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A comparative study of critical phenomena and magnetocaloric properties of ferromagnetic ternary alloys
Accepted Manuscript A comparative study of critical phenomena and magnetocaloric properties of ferromagnetic ternary alloys Yusuf Yüksel, Ümit Akıncı PII:
S0022-3697(17)31013-2
DOI:
10.1016/j.jpcs.2017.09.015
Reference:
PCS 8208
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 6 June 2017 Revised Date:
25 August 2017
Accepted Date: 4 September 2017
Please cite this article as: Y. Yüksel, Ü. Akıncı, A comparative study of critical phenomena and magnetocaloric properties of ferromagnetic ternary alloys, Journal of Physics and Chemistry of Solids (2017), doi: 10.1016/j.jpcs.2017.09.015. 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 proof before it is published in its final 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.
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A comparative study of critical phenomena and magnetocaloric properties of ferromagnetic ternary alloys ¨ Yusuf Y¨uksel, Umit Akıncı
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Department of Physics, Dokuz Eyl¨ul University, TR-35160 Izmir, Turkey
Abstract
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Magnetic and magnetocaloric properties, as well as the phase diagrams of a ferromagnetic ternary alloy system have been studied. A detailed comparison of two different methods, namely the effective field theory (EFT), and Monte Carlo (MC) simulations has been provided. Our numerical data show that the general qualitative picture presented by two methods are in a good agreement with each other. In terms of the magnetocaloric properties, our results yield that it is possible to design magnetic materials with a variety of working temperatures and magnetocaloric properties (such as large ∆S M and q values) by manipulating the magnetic phase transition via tuning the compositional factor (i.e. the mixing ratio of sublattice ions). The observed magnetocaloric effect has been found to be a direct one with ∆S M < 0 associated with a second order phase transition. Keywords: Alloy, Critical phenomena, Magnetism, Magnetocaloric, Phase transition
1. Introduction
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Magnetocaloric effect (MCE) [1, 2, 3, 4, 5, 6, 7] which is the manifestation of temperature change in a magnetic material due to adiabatically varying magnetic fields has recently attracted particular attention not only from academical point of view, but also due to its potential in technological [8, 9, 10], as well as biomedical applications [11]. The theoretical and experimental aspects regarding the phenomenon have been widely investigated in the literature [12, 13, 14, 15, 16, 17, 18]. Indeed, it is directly related to the magnetic entropy change of a magnetic material under adiabatic conditions. Fundamentally, the total entropy of a magnetic material can be written as
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S tot (T, H) = S L (T, H) + S e (T, H) + S M (T, H),
(1)
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where the first term is due to lattice vibrations, the second one comes from the electronic contribution, and the third term represents the magnetic entropy. Under adiabatic conditions, the total entropy does not change. Hence, upon application of a magnetic field on a magnetic material, the magnetic dipole moments tend to align parallel with each other which reduces the magnetic entropy. However, lattice and electronic entropy should increase, and consequently the temperature of the material increases. On the other hand, when the magnetic field is reduced towards zero, the magnetic entropy increases due to increasing magnetic disorder. In this case, in order to keep the total entropy unaltered, the electronic and lattice contributions become reduced, and the material is cooled. Using this process, one can utilize the magnetic material for cooling/heating applications. Former studies suggest that MCE is maximized around the ferromagnetic-paramagnetic transition temperature [1]. For a potential candidate to be used in magnetocaloric applications, the material should exhibit magnetic phase transition around the room temperature. Theoretically, one can calculate the MCE indirectly using the relations [19] ( ) ) ∫ ( ∫ ∂M T ∂M dH, ∆S M = dH, (2) ∆T ad = − C x ∂T H ∂T H ∗ Corresponding
author. Tel.: +90 2324119544; fax: +90 2324534188. Email address: [email protected] (Yusuf Y¨uksel)
Preprint submitted to Journal of Physics and Chemistry of Solids
August 25, 2017
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∫ q=−
T2
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which are the outcomes of the thermodynamic Maxwell relations. In Eq. (2), ∆T ad is the adiabatic temperature change whereas ∆S M is the isothermal entropy change. One should pay attention on the fact that the heat capacity C x in Eq. (2) is not independent from magnetic field (except at very low temperatures at which MCE is very weak). Hence, C x cannot be simply taken out of integration. For conventional magnetic materials exhibiting ferromagneticparamagnetic phase transitions, the magnetization is pronounced at low temperature region. However, with increasing temperature, magnetization becomes reduced, and gradually decreases. Hence, ∆S M becomes negative. In this case, the material heats up with increasing magnetic field strength whereas it cools down upon decreasing the magnetic field strength. Furthermore, antiferromagnetic or ferrimagnetic materials may exhibit positive ∆S M with increasing temperature which is known as the inverse MCE in the literature [20, 21, 22, 23]. As another physical quantity which measures the suitability of magnetic material for magnetic cooling applications, we can introduce the cooling capacity q which is given by the formula ∆S M (T )H dT,
(3)
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T1
2. Formulation
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which is a measure of the amount of heat that can be transferred from the cold end to the hot end (denoted by T 1 and T 2 , respectively) in one thermodynamical cycle during the adiabatic process. Up to now, a great many magnetic materials has been tested for their magnetocaloric features [17], and among them the rare earth Gd suits well for magnetic cooling technology with an adiabatic temperature change of ∆T ad = 14K in 7T magnetic field [24, 25], and isothermal magnetic entropy change |∆S M | = 6.1 and 10.6 (in units of Jkg−1 K−1 ) for magnetic fields 2T and 5T, respectively [12]. However, due to its finite resources, the elementary Gd cannot be used in mass production of magnetic cooling systems [26]. Hence, it is a crucial point to search for new materials with large amount of resources, room temperature magnetic phase transition, and large magnetocaloric properties around the working temperature. In addition, tuning the critical temperature is an important issue in magnetocaloric applications. It is the main motivation of the present study to theoretically elucidate the magnetic, as well as the magnetocaloric properties of ferromagnetic ternary alloys which contain three different magnetic sublattices. Conventional mean field theory (MFT) predicts an abrupt fall of magnetic specific heat around T c , however the situation is not the case in experimental systems. Hence, we use effective field theory (EFT) together with the Monte Carlo (MC) simulations in our calculations. In terms of magnetocaloric applications, MC simulation technique is successfully used in the theoretical literature [27, 28, 29, 30, 31, 32, 33, 34]. Our another aim in this study is to test the accuracy of numerical data produced by EFT in comparison with MC simulation technique. The outline of the paper is as follows: In Sec. 2 we present our model and formulation details. Numerical results and related discussions are given in Sec. 3, and finally Sec. 4 is devoted to our conclusions.
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In order to investigate the ferromagnetic ternary alloy system, consideration of the nearest neighbor exchange coupling is sufficient, and the next-nearest exchange interactions can be neglected [35]. Therefore, the Hamiltonian describing a ternary alloy with the chemical formula AB p C1−p in an external magnetic field can be written as ∑ ∑ ∑ ∑[ ] H =− JAB S iA S Bj ζ j − JAC S iA S Cj (1 − ζ j ) − H S iA − H ζ j S Bj + (1 − ζ j )S Cj , (4)
i
j
where S iA , S Bj , S Cj are the spin variables which have been selected as S iA = ±1, 0; S Bj = ± 32 , ± 12 ; S Cj = ± 12 , and H is the homogeneous magnetic field. We note that this selection of spin magnitudes is arbitrary, since the qualitative picture of theoretical results can also be obtained for different spin configurations. In Eq. (4), ζ j represents the site occupancy parameter which can take values zero or unity. If ζ j = 1 then the lattice site j is occupied by an ion B and it is the nearest neighbor of ion A at the site i. The interaction between ions A and B is ferromagnetic (JAB > 0). Similarly, if ζ j = 0 then an ion C is placed on the lattice site j. In this case, ions A and C also interact ferromagnetically with each other (JAC > 0). Our model system is schematically illustrated in Fig. 1. The pseudo spin variables are located on the nodes of a simple cubic lattice, and each ion A can interact with an ion X (X = B or C). For simplicity, the stoichiometry of the system has been selected as 1:1 which means that there 2
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are no vacant sites on the sub-lattice A (pA = 1.0) [36]. Ion concentration of sublattices B and C are given by p and (1 − p), respectively with 0 ≤ p ≤ 1. Hence, p is called the compositional factor. p = 0 and p = 1 special cases correspond to binary mixed AC and AB ferromagnetic structures, respectively. 0.5
S =1/2
1.0
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C
S =1
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A
S =3/2 B
1.5
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Figure 1: A representative visual of an arbitrary plane of the 3D ferromagnetic ternary system with p = 0.75. Sublattices with different spin magnitudes are labeled as S A = 1 (green), S B = 3/2 (yellow), and S C = 1/2 (blue).
2.1. Effective-Field Theory
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In the EFT formalism, we start by obtaining the local fields HiA , H Bj , HCj which can be written for our particular problem as follows z [ ∑ ] B C , HiA = −S iA J S ζ + J (1 − ζ )S + H AB j AC j j j j=1
H Bj
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HCj
= −S iA EiA , z ∑ B A = −S j JAB S i ζ j + Hζ j = −S Bj E Bj , i=1 z ∑ A C , (1 − ζ ) + H(1 − ζ ) = −S j J S j j AC i i=1
= −S Cj E Cj .
(5)
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With the help of Eq. (5), the expectation value (thermal average) of the nth order of the magnetic moment S k can be obtained from [37] ⟨ ⟩ Trk (S kα )n exp(−βHkα ) ⟨(S kα )n ⟩ = , α = A, B, C, (6) Trk exp(−βHkα ) which gives
⟨ ⟩ S iA
⟨ = ⟨
⟨ ⟩ (S iA )2
=
⟨ ⟩ ζ j S Bj
=
2 sinh(βEiA ) 2 cosh(βE Aj ) + 1 2 cosh(βEiA )
⟩ , ⟩
, 2 cosh(βE Aj ) + 1 ⟨ 3 1 B B ⟩ 1 3 sinh( 2 βE j ) + sinh( 2 βE j ) ζj , 2 cosh( 32 βE Bj ) + cosh( 12 βE Bj ) 3
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=
⟨ ⟩ ζ j (S Bj )3
=
⟨ ⟩ (1 − ζ j ) S Cj
=
⟨ 3 1 B B ⟩ 1 9 cosh( 2 βE j ) + cosh( 2 βE j ) ζj , 4 cosh( 32 βE Bj ) + cosh( 21 βE Bj ) ⟨ 3 1 B B ⟩ 1 27 sinh( 2 βE j ) + sinh( 2 βE j ) ζj , 8 cosh( 32 βE Bj ) + cosh( 12 βE Bj ) ⟩ 1⟨ (1 − ζ j ) tanh(βE Cj ) . 2
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⟨ ⟩ ζ j (S Bj )2
(7)
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Next, one should apply the relation exp(α∇) f (x) = f (x + α)| x=0 , i.e. the differential operator technique [38] in Eq. (7), and perform the thermal and configurational averages ⟨⟨...⟩⟩ to get ⟨⟨ ⟩⟩ Xn A = (S iA )nA ⟨⟨∏ ⟩⟩ z = ζ j exp(JAB S Bj ∇) +(1 − ζ j ) exp(JAC S Cj ∇) FnAA (x, H) x=0 , nA = 1, 2 j=1
= =
ZnC
= =
⟨
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Yn B
⟨ ⟨ ⟩⟩ ζ j (S Bj )nB ⟨ ⟨∏ ⟩⟩ z A ζj exp(JAB S i ∇) FnBB (x, H) x=0 , nB = 1, 2, 3 j=1
⟨ ⟩⟩ (1 − ζ j ) (S Cj )nC ⟩⟩ ⟨ ⟨∏ z C (1 − ζ j ) exp(JAB S j ∇) FnCC (x, H) x=0 , nC = 1. j=1
(8)
where the inner and the outer products represent the thermal and configurational averages, respectively. In obtaining the closed form of single site correlations in Eq. (8), we use the exponential relation
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exp(aζ j ) = ζ j exp(a) + 1 − ζ j ,
(9)
along with the property ζ 2j = ζ j . Moreover, the functions Fnδ (x, H) in Eq.(8) are given in Appendix. The next step is to write the exponential operators in Eq. (8) in terms of hyperbolic trigonometric functions by using the exact Van der Waerden spin identities [39, 40]. For spin-1, S iA has three components, and the identity is given by (S iA )2 cosh(a) + S iA sinh(a) + 1 − (S iA )2
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exp(aS iA ) = =
A0 (a) + A1 (a)S iA + A2 (a)(S iA )2 .
