Effect of next nearest neighbor interaction on thermodynamic properties of ultrathin magnetic films

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Effect of next nearest neighbor interaction on thermodynamic properties of ultrathin magnetic films. This work studies on the thermodynamic properties of the ultra-thin magnetic films within the framework of a transverse Ising model. Equations for free energy, entropy and specific heat of spin system are obtained by using the mean field approximation.
VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1 (2019) 76-82
Original article
Effect of Next Nearest Neighbor Interaction
on Thermodynamic Properties of Ultrathin Magnetic Films
Nguyen Tu Niem*, Bach Thanh Cong
Faculty of Physics, VNU University of Science, 334 Nguyen Trai, Thanh Xuan Hanoi, Vietnam
Received 15 January 2019
Revised 07 March 2019; Accepted 15 March 2019
Abstract:This workstudies onthe thermodynamic propertiesofthe ultra-thinmagnetic films within
the framework of a transverse Ising model. Equations for free energy, entropy and specific heat of
spin system are obtained by using the mean field approximation. We also analyze the effect of the
next nearest neighbor interaction on critical temperature, layer magnetization and specific heat of
the thin films.
Keywords: Magnetic ultra-thin films, transverse Ising model, next nearest neighbor interaction,
critical temperature.
1. Introduction
Recently the ultra-thin magnetic films have been used more and more in memory and microwave
devices.
Many researches on the magnetic ultra-thin films have been carried out both experimentally
and theoretically[1, 2]. The magnetic ultra-thin films have magnetic propertieswhich are different
from
those of the bulk [3, 4]. Several models have been used to investigate the features of the films such as:
Heisenbergmodel, transverse Ising model, XY model… Amongthem, the transverse Ising model (TIM)
is often used because its simplicityand usefulness. There are manyworks has applied TIM toinvestigate
magnetic thin films within the framework of the mean field theory and effective field theory [4-7]. In a
previous work [8], we investigated the dependence of the order-disorder phase transition temperature
on the transversal field and obtained the explicit equation for this temperature. However, we only
considerthe nearest-neighbor(n.n)exchange interaction between spins in [8]. In the real magnetic ultra-
thin films, the next nearest- neighbor (n.n.n) exchange interaction may affect strongly on properties of
________
Corresponding author.
E-mail address: nguyentuniem@gmail.com
https//doi.org/ 10.25073/2588-1124/vnumap.4314
76
' j j'
j j
j ' j'
2
F
z
2
z
Y
Y
0
s
s
c
c
N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1(2019) 76-82
77
the films. Therefore, in this paper we use the transverse Ising model to calculate the thermodynamic
properties of the ultra-thin magnetic film in which the n.n.n exchange interaction is taken into account.
All calculations are carried out by using the mean field approximation.
2.Film model and formulation
We consider a transverse Ising magnetic thin film which has simple cubic symmetry composed of n
spin layers. Each layer is defined on the xOy plane and contains N spins (see [8]). The z-direction is
perpendicular to the film surface. The Hamiltonian of the system has the form:
H = −h Sz Sx 1 J (R R )Sz Sz .
j j j, ', j'
(1)
h and Ω represent the external longitudinal and transverse fields given in energy unit (effective
dipole moment
is included in the field h).
Szj and Sxj are the z and x components of a spin operatorSj
at site j in the Oxyz coordinate; J ' is the strength of the exchange interaction between spin at site jth
and j’th, ν is the layer index (ν =1,2,...,n), Rj is the two-component vector denoting the position of jth
spin in this layer.
