Magnetization and isothermal magnetic entropy change of a mixed spin-1 and spin-2 Heisenberg superlattice

Author’s Accepted Manuscript Magnetization and isothermal magnetic entropy change of a mixed spin−1 and spin-2 Heisenberg superlattice Ping Xu, An Du www.elsevier.com/locate/physb

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S0921-4526(17)30350-2 http://dx.doi.org/10.1016/j.physb.2017.06.052 PHYSB310030

To appear in: Physica B: Physics of Condensed Matter Received date: 22 May 2017 Revised date: 16 June 2017 Accepted date: 18 June 2017 Cite this article as: Ping Xu and An Du, Magnetization and isothermal magnetic entropy change of a mixed spin−1 and spin-2 Heisenberg superlattice, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.06.052 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Magnetization and isothermal magnetic entropy change of a mixed spin-1 and spin-2 Heisenberg superlattice Ping Xu, An Du* Department of Physics, Northeastern University, Shenyang 110819, China E-mail address: [email protected] (An Du). [email protected] (Ping Xu).

*

Corresponding author.

Abstract A superlattice composed of spin-1 and spin-2 with

ABAB … structure was described with

Heisenberg model. The magnetizations and magnetic entropy changes under different magnetic fields were calculated by the Green's function method. The magnetization compensation phenomenon could be observed by altering the intralayer exchange interactions and the single-ion anisotropies of spins. Along with the temperature increasing, the system in the absence of magnetization compensation shows normal magnetic entropy change and displays a peak near the critical temperature, and yet the system with magnetization compensation shows normal magnetic entropy change near the compensation temperature but inverse magnetic entropy change near the critical temperature. Finally, we illustrated the reasons of different behaviors of magnetic entropy change by analyzing the contributions of two sublattices to the total magnetic entropy change.

Keywords: Magnetic superlattice; Magnetic entropy change; Compensation phenomenon; Green’s function.

1. Introduction With the growing awareness of energy conservation, magnetic refrigeration possessing environmental protection and energy saving more and more aroused concern. As the key content for magnetic refrigeration, the magnetocaloric effect of material has accumulated much experience from practice and theory [1-10]. One important physical quantity to characterize the magnetocaloric effect is the isothermal magnetic entropy change, which is the greater contributor in total entropy change in most cases, so it is significant to explore the factors to influence the magnetic entropy

1

change of magnetic materials. As one of important magnetic materials, the ferrimagnet may have the magnetization compensation phenomenon [11-23], and may show inverse magnetic entropy change in some cases [24-26]. The compensation behavior means that the net magnetization is zero but the magnetizations of two or more sublattices are not zero at a certain temperature in the ferrimagnetic system. The compensation phenomenon has been widely studied in theory. For example, the study concerning Ising system and Heisenberg ferrimagnets found one compensation or double compensation phenomena by the mean field method [11, 12], the Green’s function theory [13-16], the recursion method [17], the Oguchi approximation [18], the pair approximation method [19], the Monte-Carlo simulation [20, 21], the effective field theory [22, 23], etc. It is experimentally found that inverse magnetic entropy change relates to the magnetization behavior of the magnetic system [24-26]. The ferrimagnetic Ising model was found to show normal and inverse magnetic entropy change below and near the compensation temperature, respectively [27], the magnetic entropy change in magnetic multilayers was found exhibits complex magnetic entropy change behaviors [28]. In addition, the inverse magnetic entropy change usually occurs in frustration system [29]. Whether the inverse magnetic entropy change in the ferrimagnet is a common behavior, it needs to study further. In this work, we set up a superlattice Heisenberg model composed of mixed spin- 1

and

spin- 2 to study the relation between the magnetic entropy change and the magnetizing process of the system. In Section 2, Heisenberg superlattice model, formulas of magnetization and isothermal magnetic entropy change are presented. In Section 3, the temperature dependences of the total magnetization, the total magnetic entropy change and the magnetic entropy changes of sublattices are discussed. The final remarks and conclusions are given in Section 4.

2

Fig. 1. The schematic drawing of the magnetic superlattice.