(10)
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On the other hand, an exponential operator exp(aS i ) for a general spin S i is given by [38] exp(aS i ) =
2S ∑
ξn (a)(S i )n .
(11)
Bn (a)(S Bj )n ,
(12)
n=0
Hence, the identities for spin-3/2 are given as exp(aS Bj ) =
3 ∑ n=0
and the coefficients Bn (a) are given in Appendix. If the sublattice C consists of spin-1/2 ions then the Van der Waerden identity becomes exp(aS i ) = C0 (a) + C1 (a)S i , where C0 (a) = cosh(a/2) and C1 = 2 sinh(a/2). 4
(13)
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By using Eq.(11) in Eq. (8), mathematically complicated multi-site correlations appear. In this case, the problem becomes mathematically intractable, and one has to make an approximation. As the simplest approximation, one should expand Eq.(8) by neglecting the multi-site correlation identities according to [38, 41] ⟨y j (yk )2 ...(yl )3 zn ⟩ ≈ ⟨y j ⟩⟨(yk )2 ⟩...⟨(yl )3 ⟩⟨(zn )⟩, ⟨S Aj (S kA )2 ...S lA ⟩ ≈ ⟨S Aj ⟩⟨(S kA )2 ⟩...⟨S lA ⟩.
(14)
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where j , k , l , n, y j = ζ j S Bj , and z j = (1 − ζ j )S Cj . Applying approximation (14), and performing the thermal and configurational averages ⟨⟨...⟩⟩ results in [ XnA = pB0 (JAB ∇) + B1 (JAB ∇)Y1 + B2 (JAB ∇)Y2 +B3 (JAB ∇)Y3 + (1 − p)C0 (JAC ∇/2) +C1 (JAC ∇/2)Z1 ]z FnAA (x, H)| x=0 , =
p[A0 (JAB ∇) + A1 (JAB ∇)X1 + A2 (JAB ∇)X2 ]z FnBB (x, H)| x=0 ,
ZnC
=
(1 − p)[A0 (JAC ∇) + A1 (JAC ∇)X1 + A2 (JAC ∇)X2 ]z FnCC (x, H)| x=0 ,
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Yn B
(15)
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where z is the coordination number of the lattice, and ⟨⟨ ⟩⟩ ⟨⟨ ⟩⟩ X1 = S iA = mA , X2 = (S iA )2 = qA , ⟨ ⟨ ⟩⟩ ⟨ ⟨ ⟩⟩ Y1 = ζi S Bj = mB , Y2 = ζi (S Bj )2 = qB , ⟨ ⟨ ⟩⟩ ⟨ ⟨ ⟩⟩ Y3 = ζi (S Bj )3 = rB , Z1 = (1 − ζi ) S Cj = mC . Then the total magnetization of the system can be defined as MT =
1 (mA + mB + mC ). 2
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To evaluate the polynomial form of Eq. (15), one should apply binomial expansion ) n ( ∑ n (x + y)n = xn−i yi , i
(16)
(17)
i=0
in Eq.(15). In this manner, we get the following set of nonlinear equations with six unknowns in polynomial forms, Xn A
=
)( )( )( )( ) k3 ∑ k1 ∑ k2 ∑ k4 ( z ∑ ∑ z k1 k2 k3 k4 p(z−k1 ) (1 − p)k4 −k5 k1 k2 k3 k4 k5
=
AC C
Yn B
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k1 =0 k2 =0 k3 =0 k4 =0 k5 =0
×[B0 (JAB ∇)]z−k1 [B1 (JAB ∇)]k1 −k2 [B2 (JAB ∇)]k2 −k3 [B3 (JAB ∇)]k3 −k4 ×[C0 (JAC ∇/2)]k4 −k5 [C1 (JAC ∇/2)]k5 Y1k1 −k2 Y2k2 −k3 Y3k3 −k4 Z1k5 FnAA (x, H)| x=0 , )( ) k1 ( z ∑ ∑ z k1 p [A0 (JAB ∇)]z−k1 [A1 (JAB ∇)]k1 −k2 [A2 (JAB ∇)]k2 X1k1 −k2 X2k2 FnBB (x, H)| x=0 , k1 k2 k1 =0 k2 =0
ZnC
=
(1 − p)
)( ) k1 ( z ∑ ∑ z k1 [A0 (JAC ∇)]z−k1 [A1 (JAC ∇)]k1 −k2 [A2 (JAC ∇)]k2 X1k1 −k2 X2k2 FnCC (x, H)| x=0 . (18) k1 k2
k1 =0 k2 =0
where the coefficients [An (a)]α , [Bn (a)]γ , [Cn (a)]η can be found in Appendix. In order to obtain the magnetizations of the sub-lattices, as well as the total magnetization, quadrupole moments etc., one should numerically solve the system of non-linear equations given in Eq. (18) as a function of, H, p, and T . Conveniently, in order to determine the phase transition temperature at which the magnetization vanishes, one can linearize Eq. (18) [38]. However, due to the complicated form of the right hand-sides of the equations of state in Eq. (18), we prefer to utilize the zero field susceptibility per spin ) ( ∂MT , (19) χ= ∂H H=0 5
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from which the critical temperature has been determined from maxima of χ at T = T c . We also note that kB has been set to unity for simplicity, and the temperature, exchange interactions, as well as the magnetic field strength have been scaled in units of JAC in the calculations.
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2.2. Monte Carlo Simulation We have also performed Metropolis Monte Carlo simulations of Eq. (4) by imposing periodic boundary conditions in all directions. Pseudo spins are placed on the nodes of a L × L × L simple cubic lattice, and each ion A can interact with an ion X (X = B or C). We simulate systems with linear lattice dimension L = 20 (we found more or less the same results for L > 20; see Appendix). At each Monte Carlo step, lattice sites have been sequentially swept. Data were collected over 105 Monte Carlo steps after discarding the first 25% for thermalization. Following quantities have been calculated during our simulations; ⟨ NA ⟩ 1 ∑ mA = SA , NA i=1
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• the longitudinal component of thermal average of instantaneous sublattice magnetizations, ⟨ NB ⟩ 1 ∑ mB = SB , NB j=1
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where NA = pA L3 /2, NB = pL3 /2, and NC = (1 − p)L3 /2.
⟨ NC ⟩ 1 ∑ mC = SC , NC k=1
(20)
• With the help of Eq. (20), thermal and configurational averages of total magnetization are defined by MT = [NA mA + NB mB + NC mC ]) /N, where N = NA + NB + NC .
(21)
• Finally, direct susceptibility per spin at zero field can be calculated via fluctuation-dissipation theorem ) 1 ( 2 ⟨mT ⟩ − ⟨mT ⟩2 , kB T
(22)
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χ=
where ⟨mT ⟩ = MT , and the quadrupole moment ⟨m2T ⟩ can be calculated by summing up the squares of S A , S B and S C in Eq. (20).
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The location of the transition temperature has been estimated by examining the thermal variation of direct susceptibility which exhibits a prominent peak at T = T c , and the Binder cumulant U4 (L, T ) curves, (see Appendix). In the calculations, we also set kB = 1 for simplicity. 3. Results and discussion
AC C
Here, and in the following presentations, ferromagnetic exchange between ions A and C will be fixed as JAC = 1.0, and the interaction between adjacent ions A and B will be varied. On the other hand, although the present EFT and MC formulations can in principle be applied to any arbitrary coordination number z, we will discuss the results only for z = 6 which mimics the simple cubic lattice structure. When the compositional factor takes value p = 0.0 or p = 1.0, then the ternary alloy system reduces to binary mixed (1, 12 ) and (1, 32 ) ferromagnets, respectively. For instance, in terms of the EFT formalism, for the former case with z = 6, reported critical temperature value is T c = 2.1116JAC [42] whereas for the latter one we have T c = 4.846JAB [43]. For 0.0 < p < 1.0, we have a ternary alloy with three sub-lattices, and the system exhibits rich phase diagrams. Hence, before discussing the magnetic, as well as the magnetocaloric properties, it would be beneficial to present the global phase diagrams of the system. For this aim, in Fig. 2, we depict the phase diagrams in (JAB /JAC − T c /JAC ) plane corresponding to different values of the mixing ratio p. Here, the horizontal line with p = 0.0 represents the critical frontier of binary AC ferromagnet. As expected, when p = 0.0, ions B do not exist in the system and the location of the transition temperature does not depend on varying JAB values. As p increases then the magnetic interactions due to the presence of ions B originate. Depending on the strength of the exchange interaction between ions A and B, the transition temperature of the system 6
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10
p: 0.0
8
0.25
0.75 1.0
4
2
0 0.5
1.0
JAB/JAC
1.5
2.0
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0.0
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Tc/JAC
0.5 6
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Figure 2: Phase diagrams in a (JAB /JAC − T c /JAC ) plane of ferromagnetic ternary alloy system. Different symbols correspond to different compositional factor values. The dashed lines represent the EFT results whereas solid symbols correspond to MC data.
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either increases or decreases as a function of p. It is clear from Fig. 2 that there exists a critical value of JAB below which the transition temperature decreases with increasing p. Within the EFT formalism, we predict this threshold crit crit = 0.43JAC . As seen in Fig. 2, EFT and MC = 0.435JAC whereas in MC calculations, we found JAB value as JAB crit methods provide qualitatively similar results. However, EFT results for the critical temperature and JAB seem a bit larger than the MC results. This is an expected result, since thermal fluctuations are exactly taken into account in MC simulations. The system with the same spin configuration has been previously investigated in Ref. [41] for crit crit = 0.424JAC [41] for = 0.435JAC with its two dimensional counterpart JAB z = 3. By comparing our EFT result JAB crit z = 3, we can claim that EFT prediction on the spatial dimensionality dependence of JAB agrees well with MC results [36, 44]. Namely, the critical value of JAB exhibits very little variation with lattice dimensionality. Previous efforts crit as a function of several physical factors. In this regard, it have been devoted to the investigation of variation of JAB crit has been reported that JAB is independent of lattice stoichiometry [36], exhibits very little variation with spatial lattice dimensionality [36, 45] and increases with increasing exchange anisotropy [46]. Next in Fig. 3, we give the temperature dependence of total magnetization MT , as well as the zero field suscepcrit (Figs. 3c,d), and JAB = 1.0JAC (Figs. 3e,f) with several values tibility χ for JAB = 0.2JAC (Figs. 3a,b), JAB = JAB of p. As seen in Fig. 3a, when the exchange interaction between A and B type ions is weak such as JAB = 0.2JAC , increasing p, hence, increasing the density of ions B causes a decline in the transition temperature of the system, and the susceptibility exhibits a divergent behavior at the transition temperature. On the other hand, according to Fig. 3c crit which is depicted for JAB = JAB , all the MT curves merge to zero at the same transition temperature value for all p. Finally, for JAB = 1.0JAC (Fig. 3e), we have just the opposite scenario of Fig. 3a. Namely, increasing the density of ions B, causes a substantial enhancement in the ferromagnetic character of the alloy by increasing its critical temperature. Saturation value of MT at low temperature region is governed by Eqs. (16) and (21) with MA = 1.0, (1−p) MB = 3p 2 and MC = 2 . Moreover, by inspecting the susceptibility curves depicted in Figs. 3b, 3e, and 3f, we see that they exhibit a prominent divergent behavior at the paramagnetic-ferromagnetic transition temperature. We note that the type of the transition is of second order for both formulations [47]. As a general remark, we see that EFT and MC results are almost identical to each other except the vicinity of the critical region where the system undergoes a phase transition between paramagnetic and ferromagnetic phases. As we mentioned before, critical temperature predicted by EFT is observed to be slightly larger than the values obtained by MC simulations. This is a consequence of omitting the multi-spin correlation identities by using the approximation (17) in EFT calculations. However, the estimated EFT results agree well with our MC data. Up to now, we have seen that the parameters JAB and p play a significant role on the magnetic properties of the ternary alloy system. In the present work, we also consider the magnetocaloric properties of the ferromagnetic ternary alloy model by examining the isothermal variation of magnetic entropy change ∆S M . The variation of ∆S M as a function of temperature is expected to be influenced by compositional factor p, ferromagnetic exchange coupling JAB , 7
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1.4
1.0 p:
100
1.2
J
AB
=0.2J
0.75
0.50
0.25
0.0 AC
0.0
0.25
1.0
10
0.50 0.75
MT
0.8
1.0
1
0.6
p: 0.1
0.0 0.25
0.0
0.50 0.2
0.75
1.0
0.50
0.75
0.01
0.25
J
1.0 0.0 1E-3
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
T/J AC
(b)
1.4
100
p: 1.2
0.0
p: 0.0
0.75 0.8
MT
1.0
0.25
1
0.50 1.0
0.75
0.6
1.0
0.1
AB
=0.43J
AC
=0.435 J
0.01
0.0
(MC)
AC
3.0
0.0
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0.4
AB
AC
1.0
10
0.50
J
=0.2J
2.5
SC
0.25 1.0
J
AB
2.0
T/J AC
(a)
0.2
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0.4
(EFT)
0.0
J J
AB AB
=0.43J
AC
=0.435 J
(MC)
AC
(EFT)
1E-3
0.0
0.5
1.0
1.5
2.0
2.5
0.0
3.0
0.5
1.0
1.5
2.0
(c)
(d)
1.4
100
0.25
0.50
0.75
p:
1.2
0.25
1.0
10
p:
0.50
0.0
0.75
0.8
1.0
0.25
1
TE D
MT
1.0
0.0
0.0
0.4
3.0
T/J AC
T/J AC
0.6
2.5
0.50 0.75 1.0
0.1
1.0
0.50
0.2
J
AB
=1.0J
0.0
0.5
0.01
J
AC
0.0 0.0
0.75
0.25
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
T/J AC
=1.0J
0
1
2
3
4
5
AC
6
T/J AC
(f)
EP
(e)
AB
1E-3
AC C
Figure 3: Total magnetization MT (left column figures) and magnetic zero field susceptibility χ (right column) obtained for a variety of p values. crit , and for J The top, middle and bottom figures have been obtained for JAB = 0.2JAC , JAB = JAB AB = 1.0JAC , respectively. The solid symbols represent MC results whereas the dashed lines stand for data obtained by EFT.
