A generalized magnetic refrigeration scheme
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r A generalized magnetic refrigeration scheme
Ryo Tamura, a) Takahisa Ohno, b) and Hideaki Kitazawa c)1) International Center for Young Scientists, National Institute for Materials Science,1-2-1, Sengen, Tsukuba, Ibaraki 305-0047, Japan Computational Materials Science Unit, National Institute for Materials Science,1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan Quantum Beam Unit, National Institute for Materials Science, 1-2-1 Sengen,Tsukuba, Ibaraki 305-0047, Japan (Dated: 10 September 2018)
We have investigated the magnetocaloric effects in antiferromagnets and comparedthem with those in ferromagnets using Monte Carlo simulations. In antiferromagnets,the magnetic entropy reaches a maximum value at a finite magnetic field when thetemperature is fixed below the N´eel temperature. Using the fact, we proposed aprotocol for applying magnetic fields to achieve the maximum efficiency for magneticrefrigeration in antiferromagnets. In particular, we found that at low temperatures,antiferromagnets are more useful for magnetic refrigeration than ferromagnets. a) [email protected] b) [email protected] c) [email protected]
In conventionalferromagnets and paramagnets, when the applying magnetic field is turned off, the magneticentropy increases. Thus, these magnetic materials absorb an amount of heat associated withthe magnetic entropy change, and the temperature drops. This is the magnetocaloric effect(MCE), which can be applied in magnetic refrigeration. From a viewpoint of magnetic re-frigeration, magnetic materials which exhibit a large MCE under a small applied magneticfield are regarded as good materials. The large MCE was observed around the Curie tem-perature in ferromagnets using the conventional protocol for applying magnetic fields wherethe magnetic field is varied from a finite value to zero. Thus, ferromagnets are likely to besuitable materials for magnetic refrigeration.
In recent years, it was reported in many experimental researches that antiferromagnetsare also useful for magnetic refrigeration.
There are two features on the magnetic en-tropy reported in these experimental researches when the conventional protocol for applyingmagnetic fields is used. First one is that the inverse MCE, where the magnetic entropydecreases when the applying magnetic field is turned off, was observed below the N´eel tem-perature. Second one is that a large MCE similar to that in ferromagnets was obtainedaround the N´eel temperature. These complicated behaviors of the magnetic entropy werenot observed in conventional ferromagnets and paramagnets. Thus, in order to understandthe potential of antiferromagnets for magnetic refrigeration, the microscopic features of theMCE in antiferromagnets, which are not well understood, should be investigated.The purpose of this letter is to present the microscopic features of the MCE in antiferro-magnets by using Monte Carlo simulations. We show that the magnetic entropy reaches amaximum value at a finite magnetic field when the temperature is fixed below the N´eel tem-perature. Based on the fact, we propose a protocol for applying magnetic fields to achievethe maximum efficiency for magnetic refrigeration in antiferromagnets. By using our pro-posed protocol, we find that antiferromagnets exhibit a large magnetic entropy change in theordered phase below the N´eel temperature rather than around the N´eel temperature. Fur-thermore, we show that at low temperatures, antiferromagnets are more useful for magneticrefrigeration than ferromagnets.To explore the relation between the ordered magnetic structure and MCE in a unifiedway, we study MCEs in the Ising models on a simple cubic lattice with the periodic boundarycondition. Here, let N = L × L × L be the number of spins in a simple cubic lattice, where2 is the linear dimension. The model Hamiltonian is defined by H = − J ab X h i,j i ab s zi s zj − J c X h i,j i c s zi s zj − H X i s zi , s zi = ± , (1)where the first and second sums are over nearest-neighbor sites in the ab -plane and alongthe c -axis, respectively, and J ab and J c represent magnetic interactions. Furthermore, H denotes a uniform magnetic field along the z -axis, where g -factor and the Bohr magneton µ B are set to unity. When the sign of the magnetic interaction is positive, the magneticinteraction is ferromagnetic, whereas the magnetic interaction is antiferromagnetic whenthe sign is negative. For simplicity, we consider the case that the