energy, we have followed the formalism developed by Wang et al. [13]. The internal energy is defined as a statistical average of the Hamiltonian per site: U in ¼  ð〈H〉=NÞ. The internal energy is formed from two parts defined as the transverse correlation energy (UTC) and the longitudinal correlation energy (ULC). So, we obtain these terms of the internal energy as follows: U TC ¼ 〈Sz 〉Φ1 ;

ð15Þ

The coupled Eqs. (10)–(18) can be solved self-consistently for each value of temperature. Thus, the magnetization and internal energy are calculated numerically, depending on the values of the applied magnetic field. Also the specific heat and parallel susceptibility can be found numerically. We start by demonstrating results for phase diagrams of spin-2 Heisenberg ferromagnet on square lattice. In Fig. 1, we plot phase diagrams in the (J1  T) plane at J ¼5 for both positive and negative values of the single-ion anisotropy. The dotted and solid lines represent the first-order and second-order phase transition temperatures, respectively. When the next nearest neighbor interaction is zero, for all value of single-ion anisotropy, the transition becomes the first-order. Only when J1 4J and for the small values of single-ion anisotropy one observe the second-order phase transition. At the same time, these results are valid for the negative values of the single-ion anisotropy, but another discontinuity exists. This transition also becomes the first-order. We show these with the chain-dotted, chain-double-dotted, short-dashed and long-dashed lines in Fig. 1(b). For D¼  0.01 and  0.05, it disappears just after the critical temperature appears. Fig. 2 demonstrates the influence of an external magnetic field on magnetization when the value of the anisotropy is equal to D¼  0.01 at J ¼5 and J1 ¼3. At this parameters, there is no the second-order phase transition. In the case of negative anisotropy, the magnetization decreases to zero discontinuously with increasing the temperature for both h¼ 0 and h¼ 0.01. Therefore, the firstorder phase transition occurs at TD ¼17.6 (for h¼0) and TD ¼13.9 (for h ¼0.01), whereas the curve for h¼ 0.05 shows the typical behavior of ferromagnetic case without discontinuity. At the same time, similar behaviors are seen for the temperature variations of energies and specific heat. For the same values in Fig. 2, we plot the transverse correlation energy, the longitudinal correlation energy and the internal energy in Fig. 3. As seen from the figure, for both h¼ 0 and h¼0.01, the energies have discontinuities at the first-order phase transition temperature at TD ¼17.6 and TD ¼13.9, respectively.

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a

40 80

U TC

0.1 0.05

60

T

0.01

0 -20

40

-40

20 0

h=0 h = 0.01 h = 0.05

20

D=1

0

2

4

6 J1

8

10

0

20

40

60

80

0

12

-20 U LC

b 80 - 0.1 - 0.05 - 0.01

-80 0

20

40

60

0

20

40

60

T

60

-40 -60

D=-1

40

0

- 0.1

- 0.01

-20

- 0.05 0

0

2

-1 4

6 J1

8

10

12

Fig. 1. First-order (dotted, chain-dotted, chain-double-dotted, short-dashed and longdashed lines) and second-order (solid lines) phase transition temperatures as a function of J1 at J¼ 5 (a) for the positive values of D (b) for the negative values of D.

3

0 -1

0

10

20

T

30

40

-60

Fig. 3. The energies for the same values in Fig. 2(a). The transverse correlation energy (b). The longitudinal correlation energy and (c) the internal energy.

1

-2

-40

T

h=0 h = 0.01 h = 0.05

2

Uin

20

< S z>

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

3

50

Fig. 2. Temperature dependence of the magnetization for the various values of the external magnetic field h when D ¼  0.01, J¼ 5 and J1 ¼3.

At T ¼0, UTC (T ¼0) ¼0, and without the external magnetic field and anisotropy ULC (T ¼0) ¼  8 (J þJ1) for spin-2 ferromagnet on square lattice. As seen from Fig. 3(a), the transverse correlation

energy starts from zero temperature but for h¼0 and h¼ 0.01 it increases until the first-order phase transition points at TD ¼17.6 and TD ¼13.9, respectively, that is, UTC has positive values. This means that the spins are transversely parallel to each other. After phase transition points, it has negative values, whereas for h¼0.05 UTC decreases with increasing temperature and it has a minimum around the phase transition temperature. UTC is always negative for h ¼0.05. This means that the spins are transversely antiparallel to each other. It starts from ULC (T ¼0) ¼  64 and decreases until phase transition points for both h¼ 0 and h¼0, than by increasing it goes to infinite above the phase transition temperature. ULC has always become negative for all values of magnetic field. The similar effects are also seen for the internal energy. The results we obtained for h¼ 0.05 are similar to a three-dimensional results of fcc ferromagnet lattice for spin-5/2 obtained by Wang et al. [13]. For same values of parameters, we can study the specific heat of the system. This has been seen in Fig. 4. The specific heat exhibits an abnormal rise at the phase transition temperatures and at the same time it diminishes suddenly at this point. For h¼0 and