Following our previous work [8], when taking into account the n.n.n exchange interaction between
spins, we obtain the expression for the free energy of the film in the mean field approximation:
0 = N (Js (0)+ Jp (0)) S
v'
S '
N ln sh(s+1/ 2)Y
sh 2
(2a)
= , =
h2 +2 ;1 = kBT
(2b)
where Js (0)is the exchange strength between in-plane spins and given by:
J11(0) = J (0) = Jnn (0) = Js (0) = Js (Rj ) ,
j
(2c)
Jp (0) is the exchange strength between out-of-plane spinsand
J , 1(0) = J , +1(0) = Jp (0) = Jp (Rj )
j
(2d)
The critical temperature and the critical transversal field of the film can be obtained as:
a
c
0
0
...
0
0
c
a
c
0
...
0
0
0
.
c
.
a
.
c
.
...
...
0
.
. = 0,
(3a)
0
0
0
0
...
a
c
0
0
0
0
...
c
a
where
a =1b () Js (0) and c = −b () Jp (0).
(3b)
1
1 1
s
2
h
s
s
( )
( )
s p
3
2
( )
( )
S
n
2
n
h
'
( )
3
2
s +
S
x
1 x
(1
p
=
= ,
=
s
,
J
J J
s
p
s
p
s s p p s p
78
N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1(2019) 76-82
b (x) =(s+2 )coth(s+2 )x2coth x is the Brillouin function.
(3c)
The longitudinal and transversal magnetization of the films are given by:
mz = b (Y ); mx = b (Y )
(4)
Specific heat is derived from (2a) as:
12bS (Y ) n J (0)+ J (0)
C = 1 n b' Y '=1 with =kBT = 1
=1 12bS Y + bs (Y ) (Js (0)+ Jp (0))
'=1
(5a)
12
and b' (x)= 1 2 is the 1st derivative of Brillouin function.
4sh2 2 sh2 s + 2 2
(5b)
From (5a) one sees that specific heat C tend to zero in the limit T 0 .
3. Numerical result and discussion
In this section, we carry out the numerical calculations for the properties of the magnetic ultra-thin
film.The numerical calculations for the cubic spin lattice is done when the n.n.n exchange is taken into
account. We define following parameters:
J(2) Jp ) J(2)
(1) 1 (1) 2 (1)
s s s
(6)
J(1) (J(1) )
denote the exchange strength between in-plane (out-of-plane) n.n spins. J(2) (J(2) )
denote the exchange strength between in-plane (out-of-plane) n.n.n spins.
is the strength of the competition between the surface interactions of
the in-plane n.n spins and
the in-plane n.n.n spin; 2 is the parameter for the difference between the exchange interactions of the
out - of plane n.n.n spins and the in-plane n.n spins. Ratio 1 is the anisotropic parameter that shows the
difference between the exchange interactions of the in-plane and out-of plane n.n spins. We suppose
that the interactions of the n.n.n spins is smaller than the interactions of the n.n spins (that means
J(2) < J(1),J(1) and J(2) < J(1),J(1) ).
Firstly, we consider the results which are depicted in Fig.1 where the critical temperature is plotted
as a function of 2 and with different values of the transversal field. As shown in the figure, we can
found that the critical temperature TC increases with the increase of 2 and . For the film, we find that
the transversal field has a limited value Ωl. When Ω is smaller than Ωl, the film can change from
ferromagnetic phase into paramagnetic phase with all values of and 2 , but when Ω is larger than Ωl
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Effect of next nearest neighbor interaction on thermodynamic properties of ultrathin magnetic films. This work studies on the thermodynamic properties of the ultra-thin magnetic films within the framework of a transverse Ising model. Equations for free energy, entropy and specific heat of spin system are obtained by using the mean field approximation..