as well as applied magnetic field strength H. Effect of p on the thermal variation of ∆S M depends on the strength crit crit crit of JAB coupling. Namely, we have three different scenarios as JAB < JAB , JAB = JAB and JAB > JAB cases. The former situation is displayed in Fig. 4a, where the magnitude of the peak of ∆S M curve becomes greater, but the peak position slides to lower temperatures with increasing p. This is due to the fact that, as p increases towards 1, then highly anisotropic spin- 12 atoms are replaced by isotropic spin- 32 atoms. This results in an increment of degree of freedom in the system, and consequently the critical temperature decreases. On the other hand, as the density T of isotropic spin- 23 atoms increases, then the saturation magnetization becomes enhanced, and | ∂M ∂T | increases. An crit interesting behavior is depicted in Fig. 4b where we consider JAB = JAB . According to this figure, both EFT and MC results yield that although the critical temperature is not changed, maxima of ∆S M becomes enhanced with increasing p which is due to fast variation of MT with temperature, (cf. see Fig. 3c). On the other hand, according to Fig. 4c crit where JAB > JAB holds, MCE is maximized at higher temperatures, since T c increases with increasing p (see Fig.2). 8
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M AN U
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Figure 4: Temperature dependence of isothermal entropy change |∆S M | for some selected values of compositional factor p obtained for (a) JAB = crit , (c) J 0.2JAC , (b) JAB = JAB AB = 1.0JAC in the presence of an external magnetic field H = 1.0JAC . The solid symbols represent MC results whereas the dashed lines stand for data obtained by EFT.
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10
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(f)
Figure 5: Temperature dependence of isothermal entropy change |∆S M | for some selected values of external magnetic field H with p = 0.0, 0.5, and 1.0 for JAB = 0.2JAC in (a-c), and JAB = 1.0JAC in (d-f). The solid symbols represent MC results whereas the dashed lines stand for data obtained by EFT.
However, the magnitude of the peak of ∆S M curves naively varies in this case. Hence, according to our results, it is possible to design magnetic materials with a variety of working temperatures and magnetocaloric properties (such as large ∆S M values ) by manipulating the magnetic phase transition via tuning the compositional factor p in ternary alloy systems. Namely, large exchange interaction causes enhanced magnetic phase transition temperature whereas reduced exchange coupling with high magnetic moment ( 23 ions in our model) concentration means large ∆S M values, but promises reduced T c . If the longitudinal magnetic field strength is progressively increased, the temperature value at which an enhanced MCE occurs should not change, but by the definition, the magnitude of the peak of ∆S M should be pronounced. This 9
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situation is depicted in Fig. 5. From this figure, it can be seen that the maximal entropy change can be achieved with large magnetic fields, if the system is only composed of ions A and B, and if the magnetic exchange interaction JAB between these ions is relatively small (see Fig. 5c). We note that the results obtained in Fig. 5 were also qualitatively found for hexacyanochromate Prussian blue analogs [48], and again we observe a significant accuracy between EFT and MC results. Moreover, Evangelisti et al. [49] showed for Prussian Blue analogues that magnetocaloric effect in these complex structures can be controlled by intrinsic vacancies. Finally, let us present some results for cooling capacity q as a function of external magnetic field for several system parameters. This parameter measures how adequate the system is for use in magnetocaloric applications. In other words, it is a measure of the amount of heat that can be transferred from the cold end to the hot end (denoted by T 1 and T 2 , respectively) in one thermodynamic cycle. Hence, one can reveal that the materials with large q property promises high potential as candidates in magnetocaloric applications. From q(H) data of Fig. 6, we reveal that q linearly increases with H, and by comparing Fig. 6a and Fig. 6b with each other, we find that the binary mixed alloy configurations with large spin magnitudes exhibit better q values. This finding is valid for either reduced (JAB = 0.2JAC ) or enhanced (JAB = 1.0JAC ) exchange couplings, and again, there exists an apparent agreement between EFT and MC results. 3.5
3.5
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2
3
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1
5
2
3
4
5
H/J AC
(b)
TE D
(a)
4
Figure 6: Variation of cooling capacity q as a function of external magnetic field H obtained for a variety of p values for (a) JAB = 0.2JAC , (b) JAB = 1.0JAC . EFT and MC results have been presented together.
4. Concluding Remarks
AC C
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In conclusion, we have studied the magnetic and magnetocaloric properties of ferromagnetic ternary alloys containing three different spin sources such as S A = 1, S B = 3/2 and S C = 1/2. We have used effective field theory and Monte Carlo simulations in our calculations. According to our numerical data, results obtained within the framework of EFT and MC agree well with each other. Due to the introduction of the approximation given by Eq. (14), the numerical values obtained by EFT regarding the critical parameters such as the transition temperature T c and critical crit coupling JAB are found to be slightly larger than those obtained by MC simulations. However, the general qualitative picture presented by two methods is in a good agreement with each other. In terms of the magnetocaloric properties, our results yield that it is possible to design magnetic materials with a variety of working temperatures and magnetocaloric properties (such as large ∆S M values) by manipulating the magnetic phase transition via tuning the compositional factor p in ternary alloy systems. Namely, large exchange interaction causes enhanced magnetic phase transition temperature whereas reduced exchange coupling with high concentration of large magnetic moment ( 32 ions in our model) means large ∆S M values, but promises reduced T c . We also found that the binary mixed alloy configurations with large spin magnitudes exhibit better q values. Besides, effect of p on the thermal variation of ∆S M depends on the strength of JAB , however, the value of this exchange coupling has no influence on q − H curves. Finally, observed MCE in the system has been found to be a direct MCE with ∆S M < 0 associated with a second order phase transition. The EFT method presented in this work can not distinguish between the dimensionality of the system. Namely, the calculations for a two dimensional triangular lattice and a three dimensional simple cubic lattice produce the same 10
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results. This is due to the fact that these lattice types have identical number of nearest neighbors (z = 6). This is the deficiency of the EFT method. On the other hand, since the MC simulations entirely takes the thermal fluctuations into account and the lattice dimensionality can be distinguished, it would be an important task to perform atomistic simulations of real magnetic systems for describing a broader range of Fe, Co, Ni solid solutions which requires us to use more complicated lattice structures such as body centered cubic, face centered cubic and hexagonal closed packed lattices with realistic values of atomistic simulation parameters such as the real values of the exchange coupling terms and inter-atomic distances. However, this will be the subject of our forthcoming research. Acknowledgements
The numerical calculations reported in this work were performed at TUBITAK ULAKBIM High Performance and Grid Computing Center (TR-Grid e-Infrastructure).
SC
Appendix
F1A (x, H) = F2A (x, H) = F1B (x, H) =
2 sinh[β(x + H)] , 2 cosh[β(x + H) + 1] 2 cosh[β(x + H)] , 2 cosh[β(x + H) + 1] 1 3 1 3 sinh[ 2 β(x + H)] + sinh[ 2 β(x + H)] , 2 cosh[ 23 β(x + H)] + cosh[ 12 β(x + H] 3 1 1 9 cosh[ 2 β(x + H)] + cosh[ 2 β(x + H)] , 4 cosh[ 32 β(x + H)] + cosh[ 12 β(x + H)]
TE D
F2B (x, H) =
M AN U
Explicit forms of hyperbolic functions Fnδ (x, H): The hyperbolic functions in Eq.(9) can be obtained by using the identity exp(α∇) f (x) = f (x + α)| x=0 as follows
F3B (x, H) = F1C (x, H) =
3 1 1 27 sinh[ 2 β(x + H)] + sinh[ 2 β(x + H)] , 8 cosh[ 32 β(x + H)] + cosh[ 12 β(x + H)]
1 β tanh[ (x + H)]. 2 2
(23)
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Van der Waerden coefficients Bn (a) for spin 3/2: Spin-3/2 has four states, hence from Eq. (12) we obtain a set of four linear equations with four Bn unknowns as follows, 3a ) 2 a exp( ) 2 −a exp( ) 2 −3a exp( ) 2 exp(
= = = =
3 2 1 B0 + B1 2 1 B0 − B1 2 3 B0 − B1 2 B0 + B1
9 4 1 + B2 4 1 + B2 4 9 + B2 4 + B2
27 , 8 1 + B3 , 8 1 − B3 , 8 27 − B3 . 8 + B3
Solution of Eq. (24) yields B0 (a) = B1 (a) =
[ ( )] (a) 1 3a 9 cosh − cosh , 8 2 2 [ ( )] (a) 1 3a 27 sinh − sinh , 12 2 2 11
(24)
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[ ( ) ( a )] 1 3a cosh − cosh , 2 2 2 [ ( ) ] (a) 1 3a sinh − 3 sinh . 3 2 2
B2 (a) = B3 (a) =
(25)
=
[A0 (a)]z−k1 k1 −k2
[A1 (a)]
1, −(k1 −k2 )
k∑ 1 −k2
(
)
[ ] (−1) p0 exp a(k1 − k2 − 2p0 ) ,
=
2
=
)( ) p1 ( k2 ∑ ∑ [ ] k2 p1 2−p1 (−1)k2 −p1 exp a(p1 − 2p2 ) , p1 p2
SC
p0 =0
[A2 (a)]k2
k1 − k2 p0
RI PT
Explicit forms of the coefficients [An (a)]α , [Bn (a)]γ , [Cn (a)]η in Eq. (18): The explicit forms of the coefficients [An (a)]α , [Bn (a)]γ , [Cn (a)]η after applying binomial expansion have been calculated as follows,
=
[B1 (a)]k1 −k2
=
[B0 (a)]
[B2 (a)]k2 −k3