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0 -20 UTC

0.01, the specific heat has two peaks where the first-order phase transition temperatures exist, but for h¼0.05 it has a maximum. Experimentally, a specific heat study of magnetic transitions in the rare-earth metals such as terbium, holmium and dysprosium are reported by Jayasuria et al. [21–23]. Specific heats of these substances employ two peaks. The temperature corresponding to the first peak is a first-order phase transition temperature and later is a second-order phase transition temperature in these materials. In the case of the positive anisotropy (for D ¼0.01), we plot in Fig. 5 the behaviors of the magnetization curves as a function of temperature for the various values of the external magnetic field at J ¼5 and J1 ¼10. The magnetization without an external magnetic field goes to zero at Tc ¼ 58.10. The transition becomes the second-order from ferromagnet to paramagnet. When the magnetic field is applied, the transition temperatures from ferromagnet to paramagnet shift towards the high temperature region. For the same values in Fig. 5, but the several values of h, we plot the behaviors of the transverse correlation energy, the longitudinal correlation energy and the internal energy shown in Fig. 6 and specific heat shown in Fig. 7. As seen from Fig. 6, as different from results in Fig. 5, all energies have always becomes negative since

h=0 h = 0.01 h = 0.1 h = 0.5

-40 -60 -80

0

50

100

150

200

250

0

-50

-100

0.5 h=0 h = 0.01 h = 0.05

0.4

0

50

100

150

200

0

50

100

150

200

0

C

0.3 Uin

0.2 0.1 0.0

-50

-100

0

10

20

T

30

40

50

Fig. 4. The specific heat for the various values of the external magnetic field h at D ¼  0.01, J¼ 5 and J1 ¼3.

T Fig. 6. The energies for the same values in Fig. 5(a). The transverse correlation energy (b). The longitudinal correlation energy and (c) the internal energy.

0.15

2.0

1.5

0.10

1.0

5

2

h=0 0

20

40

T

0.01 60

0.1

1

0.05

1

0.5

0.0

h=0 h = 0.01 h = 0.1 h = 0.5

C



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

ULC

4

0.5

80

100

Fig. 5. Temperature dependence of the magnetization for the various values of the external magnetic field h when D¼ 0.01, J¼ 5 and J1 ¼10.

0.00

0

20

40

60

T

80

100

120

0

Fig. 7. The specific heat for the various values of the external magnetic field h at D¼ 0.01, J ¼5 and J1 ¼10.

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100

50 h = 0.5 h = 0.1 h = 0.01

80

40

60

30 χ-1

χ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

5

40

20

20

10

0

0

40

80 T

120

0

160

0

40

80

T

120

160

200

Fig. 8. The behavior of the susceptibility (a) and the reciprocal susceptibility (b) as a function of temperature for several values of h at D ¼0.01, J¼ 5 and J1 ¼ 10.

the transitions is only second-order. UTC starts from zero temperature and it has a minimum around the critical temperature. It starts from ULC (T¼ 0) ¼  120 and it increases until critical points, than it goes to infinite above the critical temperature. The internal energy goes to a fixed value with increasing temperature. These results are exactly the agreement with threedimensional results of fcc ferromagnet lattice for spin-5/2 obtained by Wang et al. [13]. Fig. 7 shows specific heat with and without the external magnetic field. The specific heat has a maximum at the critical temperature and with increasing magnetic field critical temperature shift towards the high temperature region. In Fig. 8, we demonstrate the behaviors of the parallel susceptibility and reciprocal susceptibility as a function of temperature for the several values of h. The parallel susceptibility is zero at the absolute temperature and has a maximum at critical temperature. The maximum of the susceptibility diminishes with increasing the external magnetic field. In the limit of T ¼Tc, χ  1 ¼ 0 and these values, in Fig. 8(b), correspond to the second-order phase transition. Above the critical temperature, the reciprocal susceptibility is a curved line and approaches a linear asymptote with increasing temperature.

interaction and single-ion anisotropy play an essential role in the existence of the first-order and second-order phase transition temperatures. The internal energy has a discontinuity at the firstorder phase transition temperature. The temperature dependence of the specific heat and the susceptibility under the various magnetic fields are investigated. The specific heat employs two peaks at the first-order phase transition temperature. The susceptibility has a maximum at the critical temperature. Above of which, the reciprocal susceptibility is curved.

4. Conclusion

[11] [12] [13] [14] [15] [16] [17]

In this work, we analyze the effect of the nearest and next nearest neighbor interactions, single ion anisotropy and magnetic field on the phase transition, magnetization, internal energy, specific heat and susceptibility. We have extended the formalism developed very recently by Wang et al. to include both the nearest and next nearest neighbor interactions in Heisenberg model on the square lattice. The system has the first-order phase transition properties in the case of negative single-ion anisotropy and when the next nearest neighbor interaction is greater than the nearest neighbor interaction, the second-order phase transitions exist for the small values of the single-ion anisotropy. We can say that the next nearest neighbor

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