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VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 76-82 Original article Effect of Next Nearest Neighbor Interaction on Thermodynamic Properties of Ultrathin Magnetic Films Nguyen Tu Niem*, Bach Thanh Cong Faculty of Physics, VNU University of Science, 334 Nguyen Trai, Thanh Xuan Hanoi, Vietnam Received 15 January 2019 Revised 07 March 2019; Accepted 15 March 2019 Abstract:This workstudies onthe thermodynamic propertiesofthe ultra-thinmagnetic films within the framework of a transverse Ising model. Equations for free energy, entropy and specific heat of spin system are obtained by using the mean field approximation. We also analyze the effect of the next nearest neighbor interaction on critical temperature, layer magnetization and specific heat of the thin films. Keywords: Magnetic ultra-thin films, transverse Ising model, next nearest neighbor interaction, critical temperature. 1. Introduction∗ Recently the ultra-thin magnetic films have been used more and more in memory and microwave devices. Many researches on the magnetic ultra-thin films have been carried out both experimentally and theoretically[1, 2]. The magnetic ultra-thin films have magnetic propertieswhich are different from those of the bulk [3, 4]. Several models have been used to investigate the features of the films such as: Heisenbergmodel, transverse Ising model, XY model… Amongthem, the transverse Ising model (TIM) is often used because its simplicityand usefulness. There are manyworks has applied TIM toinvestigate magnetic thin films within the framework of the mean field theory and effective field theory [4-7]. In a previous work [8], we investigated the dependence of the order-disorder phase transition temperature on the transversal field and obtained the explicit equation for this temperature. However, we only considerthe nearest-neighbor(n.n)exchange interaction between spins in [8]. In the real magnetic ultra-thin films, the next nearest- neighbor (n.n.n) exchange interaction may affect strongly on properties of ________ ∗Corresponding author. E-mail address: nguyentuniem@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4314 76 N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1(2019) 76-82 77 the films. Therefore, in this paper we use the transverse Ising model to calculate the thermodynamic properties of the ultra-thin magnetic film in which the n.n.n exchange interaction is taken into account. All calculations are carried out by using the mean field approximation. 2.Film model and formulation We consider a transverse Ising magnetic thin film which has simple cubic symmetry composed of n spin layers. Each layer is defined on the xOy plane and contains N spins (see [8]). The z-direction is perpendicular to the film surface. The Hamiltonian of the system has the form: H = −h Sz − Sx − 1  J (R −R )Sz Sz . (1)  j  j  j, ', j' h and Ω represent the external longitudinal and transverse fields given in energy unit (effective dipole moment is included in the field h). Szj and Sxj are the z and x components of a spin operatorSj at site j in the Oxyz coordinate; J ' is the strength of the exchange interaction between spin at site jth and j’th, ν is the layer index (ν =1,2,...,n), Rj is the two-component vector denoting the position of jth spin in this layer. Following our previous work [8], when taking into account the n.n.n exchange interaction between spins, we obtain the expression for the free energy of the film in the mean field approximation: 0 = N (Js (0)+ Jp (0)) S v' S ' − N ln sh(s+1/ 2)Y (2a)  sh 2  =  , = h2 +2 ;−1 = kBT (2b) where Js (0)is the exchange strength between in-plane spins and given by: J11(0) = J (0) = Jnn (0) = Js (0) = Js (Rj ) , (2c) j Jp (0) is the exchange strength between out-of-plane spinsand J , −1(0) = J , +1(0) = Jp (0) = Jp (Rj ) (2d) j The critical temperature and the critical transversal field of the film can be obtained as: a c 0 0 ... 0 c a c 0 ... 0 0 c a c ... 0 . . . . ... . 