)( )( ) p0 ∑ p1 ( ∑ z − k1 p0 p1 (−1)(z−k1 −p0 +p2 ) 2−4(z−k1 ) 32(p0 −p2 ) p0 p1 p2 p0 =0 p1 =0 p2 =0 ] [a × exp (−3z + 3k1 + 2p0 + 2p1 + 2p2 ) , 2 )( )( ) r0 ∑ k∑ r1 ( 1 −k2 ∑ k1 − k2 r0 r1 (−1)(r0 −r1 +r2 ) 2(−3k1 +3k2 ) r0 r1 r2 r0 =0 r1 =0 r2 =0 ] [a ×3(−k1 +k2 +3r0 −3r2 ) exp (−3k1 + 3k2 + 2r0 + 2r1 + 2r2 ) , 2 z−k ∑1
TE D
z−k1
=
k∑ s0 ∑ s1 ( 2 −k3 ∑ s0 =0 s1 =0 s2 =0
[a
EP × exp
k3 −k4
=
AC C
[B3 (a)]
(26)
M AN U
p1 =0 p2 =0
[C0 (a/2)]
2
)(
k2 − k3 s0
t0 =0 t1 =0 t2 =0
k3 − k4 t0
)(
×2−(k3 −k4 ) 3−(k3 −k4 −t0 +t2 ) exp
)
s1 s2
(−1)(s0 −s2 ) 2−2(k2 −k3 )
t0 t1 [a 2
)(
t1 t2
)
(−1)(k3 −k4 −t0 +t1 −t2 )
] (−3k3 + 3k4 + 2t0 + 2t1 + 2t2 ) , (27)
=
k∑ 4 −k5 u0 =0
[C1 (a/2)]k5
)(
] (−3k2 + 3k3 + 2s0 + 2s1 + 2s2 ) ,
k∑ t0 ∑ t1 ( 3 −k4 ∑
k4 −k5
s0 s1
=
(
k5 ( ∑ u1 =0
k4 − k5 u0 k5 u1
)
2−(k4 −k5 ) exp
) (−1)u1 exp
[a 2
[a 2
] (k4 − k5 − 2u0 ) ,
] (k5 − 2u1 ) . (28)
12
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Binder cumulant analysis for MC simulations obtained for several lattice size L: In order to determine the dependence of critical properties of the system on the lattice dimensionality, one can calculate the fourth order cumulant U4 (L, T ) i.e. the Binder cumulant for various lattice sizes according to ⟨MT4 ⟩
(29)
3⟨MT2 ⟩2
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U4 (L, T ) = 1 −
M AN U
SC
where ⟨MT2 ⟩ and ⟨MT4 ⟩ denote the second and fourth moments of the total magnetization MT . The Binder cumulant defined by Eq. (29) takes the value 2/3 at low temperatures (T < T c ). For sufficiently large temperature values (T > T c ) it tends to zero. At T = T c , U4 (L, T ) acquires a nontrivial value, namely the critical Binder cumulant Uc . In this case, U4 (L, T ) curves corresponding to different L intersect with each other at T = T c (cf. the points denoted by Uc in Fig. 7). As an indicator of the type of the magnetic phase transition, thermal variation of U4 (L, T ) curves corresponding to different lattice sizes can be examined. As shown in Fig. 7, based on the considered systems sizes, U4 (L, T ) curves do not exhibit any dip to negative values which means that the transitions between paramagnetic and ferromagnetic phases are of second order. From Fig. 7, it can be naively seen that the peak position of the susceptibility χ coincides with the intersection point of U4 (L, T ) curves for the whole set of the system parameters. Therefore, we can conclude that the system size L = 20 successfully simulates the behavior of the system in the thermodynamic limit. As a final remark, we should also note that, since our main motivation is not to determine the critical temperature values in high precision, in order to reduce the simulation time, we have used susceptibility data to evaluate the approximate value of the critical temperature. Of course, in a narrow temperature scale, susceptibility data would slightly become different from fourth-order cumulant analysis. However, this may cause only small differences in phase diagrams depicted in Fig. 2. References
[20] [21] [22] [23] [24] [25] [26] [27] [28]
TE D
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[12] [13] [14] [15] [16] [17] [18] [19]
A. M. Tishin, Y. I. Spichkin, The Magnetocaloric Effect and its Applications, Institute of Physics 2003. E. Warburg, Ann. Phys. 13 (1881) 141. P. Weiss, A. Piccard, J. Phys. (Paris) 7 (1917) 103. P. Debye, Ann. Phys. 81 (1926) 1154. W. F. Giauque, J. Am. Chem. Soc. 49 (1927) 1864. W. F. Giauque, D. P. MacDougall, Phys. Rev. 43 (1933) 768. A. Smith, Eur. Phys. J. H 38 (2013) 507. C. Zimm, A. Jastrab, A. Sternberg, V. Pecharsky, K. Gschneidner Jr., M. Osborne, I. Anderson, Adv. Cyrog. Eng. 43 (1998) 1759. V. Pecharsky, K. Gschneidner Jr., J. Appl. Phys. 86 (1999) 565. V. Chaudhary and R. V. Ramanujan, Sci. Rep. 6 (2016) 35156. A. M. Tishin, Magnetic therapy of malignant neoplasms by injecting material particles with high magnetocaloric effect and suitable magnetic phase transition temperature, Patent Number: EP1897590-A1, 2008 K. A. Gschneidner Jr., V. K. Pecharsky, Annu. Rev. Mater. Sci. 30 (2000) 387. K. A. Gschneidner Jr., V. K. Pecharsky, A. O. Tsokol, Rep. Prog. Phys. 68 (2005) 1479. M. H. Phan, S. C. Yu, J. Magn. Magn. Mater 308 (2007) 325. B. G. Shen, J. R. Sun, F. X. Hu, H. W. Zhang and Z. H. Cheng, Adv. Mater. 21 (2009) 4545. N. A. de Oliveira, P. J. von Ranke, Phys. Rep. 489 (2010) 89. V. Franco, J.S. Bl´azquez, B. Ingale, and A. Conde, Annu. Rev. Mater. Res. 42 (2012) 305. J. Romero Gomez, R. Ferreiro Garcia, A. De Miguel Catoira, M. Romero Gomez, Renew. Sustainable Energy Rev. 17 (2013) 74. In the thermal derivative of magnetization in Eq. (2), we have used the centered difference formula which can be given as ( dMorder ) to calculate ( ) Mi −Mi−1 1 Mi+1 −Mi T dT H = 2 T i+1 −T i + T i −T i−1 . R. Tamura, T. Ohno, H. Kitazawa, Appl. Phys. Lett. 104 (2014) 052415. R. Tamura, S. Tanaka, T. Ohno, H. Kitazawa, J. Appl. Phys. 116 (2014) 053908. J. Streˇcka, K. Karˇlov´a, T. Madaras, Physica B 466-467 (2015) 76. K. Karˇlov´a, J. Streˇcka, T. Madaras, Acta Phys. Pol. A 131 (2017) 630. G. V. Brown, J. Appl. Phys. 47 (1976) 3673. S. Yu. Dan’kov, A. M. Tishin, V. K. Pecharsky, and K. A. Gschneidner, Jr., Phys. Rev. B 57 (1998) 3478. V. K. Pecharsky, K. A. Gschneidner, Jr., Phys. Rev. Lett. 78 (1997) 4494. V. D. Buchelnikov, P. Entel, S. V. Taskaev, V. V. Sokolovskiy, A. Hucht, M. Ogura, H. Akai, M. E. Gruner, and S. K. Nayak, Phys. Rev. B 78 (2008) 184427. N. Singh, R. Arr´oyave, J. Appl. Phys. 113 (2013) 183904.
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U4(L,T)
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U4(L,T)
U4(L,T)
0.7
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J 1
AB
10
=0.2J
L=20
J
AC 1
p=0.0
0.1
AB
L=20
=0.2J
10
AC
p=0.5
1
0.1
0.01
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1.0
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2.0
2.5
0.5
1.0
T/J AC
(a)
0.00
L=30
0.3
AB
=0.43J
AC
p=0.0
UC
0.2
0.0
L=10
0.5
L=20
0.4
L=30
0.3
J
AB
=0.43J
AC
p=0.5
UC
0.2
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AB
10
=0.43J
1
0.1
L=20 AB
0.01
1.0
1.5
2.0
2.5
L=30
0.3
U4(L,T)
L=20
0.4
AC
p=0.0
UC
0.2 0.1
L=10
0.5
L=20
0.4
L=30
0.3 0.2
1.0
1.5
2.0
2.5
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100
1.0
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J
AB
=1.0J
AC
UC
p=0.5
0.5
1.0
1.5
2.0
AB
10
=1.0J
2.5
3.0
3.5
4.0
4.5
L=20
J
AC
1
p=0.0
0.1
AB
=1.0J
0.01
1.0
1.5
T/J AC
2.0
2.5
3.0
0.0
1.5
2.0
2.5
AC
0.5
1.0
T/J AC
(f)
L=10 L=20
0.4
L=30
0.3
J
AB
=1.0J
AC
UC
p=1.0
0.2
0.0
0.5
1.0
1.5
2.0
1.5
2.0
2.5
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1.0
1.5
2.5
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5.0
5.5
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J
AB
=1.0J
AC
p=1.0
0.1
0.01
1E-3
2.0
2.5
T/J AC
(h)
EP
(g)
=0.43J
0.5
1
0.01
0.5
AB
0.6
10
1E-3
0.0
2.5
p=1.0
0.0
AC
p=0.5
0.1
1E-3
2.0
100
TE D
L=20
J 1
1.5
0.0
0.0
100
10
1.0
0.1
0.0
0.5
0.5
0.7
0.6
0.1
0.0 0.0
UC
0.01
0.7
=1.0J
AC
L=20
(e)
0.6
AB
=0.43J
T/J AC
0.7
J
0.0
J
AC
0.5
(d)
L=10
AB
p=1.0
1E-3
0.0
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J
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1
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0.3
0.1
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10
=0.43J
p=0.5
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J
AC
p=0.0
2.5
U4(L,T)
J 1
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L=20
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(c)
U4(L,T)
0.4
J
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L=20
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T/J AC
0.7
L=10
0.5
U4(L,T)
U4(L,T)
2.0
(b)
0.1
U4(L,T)
1.5
T/J AC
0.6
AC
1E-3
0.0
0.7
=0.2J
0.01
1E-3
0.0
AB
p=1.0
0.1
0.01
1E-3
J
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L=20
10
3.0
3.5
4.0
4.5
0.0
0.5
1.0
T/J AC
(i)
Figure 7: Binder cumulant curves corresponding to lattice sizes L = 10, L = 20, and L = 30 with some selected values of parameters p and JAB .
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[29] V. D. Buchelnikov, V. V. Sokolovskiy, H. C. Herper, H. Ebert, M. E. Gruner, S. V. Taskaev, V. V. Khovaylo, A. Hucht, A. Dannenberg, M. Ogura, H. Akai, M. Acet, and P. Entel, Phys. Rev. B 81 (2010) 094411. [30] D. Comtesse, M. E. Gruner, M. Ogura, V. V. Sokolovskiy, V. D. Buchelnikov, A. Grnebohm, R. Arr´oyave, N. Singh, T. Gottschall, O. Gutfleisch, V. A. Chernenko, F. Albertini, S. Fhler, and P. Entel, Phys. Rev. B 89 (2014) 184403. [31] E. P. N´obrega, N. A. de Oliveira, P. J. von Ranke, and A. Troper, Phys. Rev. B 72 (2005) 134426. [32] Y. Ma, A. Du, J. Magn. Magn. Mater. 321 (2009) L65. [33] R. Arai, R. Tamura, H. Fukuda, J. Li, A. T. Saito, S. Kaji, H. Nakagome, T. Numazawa, IOP Conf. Ser. Mater. Sci. Eng. 101 (2015) 012118. ˇ [34] M. Zukoviˇ c, J. Magn. Magn. Mater. 374 (2015) 22. [35] S. Ohkoshi, K. Hashimoto, J. Am. Chem. Phys. 121 (1999) 10591. ˇ [36] M. Zukoviˇ c, A. Bob´ak, J. Magn. Magn. Mater. 322 (2010) 2868. [37] F. C. S´aBarreto, I.P. Fittipaldi, B. Zeks, Ferroelectrics 39 (1981) 1103. [38] T. Kaneyoshi, Acta Phys. Pol. A 83 (1993) 703. [39] T. Balcerzak, J. Magn. Magn. Mater. 246 (2002) 213. [40] J. W. Tucker, J. Phys. A: Math. Gen. 27 (1994) 659. [41] A. Bob´ak, O. F. Abubrig, D. Horv´ath, Physica A 312 (2002) 187. [42] A. Bob´ak, D. Horv´ath, Phys. Status Solidi B 213 (1999) 459. [43] A. Bob´ak, Physica A 258 (1998) 140. [44] E. Vatansever, Y. Y¨uksel, J. Alloys Compd. 689 (2016) 446.
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[45] G. M. Buend´ıa, J. E. Villarroel, J. Magn. Magn. Mater. 310 (2007) e495. [46] Y. Y¨uksel, J. Phys. Chem. Sol. 86 (2015) 207. [47] In EFT formulation, the type of the magnetic phase transition has been determined by examining the temperature dependence of magnetization curves defined by Eq. (16). In MC simulations however, we have used the numerical results for Binder cumulant analysis, (see Appendix). [48] E. Manuel, M. Evangelisti, M. Affronte, M. Okubo, C. Train, M. Verdaguer, Phys. Rev. B 73 (2006) 172406. [49] M. Evangelisti, E. Manuel, M. Affronte, M. Okubo, C. Train, J. Magn. Magn. Mater. 316 (2007) e569.