2. Model and formulas The magnetic superlattice (Fig. 1) composed of spin-1 and spin-2 with

ABAB … structure is

described with Heisenberg model. For clarify, the sublattice occupied by spin-1 is called sublattice

A and the other is called sublattice B . The Hamiltonian is given by

H = - J A å Si × Si ' - J B å S j × S j ' - J AB å Si × S j ii '

jj '

P

ij

P

^

(1)

- DA å ( S ) - DB å ( S ) - h å S -hå S jz . z 2 i

z 2 j

i

Where sublattice

j

L

B

z i

i

j

represents the nearest-neighbor pair, , and

i (i ') Î sublattice A

j ( j ') Î

S A = 1 , S B = 2 . The first and second terms are exchange interactions

between spins in sublattice

A and sublattice B respectively, the third term is the interlayer

exchange interaction between spins in sublattice

A and sublattice B , the fourth and fifth terms

are the single-ion anisotropy energies of spins in sublattice

3

and

A and sublattice B respectively, and

the last two terms are Zeeman energies of spins in a magnetic field. Here the Lande factors are taken as the same values for all spins. In addition,

J A (> 0) and J B (> 0) denote the ferromagnetic

nearest-neighbor interactions between spins in sublattice

J AB (< 0) sublattice

A and sublattice B respectively, and

denotes an antiferromagnetic interlayer exchange interaction between spins in

A and sublattice B . The parameters DA and DB reflect the values of the

single-ion anisotropies of spins in sublattice magnetic field

A and sublattice B respectively. The external

h is supposed to point along z direction of spins in sublattice B .

Establishing spin Green functions [13-15, 30, 31], and decoupling by random phase approximation (RPA) from Tyablikov to the exchange term [32] and by Anderson and Callen’s decoupling scheme to the single-ion anisotropy term [33], we obtain the magnetizations and thermodynamic average values of square of spins in sublattice

A and sublattice B ,

( S P - fP )(1 + fP ) 2 SP +1 + ( S P + 1 + fP )fP 2 SP +1 M P =< S >= , (1 + fP ) 2 SP +1 - fP 2 SP +1 z P

< ( S Pz ) 2 >= (2fP + 1)(1 + fP ) - {S P2fP 2 SP +1 - (1 + S Pz ) 2 (1 + fP )2 S P +1 + 2[ S PfP 2 SP +1 + (1 + fP ) 2 SP +1 (1 + S P )](1 + fP )} / [(1 + fP )2 SP +1 - fP 2 SP +1 ], where

(2)

(3)

P = A or B , and the auxiliary functions are given by,

Ek+ - Ck - h Ek- - Ck - h 2 1 [ ], + å N k Ek+ - Ek- e b Ek - 1 e b Ek - 1

(4a)

Ek+ - Ak - h Ek- - Ak - h 2 1 fB = å + [ ], + N k Ek - Ek- e b Ek - 1 e b Ek - 1

(4b)

fA =

here

N is the total number of spins, and b = 1 / k BT , and

Ek± = é Ak + Ck + 2h ± ( Ak - Ck ) 2 + 4 Bk Dk ù / 2, ë û

(5)

Ak = DAt A + 4 J A (1 - g k ) < S Az > +2 J AB < S Bz >,

(6a)

Bk = -2 J ABg k' < S Az >,

(6b)

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Ck = DBt B + 4 J B (1 - g k ) < S Bz > +2 J AB < S Az >,

(6c)

Dk = -2 J ABg k' < S Bz >,

(6d)

g k' = cos(k z c / 2),

(7a)

g k = éëcos(k x a ) + cos(k y a ) ùû / 2,

(7b)

t P = {2 - [ S P ( S P + 1)- < ( S Pz ) 2 >] S P2 } < S Pz >, The total magnetization

(8 )

M of the system is defined by

M = M A + M B.

(9)

According to Maxwell’s relations,

( ¶S

¶h )T = ( ¶M ¶T ) h ,

(10)

the magnetic entropy change in an isothermal process between the external magnetic field and

h=0

h > 0 can be calculated as follows:

n -1 h æ ö DS (T , Dh) = ò ( ¶M ¶T ) dh » d h ç d M 1 + 2å d M k + d M n ÷ 2d T , 0 hk k =2 è ø

(11)

where the partial derivative is replaced by finite difference approximation and the integration by summation approximately. 3. Results and discussion In this paper, the reduced unit was used for the purpose of simplicity, namely

| J AB | is used

as reduced coefficient. The intrinsic parameters of the system are normalized by

| J AB | , thus

J a = J A / | J AB | , J b = J B / | J AB | , Da = DA / | J AB | , and Db = DB / | J AB | , and Boltzmann constant

k B is taken as 1 .