0 0 0 0 ... a 0 0 0 0 ... c where 0 0 . = 0, (3a) c a a =1− b () Js (0) and c = −b () Jp (0). (3b) 78 N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1(2019) 76-82 b (x) =(s+2 )coth(s+2 )x− 2coth x is the Brillouin function. (3c) The longitudinal and transversal magnetization of the films are given by: mz =  b (Y ); mx =  b (Y ) (4) Specific heat is derived from (2a) as:  1− 2bS (Y ) n J (0)+ J (0)   C = 1 n    b' Y    '=1 with  =kBT = −1 (5a)  =1 1− 2bS Y + bs (Y )  (Js (0)+ Jp (0))       '=1   12 and b' (x)= 1 −  2 is the 1st derivative of Brillouin function. (5b) 4sh2  2 sh2 s + 2 2 From (5a) one sees that specific heat C tend to zero in the limit T 0 . 3. Numerical result and discussion In this section, we carry out the numerical calculations for the properties of the magnetic ultra-thin film.The numerical calculations for the cubic spin lattice is done when the n.n.n exchange is taken into account. We define following parameters: J(2) Jp ) J(2) (1) 1 (1) 2 (1) s s s (6) J(1) (J(1) ) denote the exchange strength between in-plane (out-of-plane) n.n spins. J(2) (J(2) ) denote the exchange strength between in-plane (out-of-plane) n.n.n spins.  is the strength of the competition between the surface interactions of the in-plane n.n spins and the in-plane n.n.n spin; 2 is the parameter for the difference between the exchange interactions of the out - of plane n.n.n spins and the in-plane n.n spins. Ratio 1 is the anisotropic parameter that shows the difference between the exchange interactions of the in-plane and out-of plane n.n spins. We suppose that the interactions of the n.n.n spins is smaller than the interactions of the n.n spins (that means J(2) < J(1),J(1) and J(2) < J(1),J(1) ). Firstly, we consider the results which are depicted in Fig.1 where the critical temperature is plotted as a function of 2 and  with different values of the transversal field. As shown in the figure, we can found that the critical temperature TC increases with the increase of 2 and . For the film, we find that the transversal field has a limited value Ωl. When Ω is smaller than Ωl, the film can change from ferromagnetic phase into paramagnetic phase with all values of  and 2 , but when Ω is larger than Ωl N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1(2019) 76-82 79 we can only observe the transition from ferromagnetic phase into paramagnetic phase with some values of  and2 . (a) (b) Fig1. The dependence of the critical temperature of 3-layer film on the strength of the competition between the interactions of the n.n spins and the then.n.n spins out-of plane 2 = J(2) / J(1) (a) and in-plane  = J(2) / J(1) (b) with various transversal fields.Here 1 = J(1) / J(1) =0.5and s=1. Fig. 2 shows the effect of n.n.n exchange interaction on the critical temperature. The dotted line is plotted with 2 = 0.05.The dashed line is plotted with 2 = −0.1. It is obvious that there is a great difference of the critical of the films when n.n.n exchange interaction is taken account. Fig.2. The effect of then.n.n exchange interaction on Tc for differenent thickness of the film. Here  =1.0 ,1 = 0.5 , = 0.1, s=1. Fig.3. The effect of then.n.n exchange interaction on the phase diagram of 4-layer film. Here 1 = 0.5 , 2 = 0.05, = 0.1, s=1. Comparing between dash dotted line with dotted linein Fig.2,we see that the critical temperature of the films with the n.n.n interaction increases more rapidly than that with the n.n interaction. Besides, if 2 < 0, we also see that the critical temperature of the film with n.n.n interaction is much smaller than that when 2 > 0. It is obvious that strength of the n.n.n interaction affects quite strongly on the critical temperature TC of the film. Fig.3 indicates the dependence of the phase diagram on the strength of the 80 N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1(2019) 76-82 n.n.n exchange interaction of the films with 4 layers. Obviously, when taking into the n.n.n exchange interaction, the range of the ferromagnetic phase in the phase diagram is larger. Next we investigate the nature of the magnetization components with the existence of the n.n.n exchange. For simplicity, in the numerical calculations we note that the film is symmetric. Fig.4a. presents dependence on temperature of the magnetization components of the