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Ferromagnetic ternary alloy system has been investigated. Magnetic and magnetocaloric properties have been studied. Effective field theory and Monte Carlo simulations have been implemented. A direct magnetocaloric effect has been observed. Magnetocaloric properties are strongly dependent on the compositional factor p.
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S0022-3697(17)31013-2
DOI:
10.1016/j.jpcs.2017.09.015
Reference:
PCS 8208
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 6 June 2017 Revised Date:
25 August 2017
Accepted Date: 4 September 2017
Please cite this article as: Y. Yüksel, Ü. Akıncı, A comparative study of critical phenomena and magnetocaloric properties of ferromagnetic ternary alloys, Journal of Physics and Chemistry of Solids (2017), doi: 10.1016/j.jpcs.2017.09.015. 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 proof before it is published in its final 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.
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A comparative study of critical phenomena and magnetocaloric properties of ferromagnetic ternary alloys ¨ Yusuf Y¨uksel, Umit Akıncı
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Department of Physics, Dokuz Eyl¨ul University, TR-35160 Izmir, Turkey
Abstract
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Magnetic and magnetocaloric properties, as well as the phase diagrams of a ferromagnetic ternary alloy system have been studied. A detailed comparison of two different methods, namely the effective field theory (EFT), and Monte Carlo (MC) simulations has been provided. Our numerical data show that the general qualitative picture presented by two methods are in a good agreement with each other. In terms of the magnetocaloric properties, our results yield that it is possible to design magnetic materials with a variety of working temperatures and magnetocaloric properties (such as large ∆S M and q values) by manipulating the magnetic phase transition via tuning the compositional factor (i.e. the mixing ratio of sublattice ions). The observed magnetocaloric effect has been found to be a direct one with ∆S M < 0 associated with a second order phase transition. Keywords: Alloy, Critical phenomena, Magnetism, Magnetocaloric, Phase transition
1. Introduction
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Magnetocaloric effect (MCE) [1, 2, 3, 4, 5, 6, 7] which is the manifestation of temperature change in a magnetic material due to adiabatically varying magnetic fields has recently attracted particular attention not only from academical point of view, but also due to its potential in technological [8, 9, 10], as well as biomedical applications [11]. The theoretical and experimental aspects regarding the phenomenon have been widely investigated in the literature [12, 13, 14, 15, 16, 17, 18]. Indeed, it is directly related to the magnetic entropy change of a magnetic material under adiabatic conditions. Fundamentally, the total entropy of a magnetic material can be written as
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S tot (T, H) = S L (T, H) + S e (T, H) + S M (T, H),
(1)
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where the first term is due to lattice vibrations, the second one comes from the electronic contribution, and the third term represents the magnetic entropy. Under adiabatic conditions, the total entropy does not change. Hence, upon application of a magnetic field on a magnetic material, the magnetic dipole moments tend to align parallel with each other which reduces the magnetic entropy. However, lattice and electronic entropy should increase, and consequently the temperature of the material increases. On the other hand, when the magnetic field is reduced towards zero, the magnetic entropy increases due to increasing magnetic disorder. In this case, in order to keep the total entropy unaltered, the electronic and lattice contributions become reduced, and the material is cooled. Using this process, one can utilize the magnetic material for cooling/heating applications. Former studies suggest that MCE is maximized around the ferromagnetic-paramagnetic transition temperature [1]. For a potential candidate to be used in magnetocaloric applications, the material should exhibit magnetic phase transition around the room temperature. Theoretically, one can calculate the MCE indirectly using the relations [19] ( ) ) ∫ ( ∫ ∂M T ∂M dH, ∆S M = dH, (2) ∆T ad = − C x ∂T H ∂T H ∗ Corresponding
author. Tel.: +90 2324119544; fax: +90 2324534188. Email address: [email protected] (Yusuf Y¨uksel)
Preprint submitted to Journal of Physics and Chemistry of Solids
August 25, 2017
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∫ q=−
T2
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which are the outcomes of the thermodynamic Maxwell relations. In Eq. (2), ∆T ad is the adiabatic temperature change whereas ∆S M is the isothermal entropy change. One should pay attention on the fact that the heat capacity C x in Eq. (2) is not independent from magnetic field (except at very low temperatures at which MCE is very weak). Hence, C x cannot be simply taken out of integration. For conventional magnetic materials exhibiting ferromagneticparamagnetic phase transitions, the magnetization is pronounced at low temperature region. However, with increasing temperature, magnetization becomes reduced, and gradually decreases. Hence, ∆S M becomes negative. In this case, the material heats up with increasing magnetic field strength whereas it cools down upon decreasing the magnetic field strength. Furthermore, antiferromagnetic or ferrimagnetic materials may exhibit positive ∆S M with increasing temperature which is known as the inverse MCE in the literature [20, 21, 22, 23]. As another physical quantity which measures the suitability of magnetic material for magnetic cooling applications, we can introduce the cooling capacity q which is given by the formula ∆S M (T )H dT,
(3)
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T1
2. Formulation
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which is a measure of the amount of heat that can be transferred from the cold end to the hot end (denoted by T 1 and T 2 , respectively) in one thermodynamical cycle during the adiabatic process. Up to now, a great many magnetic materials has been tested for their magnetocaloric features [17], and among them the rare earth Gd suits well for magnetic cooling technology with an adiabatic temperature change of ∆T ad = 14K in 7T magnetic field [24, 25], and isothermal magnetic entropy change |∆S M | = 6.1 and 10.6 (in units of Jkg−1 K−1 ) for magnetic fields 2T and 5T, respectively [12]. However, due to its finite resources, the elementary Gd cannot be used in mass production of magnetic cooling systems [26]. Hence, it is a crucial point to search for new materials with large amount of resources, room temperature magnetic phase transition, and large magnetocaloric properties around the working temperature. In addition, tuning the critical temperature is an important issue in magnetocaloric applications. It is the main motivation of the present study to theoretically elucidate the magnetic, as well as the magnetocaloric properties of ferromagnetic ternary alloys which contain three different magnetic sublattices. Conventional mean field theory (MFT) predicts an abrupt fall of magnetic specific heat around T c , however the situation is not the case in experimental systems. Hence, we use effective field theory (EFT) together with the Monte Carlo (MC) simulations in our calculations. In terms of magnetocaloric applications, MC simulation technique is successfully used in the theoretical literature [27, 28, 29, 30, 31, 32, 33, 34]. Our another aim in this study is to test the accuracy of numerical data produced by EFT in comparison with MC simulation technique. The outline of the paper is as follows: In Sec. 2 we present our model and formulation details. Numerical results and related discussions are given in Sec. 3, and finally Sec. 4 is devoted to our conclusions.
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In order to investigate the ferromagnetic ternary alloy system, consideration of the nearest neighbor exchange coupling is sufficient, and the next-nearest exchange interactions can be neglected [35]. Therefore, the Hamiltonian describing a ternary alloy with the chemical formula AB p C1−p in an external magnetic field can be written as ∑ ∑ ∑ ∑[ ] H =− JAB S iA S Bj ζ j − JAC S iA S Cj (1 − ζ j ) − H S iA − H ζ j S Bj + (1 − ζ j )S Cj , (4)
i
j
where S iA , S Bj , S Cj are the spin variables which have been selected as S iA = ±1, 0; S Bj = ± 32 , ± 12 ; S Cj = ± 12 , and H is the homogeneous magnetic field. We note that this selection of spin magnitudes is arbitrary, since the qualitative picture of theoretical results can also be obtained for different spin configurations. In Eq. (4), ζ j represents the site occupancy parameter which can take values zero or unity. If ζ j = 1 then the lattice site j is occupied by an ion B and it is the nearest neighbor of ion A at the site i. The interaction between ions A and B is ferromagnetic (JAB > 0). Similarly, if ζ j = 0 then an ion C is placed on the lattice site j. In this case, ions A and C also interact ferromagnetically with each other (JAC > 0). Our model system is schematically illustrated in Fig. 1. The pseudo spin variables are located on the nodes of a simple cubic lattice, and each ion A can interact with an ion X (X = B or C). For simplicity, the stoichiometry of the system has been selected as 1:1 which means that there 2
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are no vacant sites on the sub-lattice A (pA = 1.0) [36]. Ion concentration of sublattices B and C are given by p and (1 − p), respectively with 0 ≤ p ≤ 1. Hence, p is called the compositional factor. p = 0 and p = 1 special cases correspond to binary mixed AC and AB ferromagnetic structures, respectively. 0.5
S =1/2
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C
S =1
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A
S =3/2 B
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Figure 1: A representative visual of an arbitrary plane of the 3D ferromagnetic ternary system with p = 0.75. Sublattices with different spin magnitudes are labeled as S A = 1 (green), S B = 3/2 (yellow), and S C = 1/2 (blue).
2.1. Effective-Field Theory
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In the EFT formalism, we start by obtaining the local fields HiA , H Bj , HCj which can be written for our particular problem as follows z [ ∑ ] B C , HiA = −S iA J S ζ + J (1 − ζ )S + H AB j AC j j j j=1
H Bj
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HCj
= −S iA EiA , z ∑ B A = −S j JAB S i ζ j + Hζ j = −S Bj E Bj , i=1 z ∑ A C , (1 − ζ ) + H(1 − ζ ) = −S j J S j j AC i i=1
= −S Cj E Cj .
(5)
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With the help of Eq. (5), the expectation value (thermal average) of the nth order of the magnetic moment S k can be obtained from [37] ⟨ ⟩ Trk (S kα )n exp(−βHkα ) ⟨(S kα )n ⟩ = , α = A, B, C, (6) Trk exp(−βHkα ) which gives
⟨ ⟩ S iA
⟨ = ⟨
⟨ ⟩ (S iA )2
=
⟨ ⟩ ζ j S Bj
=
2 sinh(βEiA ) 2 cosh(βE Aj ) + 1 2 cosh(βEiA )
⟩ , ⟩
, 2 cosh(βE Aj ) + 1 ⟨ 3 1 B B ⟩ 1 3 sinh( 2 βE j ) + sinh( 2 βE j ) ζj , 2 cosh( 32 βE Bj ) + cosh( 12 βE Bj ) 3
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=
⟨ ⟩ ζ j (S Bj )3
=
⟨ ⟩ (1 − ζ j ) S Cj
=
⟨ 3 1 B B ⟩ 1 9 cosh( 2 βE j ) + cosh( 2 βE j ) ζj , 4 cosh( 32 βE Bj ) + cosh( 21 βE Bj ) ⟨ 3 1 B B ⟩ 1 27 sinh( 2 βE j ) + sinh( 2 βE j ) ζj , 8 cosh( 32 βE Bj ) + cosh( 12 βE Bj ) ⟩ 1⟨ (1 − ζ j ) tanh(βE Cj ) . 2
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⟨ ⟩ ζ j (S Bj )2
(7)
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Next, one should apply the relation exp(α∇) f (x) = f (x + α)| x=0 , i.e. the differential operator technique [38] in Eq. (7), and perform the thermal and configurational averages ⟨⟨...⟩⟩ to get ⟨⟨ ⟩⟩ Xn A = (S iA )nA ⟨⟨∏ ⟩⟩ z = ζ j exp(JAB S Bj ∇) +(1 − ζ j ) exp(JAC S Cj ∇) FnAA (x, H) x=0 , nA = 1, 2 j=1
= =
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= =
⟨
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Yn B
⟨ ⟨ ⟩⟩ ζ j (S Bj )nB ⟨ ⟨∏ ⟩⟩ z A ζj exp(JAB S i ∇) FnBB (x, H) x=0 , nB = 1, 2, 3 j=1
⟨ ⟩⟩ (1 − ζ j ) (S Cj )nC ⟩⟩ ⟨ ⟨∏ z C (1 − ζ j ) exp(JAB S j ∇) FnCC (x, H) x=0 , nC = 1. j=1
(8)
where the inner and the outer products represent the thermal and configurational averages, respectively. In obtaining the closed form of single site correlations in Eq. (8), we use the exponential relation
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exp(aζ j ) = ζ j exp(a) + 1 − ζ j ,
(9)
along with the property ζ 2j = ζ j . Moreover, the functions Fnδ (x, H) in Eq.(8) are given in Appendix. The next step is to write the exponential operators in Eq. (8) in terms of hyperbolic trigonometric functions by using the exact Van der Waerden spin identities [39, 40]. For spin-1, S iA has three components, and the identity is given by (S iA )2 cosh(a) + S iA sinh(a) + 1 − (S iA )2
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exp(aS iA ) = =
A0 (a) + A1 (a)S iA + A2 (a)(S iA )2 .