3.1 The total magnetization and magnetic entropy change As shown in Fig. 2(a) and (c), the total magnetization temperature gets to the critical temperature 5

M falls monotonically to zero as the

Tc when the exchange interaction J a or the single-i-

Fig. 2. Four groups of magnetization ( h ( h/ |

J a and Da . (a) and (c). Temperature dependences of the total spontaneous

= 0 ). (b) and (d). Temperature dependences of the total magnetic entropy change

J AB |= 1.7 ).

on anisotropy of spins

Da of sublattice A is relatively small. As J a or Da increases, the

compensation phenomenon of the magnetization arises before the magnetization is zero but the magnetizations of sublattice

A and sublattice B are not zero and

the temperature at this point is called the compensation temperature

Tcomp . Subsequently, M

firstly increases along negative direction and then drops to zero around the directly influence the energy of spins in sublattice magnetic structure of sublattice from

Tc , that is the total

Tc . Both J a and Da

A . Increasing any term will stabilize the

A , thus the speed of the magnetization decreasing in sublattice A

1 with the temperature rise slows down and magnetization compensation phenomenon is

more likely to occur. In a certain range of values, the critical temperature increases evidently with 6

increasing

J a or Da but the compensation temperature changes little.

Applying external magnetic field to the system, the total magnetic entropy changes varying with the temperature are shown in Fig. 2(b) and (d). For the system without magnetization compensation, DS shows normal magnetic entropy change behavior in agreement with that of the common magnetic materials, displaying negative maximum near the critical temperature. However, the case of the system with magnetization compensation is different. DS increases to a positive value after firstly decreases from zero and drops to negative value finally with increasing temperature. There is a negative peak near the

Tcomp but a positive peak near the Tc . In the case

of the system with unobvious compensation behavior, the interval between the

Tcomp and the Tc

is lesser and the two peaks of magnetic entropy change very close to each other. Indeed, the ferrimagnetic compounds such as Gd3Go11B4 and (Gd0.8Y0.2)3Go11B4 demonstrate these magnetic entropy change characteristics [26]. Similarly, as shown in Fig. 3, by adjusting the exchange interaction anisotropy

J b or the single-ion

Db in sublattice B , the effect of affecting thermal fluctuation of spins in sublattice

B will change, thereby the Tcomp and the Tc of the system will change. In the absence of magnetization compensation, the system shows normal magnetic entropy changes with a peak near the

Tc . However, when the compensation phenomenon exists in the system, there is a negative

peak near the

Tcomp and a positive peak near the Tc for DS .

3.2 The total magnetic entropy change in different external magnetic fields In order to clearly reflect the behavior of the total magnetic entropy change, we present the magnetic entropy change in different external magnetic fields in Fig. 4 and 5, where the external m-

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Fig. 3. Four groups of

Jb

spontaneous magnetization ( h entropy change ( h / |

and

Db . (a) and (c). Temperature dependences of the total

= 0 ). (b) and (d). Temperature dependences of the total magnetic

J AB |= 1.7 ).

agnetic field increases from

0.1

to

1.7

with

0.1

step. Whether the compensation

phenomenon exists or not in the system, the overall trends of DS are almost invariant in the range of given external magnetic field. Without compensation phenomenon, the system shows normal magnetic entropy change in the whole range of temperature, and presents a greater value (Fig.4 (b)) or maximum (Fig.4 (d)) near the

Tc . Along with increasing magnetic field, the forms of DS are invariant but the value

amplifies. In particular, a tiny peak occurs in Fig.4 (a) and (c) during the temperature increasing when the external magnetic field is greater than a certain value. And the bigger the external magnetic field, the more prominent this phenomenon is. It is found that these two systems are adjac8

Fig. 4. The system without compensation, the temperature dependences of total magnetic entropy

change

in

various

external

magnetic

fields

(

h / | J AB |= 0.1 ~ 1.7 ,

Dh / | J AB |= 0.1 ). (The magnetization behaviors are given in Fig. 1 and 2.) The red arrows represent the direction of increasing external magnetic field. 9

Fig. 5. The system with compensation, (a) the temperature dependence of the total spontaneous magnetization ( h

= 0 ); (b) the temperature dependences of the total magnetic entropy change

in various external magnetic fields (

h / | J AB |= 0.1 ~ 1.7 , Dh / | J AB |= 0.1 ). The red

arrows represent the direction of increasing external magnetic field. ent to the system with magnetization compensation, so the magnetic entropy change characteristic of the system with magnetization compensation emerges indistinctly. Additionally, sublattice

A

has easy-plane anisotropy in Fig.4 (b), and the numerical value of DS is smaller. The system in Fig.4 (d) is far away from the system with magnetization compensation, and the absolute value of the magnetic entropy change is greater. The magnetic entropy change result as the compensation phenomenon exists in the system is given in Fig. 5. The total magnetic entropy change DS and an inverse peak near the

shows a normal peak near the

Tcomp

Tc , turning into negative value when the temperature exceeds the

critical temperature. Likewise, with increasing external magnetic field, the form of DS

is 10

invariant but the value amplifies, and augment of the negative peak is visible, but the positive peak is not and almost unchanged when the field exceeds

1.4 .