symmetric two layer film for a given transversal field. The existence of the n.n.n exchange leads to the extension of the longitudinal magnetization mz but the reduction oftransversal magnetization mx. At a given temperature, the larger 2 leads to the decrease of mz more rapidly as shown in Fig.4b. 1.0 0.8 mz(NNN exchange) mx(NNN exchange) mz(NN exchange) mx(NN exchange) 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 kBT/J(1) Fig.4a. The thermal variation of the spin components for symmetric double layer thin film. Here ξ=0.1, s=1,1 = 0.5 . Fig.4b. The dependence of the longitudinal spin of symmetric double layer thin film on the transversal field with different 2 at given temperature kBT / J(1) =1.8 , ξ=0.1, s=1,1 = 0.5 . The influence of  and 2 on the temperature dependence of the longitudinal magnetization and the specific heat of the symmetric three layer film is shown in Fig.5 and Fig.6. m(1), m(2) and m(3) are longitudinal magnetizations of the first layer (the surface), the second layer, and the third layer (the surface). Because the film is symmetric, m(1) is equal to m(3). (a) (b) N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1(2019) 76-82 81 Fig.5. The thermal variation of the longitudinal magnetization of three-layer thin film with different values of  for 1 = 0.5, 2 = 0.01(a) and different values of 2 for 1 = 0.5,  = 0.1(b). Parameters is chosen: s=1; / J(1) =1.5 It can be seen fromFig.5 and Fig.6 that the layer magnetization and peakof the specific heat increase with the increase of the n.n.n exchange interaction. We find that the layer magnetizations are m(2) > m(1) for a fixed value of 2 and  , namely the inner layer magnetization is larger than the surface magnetization. Fig.6 also reveals the effect of 2 and  on the peak of the specific heat. The increase of  leadstotheshiftofthepeakofthespecificheattowardthehighertemperature(Fig.6a),whiletheincrease of 2 onlyleads to the increase of the value of the specific heat of the film(Fig. 6b). (a) (b) Fig.6. The thermal variation of the specific heat of three-layer thin film with different values of  for 1 = 0.3, 2 = 0.01(a) and different values of 2 for 1 = 0.3,  = 0.1(b). All the curves are plotted with s=1; / J(1) =2.0. 4. Conclusion In conclusion, we have considered features of the magnetic ultra-thin film with the n.n.n interaction. The contribution of the n.n.n interaction causes the increase of the critical temperature, the transversal field, the longitudinal magnetization and the specific heat. The numerical calculations imply that the n.n.n interaction has an influence quite clearly on the magnetic properties of the thin films. Acknowledgement This research is funded by the VNU University of Science under project number TN.17.04. 82 N.T. Niem, B.T.Cong / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1(2019) 76-82 References [1] C.L. Wang, W.L. Zhongand P.L. Zhang, The Curie temperature of ultra-thinferroelectric films, Journal of Physics: Condensed Matter 4 (1992) 4743-4749 doi: 10.1088/0953-8984/4/19/014 [2] B. H. Teng and H. K. Sy, Phase diagrams of the transverse Ising model with a four-spin interaction, Europhysics Letters 73 (4) (2006) 601-606. https://doi.org/10.1209/epl/i2005-10437-y [3] T. Kaneyoshi, New aspects of magnetic properties in a transverse Ising thin film, Physica A 328 (2003) 174-184. doi:10.1016/S0378-4371(03)00543-0. [4] T. Kaneyoshi, The effective susceptibility exponent in transverse Ising thin films, Physica Status Solidi (b) 241 (2004) 213-218. doi: 10.1002/pssb.200301917 [5] T. Kaneyoshi, Effective critical exponents of magnetization in transverse Ising thin films, Physica A 332 (2004) 367-379. doi:10.1016/j.physa.2003.10.016. [6] T. Kaneyoshi, Ferrimagnetic magnetizations in a thin film described by the transverse Ising model, Physica Status Solidi B 246 (2009) 2359-2365. doi: 10.1002/pssb.200945176. [7] ÜmitAkınci, Effects of the randomly distributed magnetic field on the phase diagrams of the transverse Ising thin film, Journal of Magnetism and Magnetic Materials 329 (2013) 178-187. https://doi.org/10.1016/j.jmmm.2012.10.034. 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