(10)
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On the other hand, an exponential operator exp(aS i ) for a general spin S i is given by [38] exp(aS i ) =
2S ∑
ξn (a)(S i )n .
(11)
Bn (a)(S Bj )n ,
(12)
n=0
Hence, the identities for spin-3/2 are given as exp(aS Bj ) =
3 ∑ n=0
and the coefficients Bn (a) are given in Appendix. If the sublattice C consists of spin-1/2 ions then the Van der Waerden identity becomes exp(aS i ) = C0 (a) + C1 (a)S i , where C0 (a) = cosh(a/2) and C1 = 2 sinh(a/2). 4
(13)
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By using Eq.(11) in Eq. (8), mathematically complicated multi-site correlations appear. In this case, the problem becomes mathematically intractable, and one has to make an approximation. As the simplest approximation, one should expand Eq.(8) by neglecting the multi-site correlation identities according to [38, 41] ⟨y j (yk )2 ...(yl )3 zn ⟩ ≈ ⟨y j ⟩⟨(yk )2 ⟩...⟨(yl )3 ⟩⟨(zn )⟩, ⟨S Aj (S kA )2 ...S lA ⟩ ≈ ⟨S Aj ⟩⟨(S kA )2 ⟩...⟨S lA ⟩.
(14)
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where j , k , l , n, y j = ζ j S Bj , and z j = (1 − ζ j )S Cj . Applying approximation (14), and performing the thermal and configurational averages ⟨⟨...⟩⟩ results in [ XnA = pB0 (JAB ∇) + B1 (JAB ∇)Y1 + B2 (JAB ∇)Y2 +B3 (JAB ∇)Y3 + (1 − p)C0 (JAC ∇/2) +C1 (JAC ∇/2)Z1 ]z FnAA (x, H)| x=0 , =
p[A0 (JAB ∇) + A1 (JAB ∇)X1 + A2 (JAB ∇)X2 ]z FnBB (x, H)| x=0 ,
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=
(1 − p)[A0 (JAC ∇) + A1 (JAC ∇)X1 + A2 (JAC ∇)X2 ]z FnCC (x, H)| x=0 ,
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Yn B
(15)
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where z is the coordination number of the lattice, and ⟨⟨ ⟩⟩ ⟨⟨ ⟩⟩ X1 = S iA = mA , X2 = (S iA )2 = qA , ⟨ ⟨ ⟩⟩ ⟨ ⟨ ⟩⟩ Y1 = ζi S Bj = mB , Y2 = ζi (S Bj )2 = qB , ⟨ ⟨ ⟩⟩ ⟨ ⟨ ⟩⟩ Y3 = ζi (S Bj )3 = rB , Z1 = (1 − ζi ) S Cj = mC . Then the total magnetization of the system can be defined as MT =
1 (mA + mB + mC ). 2
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To evaluate the polynomial form of Eq. (15), one should apply binomial expansion ) n ( ∑ n (x + y)n = xn−i yi , i
(16)
(17)
i=0
in Eq.(15). In this manner, we get the following set of nonlinear equations with six unknowns in polynomial forms, Xn A
=
)( )( )( )( ) k3 ∑ k1 ∑ k2 ∑ k4 ( z ∑ ∑ z k1 k2 k3 k4 p(z−k1 ) (1 − p)k4 −k5 k1 k2 k3 k4 k5
=
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Yn B
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k1 =0 k2 =0 k3 =0 k4 =0 k5 =0
×[B0 (JAB ∇)]z−k1 [B1 (JAB ∇)]k1 −k2 [B2 (JAB ∇)]k2 −k3 [B3 (JAB ∇)]k3 −k4 ×[C0 (JAC ∇/2)]k4 −k5 [C1 (JAC ∇/2)]k5 Y1k1 −k2 Y2k2 −k3 Y3k3 −k4 Z1k5 FnAA (x, H)| x=0 , )( ) k1 ( z ∑ ∑ z k1 p [A0 (JAB ∇)]z−k1 [A1 (JAB ∇)]k1 −k2 [A2 (JAB ∇)]k2 X1k1 −k2 X2k2 FnBB (x, H)| x=0 , k1 k2 k1 =0 k2 =0
ZnC
=
(1 − p)
)( ) k1 ( z ∑ ∑ z k1 [A0 (JAC ∇)]z−k1 [A1 (JAC ∇)]k1 −k2 [A2 (JAC ∇)]k2 X1k1 −k2 X2k2 FnCC (x, H)| x=0 . (18) k1 k2
k1 =0 k2 =0
where the coefficients [An (a)]α , [Bn (a)]γ , [Cn (a)]η can be found in Appendix. In order to obtain the magnetizations of the sub-lattices, as well as the total magnetization, quadrupole moments etc., one should numerically solve the system of non-linear equations given in Eq. (18) as a function of, H, p, and T . Conveniently, in order to determine the phase transition temperature at which the magnetization vanishes, one can linearize Eq. (18) [38]. However, due to the complicated form of the right hand-sides of the equations of state in Eq. (18), we prefer to utilize the zero field susceptibility per spin ) ( ∂MT , (19) χ= ∂H H=0 5
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from which the critical temperature has been determined from maxima of χ at T = T c . We also note that kB has been set to unity for simplicity, and the temperature, exchange interactions, as well as the magnetic field strength have been scaled in units of JAC in the calculations.
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2.2. Monte Carlo Simulation We have also performed Metropolis Monte Carlo simulations of Eq. (4) by imposing periodic boundary conditions in all directions. Pseudo spins are placed on the nodes of a L × L × L simple cubic lattice, and each ion A can interact with an ion X (X = B or C). We simulate systems with linear lattice dimension L = 20 (we found more or less the same results for L > 20; see Appendix). At each Monte Carlo step, lattice sites have been sequentially swept. Data were collected over 105 Monte Carlo steps after discarding the first 25% for thermalization. Following quantities have been calculated during our simulations; ⟨ NA ⟩ 1 ∑ mA = SA , NA i=1
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• the longitudinal component of thermal average of instantaneous sublattice magnetizations, ⟨ NB ⟩ 1 ∑ mB = SB , NB j=1
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where NA = pA L3 /2, NB = pL3 /2, and NC = (1 − p)L3 /2.
⟨ NC ⟩ 1 ∑ mC = SC , NC k=1
(20)
• With the help of Eq. (20), thermal and configurational averages of total magnetization are defined by MT = [NA mA + NB mB + NC mC ]) /N, where N = NA + NB + NC .
(21)
• Finally, direct susceptibility per spin at zero field can be calculated via fluctuation-dissipation theorem ) 1 ( 2 ⟨mT ⟩ − ⟨mT ⟩2 , kB T
(22)
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χ=
where ⟨mT ⟩ = MT , and the quadrupole moment ⟨m2T ⟩ can be calculated by summing up the squares of S A , S B and S C in Eq. (20).
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The location of the transition temperature has been estimated by examining the thermal variation of direct susceptibility which exhibits a prominent peak at T = T c , and the Binder cumulant U4 (L, T ) curves, (see Appendix). In the calculations, we also set kB = 1 for simplicity. 3. Results and discussion
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Here, and in the following presentations, ferromagnetic exchange between ions A and C will be fixed as JAC = 1.0, and the interaction between adjacent ions A and B will be varied. On the other hand, although the present EFT and MC formulations can in principle be applied to any arbitrary coordination number z, we will discuss the results only for z = 6 which mimics the simple cubic lattice structure. When the compositional factor takes value p = 0.0 or p = 1.0, then the ternary alloy system reduces to binary mixed (1, 12 ) and (1, 32 ) ferromagnets, respectively. For instance, in terms of the EFT formalism, for the former case with z = 6, reported critical temperature value is T c = 2.1116JAC [42] whereas for the latter one we have T c = 4.846JAB [43]. For 0.0 < p < 1.0, we have a ternary alloy with three sub-lattices, and the system exhibits rich phase diagrams. Hence, before discussing the magnetic, as well as the magnetocaloric properties, it would be beneficial to present the global phase diagrams of the system. For this aim, in Fig. 2, we depict the phase diagrams in (JAB /JAC − T c /JAC ) plane corresponding to different values of the mixing ratio p. Here, the horizontal line with p = 0.0 represents the critical frontier of binary AC ferromagnet. As expected, when p = 0.0, ions B do not exist in the system and the location of the transition temperature does not depend on varying JAB values. As p increases then the magnetic interactions due to the presence of ions B originate. Depending on the strength of the exchange interaction between ions A and B, the transition temperature of the system 6
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p: 0.0
8
0.25
0.75 1.0
4
2
0 0.5
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JAB/JAC
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Tc/JAC
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Figure 2: Phase diagrams in a (JAB /JAC − T c /JAC ) plane of ferromagnetic ternary alloy system. Different symbols correspond to different compositional factor values. The dashed lines represent the EFT results whereas solid symbols correspond to MC data.
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either increases or decreases as a function of p. It is clear from Fig. 2 that there exists a critical value of JAB below which the transition temperature decreases with increasing p. Within the EFT formalism, we predict this threshold crit crit = 0.43JAC . As seen in Fig. 2, EFT and MC = 0.435JAC whereas in MC calculations, we found JAB value as JAB crit methods provide qualitatively similar results. However, EFT results for the critical temperature and JAB seem a bit larger than the MC results. This is an expected result, since thermal fluctuations are exactly taken into account in MC simulations. The system with the same spin configuration has been previously investigated in Ref. [41] for crit crit = 0.424JAC [41] for = 0.435JAC with its two dimensional counterpart JAB z = 3. By comparing our EFT result JAB crit z = 3, we can claim that EFT prediction on the spatial dimensionality dependence of JAB agrees well with MC results [36, 44]. Namely, the critical value of JAB exhibits very little variation with lattice dimensionality. Previous efforts crit as a function of several physical factors. In this regard, it have been devoted to the investigation of variation of JAB crit has been reported that JAB is independent of lattice stoichiometry [36], exhibits very little variation with spatial lattice dimensionality [36, 45] and increases with increasing exchange anisotropy [46]. Next in Fig. 3, we give the temperature dependence of total magnetization MT , as well as the zero field suscepcrit (Figs. 3c,d), and JAB = 1.0JAC (Figs. 3e,f) with several values tibility χ for JAB = 0.2JAC (Figs. 3a,b), JAB = JAB of p. As seen in Fig. 3a, when the exchange interaction between A and B type ions is weak such as JAB = 0.2JAC , increasing p, hence, increasing the density of ions B causes a decline in the transition temperature of the system, and the susceptibility exhibits a divergent behavior at the transition temperature. On the other hand, according to Fig. 3c crit which is depicted for JAB = JAB , all the MT curves merge to zero at the same transition temperature value for all p. Finally, for JAB = 1.0JAC (Fig. 3e), we have just the opposite scenario of Fig. 3a. Namely, increasing the density of ions B, causes a substantial enhancement in the ferromagnetic character of the alloy by increasing its critical temperature. Saturation value of MT at low temperature region is governed by Eqs. (16) and (21) with MA = 1.0, (1−p) MB = 3p 2 and MC = 2 . Moreover, by inspecting the susceptibility curves depicted in Figs. 3b, 3e, and 3f, we see that they exhibit a prominent divergent behavior at the paramagnetic-ferromagnetic transition temperature. We note that the type of the transition is of second order for both formulations [47]. As a general remark, we see that EFT and MC results are almost identical to each other except the vicinity of the critical region where the system undergoes a phase transition between paramagnetic and ferromagnetic phases. As we mentioned before, critical temperature predicted by EFT is observed to be slightly larger than the values obtained by MC simulations. This is a consequence of omitting the multi-spin correlation identities by using the approximation (17) in EFT calculations. However, the estimated EFT results agree well with our MC data. Up to now, we have seen that the parameters JAB and p play a significant role on the magnetic properties of the ternary alloy system. In the present work, we also consider the magnetocaloric properties of the ferromagnetic ternary alloy model by examining the isothermal variation of magnetic entropy change ∆S M . The variation of ∆S M as a function of temperature is expected to be influenced by compositional factor p, ferromagnetic exchange coupling JAB , 7
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(e)
AB
1E-3
AC C
Figure 3: Total magnetization MT (left column figures) and magnetic zero field susceptibility χ (right column) obtained for a variety of p values. crit , and for J The top, middle and bottom figures have been obtained for JAB = 0.2JAC , JAB = JAB AB = 1.0JAC , respectively. The solid symbols represent MC results whereas the dashed lines stand for data obtained by EFT.