In molecular field assumption, Ref. [27] summarized the magnetic entropy change characteristic in the ferrimagnetic system, exhibiting a negative peak below the positive peak near the

Tcomp but a

Tcomp ( h = 0 ) and a negative peak near the Tc again. The results of the

Ref. [28], referring to the magnetic entropy change of ferrimagnetic multilayers, coincide with our results under some parameters. 3.3 Magnetic entropy changes of sublattice A and sublattice B To analyze the behavior of the magnetic entropy change in ferrimagnetic system, we calculated the contributions to the total magnetic entropy change of sublattice A and sublattice

B . As shown in Fig. 6, the magnetic entropy change of sublattice A denoted as DS A is positive while sublattice B denoted as DS B is negative in overall temperature. No matter whether the compensation phenomenon exists or not in the system, increases to a peak near the

DS A

Tc and then decreases toward zero. In the present system, the

external magnetic field is antiparallel to the direction of spontaneous magnetization of sublattice

A , the magnetic field makes the magnetic moments of sublattice A tend to rotate toward the direction of the magnetization of sublattice B . Thus the competition between interlayer exchange energy and Zeeman energy for spins in sublattice A

appears, which results in unstable and

disordered magnetic structure and positive magnetic entropy change in sublattice A . As the temperature goes up, the thermal fluctuation of spins will augment and DS A

will increase

continually. When the temperature exceeds the critical temperature, the magnetic ordering vanishes and therefore DS A decreases. The inverse magnetic entropy change appearing in frustr-

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Fig. 6. Sublattice magnetic entropy changes DS A and DS B versus temperature at

h / | J AB |= 1.7 , in which the diverse parameters are the same as in Fig. 2 and Fig. 3.

12

ation system [29] may be similar to the above mechanism. The result of the magnetic entropy change in sublattice B is different with that of sublattice

A evidently. When the system has no magnetization compensation, DS B has a negative peak near the

Tc , which is analogous to the normal magnetic entropy change of a ferromagnet.

Because of the external magnetic field paralleling to the direction of magnetization in sublattice

B , increasing the external magnetic field makes the structure of sublattice B more stable and so

DS B is negative. In the case of the system with magnetization compensation, negative DS B shows a big numerical value near the near the

Tcomp . It is visible evident that the numerical value of DS B

Tcomp is bigger than that near the Tc when the interval between the Tcomp and the

Tc is large enough. It seems that the peak of DS B shifts from the Tc to the Tcomp . As the temperature exceeds the

Tcomp , the magnetization of sublattice B

becomes small and

approaches zero, so the numerical value of the magnetic entropy change begins to decrease. On the whole, the case without the magnetization compensation phenomenon, the system undergoes normal magnetic entropy change mainly deriving from a greater negative value of

DS B . But when the compensation phenomenon exists in the system, the normal peak of DS B generates negative DS

near the

Tcomp , and numerical value of positive DS A bigger than

negative DS B generates a positive DS near the from the

Tc . Actually, that is the peak of DS B shift

Tc to the Tcomp accounts for inverse magnetic entropy change near the Tc in the

system with magnetization compensation. 4. Conclusions To conclude, we investigate the magnetic entropy change of a layered superlattice with and without magnetization compensation phenomenon. The external magnetic field is parallel to the

13

direction of magnetization in sublattice B as a condition, the magnetic entropy change of the sublattice A is positive but that of the sublattice B is negative in the whole temperature region. For the system without compensation, both DS A

and DS B

have a peak near the critical

temperature and the numerical value of DS B is greater than that of DS A , therefore the total magnetic change DS is negative all the time and displays a larger value or peak near the critical temperature. While for the system with compensation, DS A

shows a peak near the critical

temperature, but the peak of DS B appears near the compensation temperature. Thus the total magnetic entropy change shows a negative peak near the compensation temperature and a positive peak near the critical temperature namely inverse magnetic entropy change. Moreover, the overall trends of DS are almost invariant in the range of given external magnetic field.

Funding: This work was supported by the Ministry of Science and Technology of China [grant number 2011CB606404].

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