as well as applied magnetic field strength H. Effect of p on the thermal variation of ∆S M depends on the strength crit crit crit of JAB coupling. Namely, we have three different scenarios as JAB < JAB , JAB = JAB and JAB > JAB cases. The former situation is displayed in Fig. 4a, where the magnitude of the peak of ∆S M curve becomes greater, but the peak position slides to lower temperatures with increasing p. This is due to the fact that, as p increases towards 1, then highly anisotropic spin- 12 atoms are replaced by isotropic spin- 32 atoms. This results in an increment of degree of freedom in the system, and consequently the critical temperature decreases. On the other hand, as the density T of isotropic spin- 23 atoms increases, then the saturation magnetization becomes enhanced, and | ∂M ∂T | increases. An crit interesting behavior is depicted in Fig. 4b where we consider JAB = JAB . According to this figure, both EFT and MC results yield that although the critical temperature is not changed, maxima of ∆S M becomes enhanced with increasing p which is due to fast variation of MT with temperature, (cf. see Fig. 3c). On the other hand, according to Fig. 4c crit where JAB > JAB holds, MCE is maximized at higher temperatures, since T c increases with increasing p (see Fig.2). 8
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Figure 4: Temperature dependence of isothermal entropy change |∆S M | for some selected values of compositional factor p obtained for (a) JAB = crit , (c) J 0.2JAC , (b) JAB = JAB AB = 1.0JAC in the presence of an external magnetic field H = 1.0JAC . The solid symbols represent MC results whereas the dashed lines stand for data obtained by EFT.
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Figure 5: Temperature dependence of isothermal entropy change |∆S M | for some selected values of external magnetic field H with p = 0.0, 0.5, and 1.0 for JAB = 0.2JAC in (a-c), and JAB = 1.0JAC in (d-f). The solid symbols represent MC results whereas the dashed lines stand for data obtained by EFT.
However, the magnitude of the peak of ∆S M curves naively varies in this case. Hence, according to our results, it is possible to design magnetic materials with a variety of working temperatures and magnetocaloric properties (such as large ∆S M values ) by manipulating the magnetic phase transition via tuning the compositional factor p in ternary alloy systems. Namely, large exchange interaction causes enhanced magnetic phase transition temperature whereas reduced exchange coupling with high magnetic moment ( 23 ions in our model) concentration means large ∆S M values, but promises reduced T c . If the longitudinal magnetic field strength is progressively increased, the temperature value at which an enhanced MCE occurs should not change, but by the definition, the magnitude of the peak of ∆S M should be pronounced. This 9
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situation is depicted in Fig. 5. From this figure, it can be seen that the maximal entropy change can be achieved with large magnetic fields, if the system is only composed of ions A and B, and if the magnetic exchange interaction JAB between these ions is relatively small (see Fig. 5c). We note that the results obtained in Fig. 5 were also qualitatively found for hexacyanochromate Prussian blue analogs [48], and again we observe a significant accuracy between EFT and MC results. Moreover, Evangelisti et al. [49] showed for Prussian Blue analogues that magnetocaloric effect in these complex structures can be controlled by intrinsic vacancies. Finally, let us present some results for cooling capacity q as a function of external magnetic field for several system parameters. This parameter measures how adequate the system is for use in magnetocaloric applications. In other words, it is a measure of the amount of heat that can be transferred from the cold end to the hot end (denoted by T 1 and T 2 , respectively) in one thermodynamic cycle. Hence, one can reveal that the materials with large q property promises high potential as candidates in magnetocaloric applications. From q(H) data of Fig. 6, we reveal that q linearly increases with H, and by comparing Fig. 6a and Fig. 6b with each other, we find that the binary mixed alloy configurations with large spin magnitudes exhibit better q values. This finding is valid for either reduced (JAB = 0.2JAC ) or enhanced (JAB = 1.0JAC ) exchange couplings, and again, there exists an apparent agreement between EFT and MC results. 3.5
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Figure 6: Variation of cooling capacity q as a function of external magnetic field H obtained for a variety of p values for (a) JAB = 0.2JAC , (b) JAB = 1.0JAC . EFT and MC results have been presented together.
4. Concluding Remarks
AC C
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In conclusion, we have studied the magnetic and magnetocaloric properties of ferromagnetic ternary alloys containing three different spin sources such as S A = 1, S B = 3/2 and S C = 1/2. We have used effective field theory and Monte Carlo simulations in our calculations. According to our numerical data, results obtained within the framework of EFT and MC agree well with each other. Due to the introduction of the approximation given by Eq. (14), the numerical values obtained by EFT regarding the critical parameters such as the transition temperature T c and critical crit coupling JAB are found to be slightly larger than those obtained by MC simulations. However, the general qualitative picture presented by two methods is in a good agreement with each other. In terms of the magnetocaloric properties, our results yield that it is possible to design magnetic materials with a variety of working temperatures and magnetocaloric properties (such as large ∆S M values) by manipulating the magnetic phase transition via tuning the compositional factor p in ternary alloy systems. Namely, large exchange interaction causes enhanced magnetic phase transition temperature whereas reduced exchange coupling with high concentration of large magnetic moment ( 32 ions in our model) means large ∆S M values, but promises reduced T c . We also found that the binary mixed alloy configurations with large spin magnitudes exhibit better q values. Besides, effect of p on the thermal variation of ∆S M depends on the strength of JAB , however, the value of this exchange coupling has no influence on q − H curves. Finally, observed MCE in the system has been found to be a direct MCE with ∆S M < 0 associated with a second order phase transition. The EFT method presented in this work can not distinguish between the dimensionality of the system. Namely, the calculations for a two dimensional triangular lattice and a three dimensional simple cubic lattice produce the same 10
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results. This is due to the fact that these lattice types have identical number of nearest neighbors (z = 6). This is the deficiency of the EFT method. On the other hand, since the MC simulations entirely takes the thermal fluctuations into account and the lattice dimensionality can be distinguished, it would be an important task to perform atomistic simulations of real magnetic systems for describing a broader range of Fe, Co, Ni solid solutions which requires us to use more complicated lattice structures such as body centered cubic, face centered cubic and hexagonal closed packed lattices with realistic values of atomistic simulation parameters such as the real values of the exchange coupling terms and inter-atomic distances. However, this will be the subject of our forthcoming research. Acknowledgements
The numerical calculations reported in this work were performed at TUBITAK ULAKBIM High Performance and Grid Computing Center (TR-Grid e-Infrastructure).
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Appendix
F1A (x, H) = F2A (x, H) = F1B (x, H) =
2 sinh[β(x + H)] , 2 cosh[β(x + H) + 1] 2 cosh[β(x + H)] , 2 cosh[β(x + H) + 1] 1 3 1 3 sinh[ 2 β(x + H)] + sinh[ 2 β(x + H)] , 2 cosh[ 23 β(x + H)] + cosh[ 12 β(x + H] 3 1 1 9 cosh[ 2 β(x + H)] + cosh[ 2 β(x + H)] , 4 cosh[ 32 β(x + H)] + cosh[ 12 β(x + H)]
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F2B (x, H) =
M AN U
Explicit forms of hyperbolic functions Fnδ (x, H): The hyperbolic functions in Eq.(9) can be obtained by using the identity exp(α∇) f (x) = f (x + α)| x=0 as follows
F3B (x, H) = F1C (x, H) =
3 1 1 27 sinh[ 2 β(x + H)] + sinh[ 2 β(x + H)] , 8 cosh[ 32 β(x + H)] + cosh[ 12 β(x + H)]
1 β tanh[ (x + H)]. 2 2
(23)
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Van der Waerden coefficients Bn (a) for spin 3/2: Spin-3/2 has four states, hence from Eq. (12) we obtain a set of four linear equations with four Bn unknowns as follows, 3a ) 2 a exp( ) 2 −a exp( ) 2 −3a exp( ) 2 exp(
= = = =
3 2 1 B0 + B1 2 1 B0 − B1 2 3 B0 − B1 2 B0 + B1
9 4 1 + B2 4 1 + B2 4 9 + B2 4 + B2
27 , 8 1 + B3 , 8 1 − B3 , 8 27 − B3 . 8 + B3
Solution of Eq. (24) yields B0 (a) = B1 (a) =
[ ( )] (a) 1 3a 9 cosh − cosh , 8 2 2 [ ( )] (a) 1 3a 27 sinh − sinh , 12 2 2 11
(24)
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[ ( ) ( a )] 1 3a cosh − cosh , 2 2 2 [ ( ) ] (a) 1 3a sinh − 3 sinh . 3 2 2
B2 (a) = B3 (a) =
(25)
=
[A0 (a)]z−k1 k1 −k2
[A1 (a)]
1, −(k1 −k2 )
k∑ 1 −k2
(
)
[ ] (−1) p0 exp a(k1 − k2 − 2p0 ) ,
=
2
=
)( ) p1 ( k2 ∑ ∑ [ ] k2 p1 2−p1 (−1)k2 −p1 exp a(p1 − 2p2 ) , p1 p2
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p0 =0
[A2 (a)]k2
k1 − k2 p0
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Explicit forms of the coefficients [An (a)]α , [Bn (a)]γ , [Cn (a)]η in Eq. (18): The explicit forms of the coefficients [An (a)]α , [Bn (a)]γ , [Cn (a)]η after applying binomial expansion have been calculated as follows,
=
[B1 (a)]k1 −k2
=
[B0 (a)]
[B2 (a)]k2 −k3
)( )( ) p0 ∑ p1 ( ∑ z − k1 p0 p1 (−1)(z−k1 −p0 +p2 ) 2−4(z−k1 ) 32(p0 −p2 ) p0 p1 p2 p0 =0 p1 =0 p2 =0 ] [a × exp (−3z + 3k1 + 2p0 + 2p1 + 2p2 ) , 2 )( )( ) r0 ∑ k∑ r1 ( 1 −k2 ∑ k1 − k2 r0 r1 (−1)(r0 −r1 +r2 ) 2(−3k1 +3k2 ) r0 r1 r2 r0 =0 r1 =0 r2 =0 ] [a ×3(−k1 +k2 +3r0 −3r2 ) exp (−3k1 + 3k2 + 2r0 + 2r1 + 2r2 ) , 2 z−k ∑1
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z−k1
=
k∑ s0 ∑ s1 ( 2 −k3 ∑ s0 =0 s1 =0 s2 =0
[a
EP × exp
k3 −k4
=
AC C
[B3 (a)]
(26)
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p1 =0 p2 =0
[C0 (a/2)]
2
)(
k2 − k3 s0
t0 =0 t1 =0 t2 =0
k3 − k4 t0
)(
×2−(k3 −k4 ) 3−(k3 −k4 −t0 +t2 ) exp
)
s1 s2
(−1)(s0 −s2 ) 2−2(k2 −k3 )
t0 t1 [a 2
)(
t1 t2
)
(−1)(k3 −k4 −t0 +t1 −t2 )
] (−3k3 + 3k4 + 2t0 + 2t1 + 2t2 ) , (27)
=
k∑ 4 −k5 u0 =0
[C1 (a/2)]k5
)(
] (−3k2 + 3k3 + 2s0 + 2s1 + 2s2 ) ,
k∑ t0 ∑ t1 ( 3 −k4 ∑
k4 −k5
s0 s1
=
(
k5 ( ∑ u1 =0
k4 − k5 u0 k5 u1
)
2−(k4 −k5 ) exp
) (−1)u1 exp
[a 2
[a 2
] (k4 − k5 − 2u0 ) ,
] (k5 − 2u1 ) . (28)
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Binder cumulant analysis for MC simulations obtained for several lattice size L: In order to determine the dependence of critical properties of the system on the lattice dimensionality, one can calculate the fourth order cumulant U4 (L, T ) i.e. the Binder cumulant for various lattice sizes according to ⟨MT4 ⟩
(29)
3⟨MT2 ⟩2
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U4 (L, T ) = 1 −
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where ⟨MT2 ⟩ and ⟨MT4 ⟩ denote the second and fourth moments of the total magnetization MT . The Binder cumulant defined by Eq. (29) takes the value 2/3 at low temperatures (T < T c ). For sufficiently large temperature values (T > T c ) it tends to zero. At T = T c , U4 (L, T ) acquires a nontrivial value, namely the critical Binder cumulant Uc . In this case, U4 (L, T ) curves corresponding to different L intersect with each other at T = T c (cf. the points denoted by Uc in Fig. 7). As an indicator of the type of the magnetic phase transition, thermal variation of U4 (L, T ) curves corresponding to different lattice sizes can be examined. As shown in Fig. 7, based on the considered systems sizes, U4 (L, T ) curves do not exhibit any dip to negative values which means that the transitions between paramagnetic and ferromagnetic phases are of second order. From Fig. 7, it can be naively seen that the peak position of the susceptibility χ coincides with the intersection point of U4 (L, T ) curves for the whole set of the system parameters. Therefore, we can conclude that the system size L = 20 successfully simulates the behavior of the system in the thermodynamic limit. As a final remark, we should also note that, since our main motivation is not to determine the critical temperature values in high precision, in order to reduce the simulation time, we have used susceptibility data to evaluate the approximate value of the critical temperature. Of course, in a narrow temperature scale, susceptibility data would slightly become different from fourth-order cumulant analysis. However, this may cause only small differences in phase diagrams depicted in Fig. 2. References
[20] [21] [22] [23] [24] [25] [26] [27] [28]
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[12] [13] [14] [15] [16] [17] [18] [19]
A. M. Tishin, Y. I. Spichkin, The Magnetocaloric Effect and its Applications, Institute of Physics 2003. E. Warburg, Ann. Phys. 13 (1881) 141. P. Weiss, A. Piccard, J. Phys. (Paris) 7 (1917) 103. P. Debye, Ann. Phys. 81 (1926) 1154. W. F. Giauque, J. Am. Chem. Soc. 49 (1927) 1864. W. F. Giauque, D. P. MacDougall, Phys. Rev. 43 (1933) 768. A. Smith, Eur. Phys. J. H 38 (2013) 507. C. Zimm, A. Jastrab, A. Sternberg, V. Pecharsky, K. Gschneidner Jr., M. Osborne, I. Anderson, Adv. Cyrog. Eng. 43 (1998) 1759. V. Pecharsky, K. Gschneidner Jr., J. Appl. Phys. 86 (1999) 565. V. Chaudhary and R. V. Ramanujan, Sci. Rep. 6 (2016) 35156. A. M. Tishin, Magnetic therapy of malignant neoplasms by injecting material particles with high magnetocaloric effect and suitable magnetic phase transition temperature, Patent Number: EP1897590-A1, 2008 K. A. Gschneidner Jr., V. K. Pecharsky, Annu. Rev. Mater. Sci. 30 (2000) 387. K. A. Gschneidner Jr., V. K. Pecharsky, A. O. Tsokol, Rep. Prog. Phys. 68 (2005) 1479. M. H. Phan, S. C. Yu, J. Magn. Magn. Mater 308 (2007) 325. B. G. Shen, J. R. Sun, F. X. Hu, H. W. Zhang and Z. H. Cheng, Adv. Mater. 21 (2009) 4545. N. A. de Oliveira, P. J. von Ranke, Phys. Rep. 489 (2010) 89. V. Franco, J.S. Bl´azquez, B. Ingale, and A. Conde, Annu. Rev. Mater. Res. 42 (2012) 305. J. Romero Gomez, R. Ferreiro Garcia, A. De Miguel Catoira, M. Romero Gomez, Renew. Sustainable Energy Rev. 17 (2013) 74. In the thermal derivative of magnetization in Eq. (2), we have used the centered difference formula which can be given as ( dMorder ) to calculate ( ) Mi −Mi−1 1 Mi+1 −Mi T dT H = 2 T i+1 −T i + T i −T i−1 . R. Tamura, T. Ohno, H. Kitazawa, Appl. Phys. Lett. 104 (2014) 052415. R. Tamura, S. Tanaka, T. Ohno, H. Kitazawa, J. Appl. Phys. 116 (2014) 053908. J. Streˇcka, K. Karˇlov´a, T. Madaras, Physica B 466-467 (2015) 76. K. Karˇlov´a, J. Streˇcka, T. Madaras, Acta Phys. Pol. A 131 (2017) 630. G. V. Brown, J. Appl. Phys. 47 (1976) 3673. S. Yu. Dan’kov, A. M. Tishin, V. K. Pecharsky, and K. A. Gschneidner, Jr., Phys. Rev. B 57 (1998) 3478. V. K. Pecharsky, K. A. Gschneidner, Jr., Phys. Rev. Lett. 78 (1997) 4494. V. D. Buchelnikov, P. Entel, S. V. Taskaev, V. V. Sokolovskiy, A. Hucht, M. Ogura, H. Akai, M. E. Gruner, and S. K. Nayak, Phys. Rev. B 78 (2008) 184427. N. Singh, R. Arr´oyave, J. Appl. Phys. 113 (2013) 183904.
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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0.7
L=10
0.5
L=20
0.4
L=30
0.3
J
AB
=0.2J
AC
UC
p=0.0
0.2 0.1
0.7
0.6
L=10
0.5
J
L=20
0.4
=0.2J
AC
p=0.5
L=30
0.3
AB
U4(L,T)
0.6
U4(L,T)
U4(L,T)
0.7
UC
0.2 0.1
0.0 0.5
1.0
1.5
2.0
2.5
L=10
0.5
L=20
0.4
L=30
0.3
J
AB
=0.2J
AC
p=1.0
UC
0.2 0.1
0.0
0.0
100
0.6
0.0
0.0
0.5
1.0
1.5
2.0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
100 100
J 1
AB
10
=0.2J
L=20
J
AC 1
p=0.0
0.1
AB
L=20
=0.2J
10
AC
p=0.5
1
0.1
0.01
0.5
1.0
1.5
2.0
2.5
0.5
1.0
T/J AC
(a)
0.00
L=30
0.3
AB
=0.43J
AC
p=0.0
UC
0.2
0.0
L=10
0.5
L=20
0.4
L=30
0.3
J
AB
=0.43J
AC
p=0.5
UC
0.2
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
AB
10
=0.43J
1
0.1
L=20 AB
0.01
1.0
1.5
2.0
2.5
L=30
0.3
U4(L,T)
L=20
0.4
AC
p=0.0
UC
0.2 0.1
L=10
0.5
L=20
0.4
L=30
0.3 0.2
1.0
1.5
2.0
2.5
3.0
100
1.0
1.5
2.0
2.5
J
AB
=1.0J
AC
UC
p=0.5
0.5
1.0
1.5
2.0
AB
10
=1.0J
2.5
3.0
3.5
4.0
4.5
L=20
J
AC
1
p=0.0
0.1
AB
=1.0J
0.01
1.0
1.5
T/J AC
2.0
2.5
3.0
0.0
1.5
2.0
2.5
AC
0.5
1.0
T/J AC
(f)
L=10 L=20
0.4
L=30
0.3
J
AB
=1.0J
AC
UC
p=1.0
0.2
0.0
0.5
1.0
1.5
2.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
0.5
1.0
1.5
2.5
3.0
3.5
4.0
4.5
5.0
5.5
L=20
J
AB
=1.0J
AC
p=1.0
0.1
0.01
1E-3
2.0
2.5
T/J AC
(h)
EP
(g)
=0.43J
0.5
1
0.01
0.5
AB
0.6
10
1E-3
0.0
2.5
p=1.0
0.0
AC
p=0.5
0.1
1E-3
2.0
100
TE D
L=20
J 1
1.5
0.0
0.0
100
10
1.0
0.1
0.0
0.5
0.5
0.7
0.6
0.1
0.0 0.0
UC
0.01
0.7
=1.0J
AC
L=20
(e)
0.6
AB
=0.43J
T/J AC
0.7
J
0.0
J
AC
0.5
(d)
L=10
AB
p=1.0
1E-3
0.0
T/J AC
0.5
J
0.2
1
0.01
0.5
L=30
0.3
0.1
1E-3
0.0
L=20
0.4
10
=0.43J
p=0.5
0.1
1E-3
L=10
0.5
100
J
AC
p=0.0
2.5
U4(L,T)
J 1
0.6
M AN U
10
1.50
0.0
0.0
100
L=20
1.25
0.1
0.0
0.0
1.00
0.7
0.6
0.1
100
0.75
(c)
U4(L,T)
0.4
J
0.50
SC
L=20
0.25
T/J AC
0.7
L=10
0.5
U4(L,T)
U4(L,T)
2.0
(b)
0.1
U4(L,T)
1.5
T/J AC
0.6
AC
1E-3
0.0
0.7
=0.2J
0.01
1E-3
0.0
AB
p=1.0
0.1
0.01
1E-3
J
RI PT
L=20
10
3.0
3.5
4.0
4.5
0.0
0.5
1.0
T/J AC
(i)
Figure 7: Binder cumulant curves corresponding to lattice sizes L = 10, L = 20, and L = 30 with some selected values of parameters p and JAB .
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[29] V. D. Buchelnikov, V. V. Sokolovskiy, H. C. Herper, H. Ebert, M. E. Gruner, S. V. Taskaev, V. V. Khovaylo, A. Hucht, A. Dannenberg, M. Ogura, H. Akai, M. Acet, and P. Entel, Phys. Rev. B 81 (2010) 094411. [30] D. Comtesse, M. E. Gruner, M. Ogura, V. V. Sokolovskiy, V. D. Buchelnikov, A. Grnebohm, R. Arr´oyave, N. Singh, T. Gottschall, O. Gutfleisch, V. A. Chernenko, F. Albertini, S. Fhler, and P. Entel, Phys. Rev. B 89 (2014) 184403. [31] E. P. N´obrega, N. A. de Oliveira, P. J. von Ranke, and A. Troper, Phys. Rev. B 72 (2005) 134426. [32] Y. Ma, A. Du, J. Magn. Magn. Mater. 321 (2009) L65. [33] R. Arai, R. Tamura, H. Fukuda, J. Li, A. T. Saito, S. Kaji, H. Nakagome, T. Numazawa, IOP Conf. Ser. Mater. Sci. Eng. 101 (2015) 012118. ˇ [34] M. Zukoviˇ c, J. Magn. Magn. Mater. 374 (2015) 22. [35] S. Ohkoshi, K. Hashimoto, J. Am. Chem. Phys. 121 (1999) 10591. ˇ [36] M. Zukoviˇ c, A. Bob´ak, J. Magn. Magn. Mater. 322 (2010) 2868. [37] F. C. S´aBarreto, I.P. Fittipaldi, B. Zeks, Ferroelectrics 39 (1981) 1103. [38] T. Kaneyoshi, Acta Phys. Pol. A 83 (1993) 703. [39] T. Balcerzak, J. Magn. Magn. Mater. 246 (2002) 213. [40] J. W. Tucker, J. Phys. A: Math. Gen. 27 (1994) 659. [41] A. Bob´ak, O. F. Abubrig, D. Horv´ath, Physica A 312 (2002) 187. [42] A. Bob´ak, D. Horv´ath, Phys. Status Solidi B 213 (1999) 459. [43] A. Bob´ak, Physica A 258 (1998) 140. [44] E. Vatansever, Y. Y¨uksel, J. Alloys Compd. 689 (2016) 446.
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[45] G. M. Buend´ıa, J. E. Villarroel, J. Magn. Magn. Mater. 310 (2007) e495. [46] Y. Y¨uksel, J. Phys. Chem. Sol. 86 (2015) 207. [47] In EFT formulation, the type of the magnetic phase transition has been determined by examining the temperature dependence of magnetization curves defined by Eq. (16). In MC simulations however, we have used the numerical results for Binder cumulant analysis, (see Appendix). [48] E. Manuel, M. Evangelisti, M. Affronte, M. Okubo, C. Train, M. Verdaguer, Phys. Rev. B 73 (2006) 172406. [49] M. Evangelisti, E. Manuel, M. Affronte, M. Okubo, C. Train, J. Magn. Magn. Mater. 316 (2007) e569.
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Ferromagnetic ternary alloy system has been investigated. Magnetic and magnetocaloric properties have been studied. Effective field theory and Monte Carlo simulations have been implemented. A direct magnetocaloric effect has been observed. Magnetocaloric properties are strongly dependent on the compositional factor p.
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