absolute values of J ab and J c are the same, that is J := | J ab | = | J c | , where J is the energy unit. At zero magnetic field( H/J = 0), the system exhibits a second-order phase transition at the critical temperature T c /J = 1 . · · · independent of the signs of J ab and J c , where the Boltzmann constant k B is set to unity. In this letter, we focus on four combinations of interactions (orderedmagnetic structures): (i) J ab > J c > J ab > J c < J ab < J c > J ab < J c < in Monte Carlo simulations to calculate the tem-perature T and H dependence of the magnetic entropy with high accuracy. In the Wang-Landau method, we use a random walk in the energy space to obtain the absolute densityof states. Then, we can directly calculate the magnetic entropy without integrating magne-tization or specific heat. For details of the Wang-Landau method, see Refs. [35,36].Figure 1 (b) shows the magnetic entropy per spin S M ( T, H ) as a function of
T /T c for H/J = 0 - 5 when the lattice size is L = 16. Here, the unit of S M ( T, H ) is the Boltzmannconstant k B , and thus the magnetic entropy per mol is obtained by k B N A S M ( T, H ) [J / mol K]where N A is the Avogadro’s number. Furthermore, since the spin degree of freedom is twoin the Ising model, the maximum value of S M ( T, H ) is ln 2 = 0 . · · · . When H/J = 0,the results do not depend on the magnetic structure. The magnetic entropies for L = 8,12, and 16 overlap within the line width in Fig. 1 (b), and thus the lattice size dependence3 erromagnet A-type antiferromagnet G-type antiferromagnetC-type antiferromagnet (c)(a)(b) abc
0 1 2 0 1 2 3
0 1 2 0 1 2 3 4 5
0 1 2 0 1 2 3
0 1 2 0 1 2 3 4 5
0 1 2 0 1 2 3
0 1 2 0 1 2 3 4 5
0 1 2 0 1 2 3 4 5
FIG. 1. (Color online) (a) Schematic of ordered magnetic structures in the Ising model on a simplecubic lattice. The signs of the magnetic interactions in the ab -plane, J ab , and along the c -axis, J c ,where each magnetic structure is in the ground state are shown. (b) Temperature T /T c dependenceof the magnetic entropy per spin S M ( T, H ) for L = 16 under several magnetic fields. Arrow is theisothermal demagnetization process in which the magnetic entropy change becomes a maximumfrom T /T c = 0 .
85 and
H/J = 5. Inset shows
T /T c dependence of H max ( T ) /J at which S M ( T, H )reaches a maximum value. (c) Contour map of the magnetic entropy change ∆ S M ( T, H ′ ) definedin Eq. (2). of the magnetic entropy is negligibly small. Therefore, we use a lattice size of L = 16throughout this paper. In the ferromagnet, magnetic entropy increases as the magnetic fielddecreases at any temperature. The same behaviors are obtained in the paramagnetic phaseof antiferromagnets above T c /J . In contrast, below T c /J for antiferromagnets, there is the4ase that the magnetic entropy decreases as the magnetic field decreases, which is the originof the inverse MCE. Because the magnetic field does not favor the spin configuration inthe antiferromagnetically ordered phase, the antiferromagnetic state is destroyed by apply-ing the magnetic field. This behavior suggests that in antiferromagnets, there is a finitemagnetic field H max ( T ) at which the magnetic entropy reaches a maximum value when thetemperature is fixed below T c /J . The inset of Fig. 1 (b) shows T /T c dependence of H max ( T )for each antiferromagnetic structure. Note that H max ( T ) is always zero in the ferromagnetas mentioned above. The behavior of H max ( T ) below T c /J in antiferromagnets indicatesthat we should use a new protocol for applying magnetic fields to obtain the maximummagnetic entropy change. That is, the magnetic field should be varied from a finite value to H max ( T ) instead of zero below T c /J in antiferromagnets. For example, suppose we considerthe isothermal demagnetization process from T /T c = 0 .
85 and
H/J = 5. The processes inwhich the magnetic entropy change becomes a maximum are denoted by arrows in Fig. 1(b). In the ferromagnet, when the magnetic field is turned off, the maximum magnetic en-tropy change is obtained. In contrast, in each antiferromagnet, the magnetic entropy changeobtained from our proposed protocol where
H/J is varied from 5 to H max ( T ) /J is largerthan that obtained from the conventional protocol where H/J is varied from 5 to 0.Next, we consider a temperature region in which a large magnetic entropy change isobtained in each magnetic structure. Here, we define the magnetic entropy change by∆ S M ( T, H ′ ) := max { S M ( T, H ) | H ≤ H ′ } − min { S M ( T, H ) | H ≤ H ′ } , (2)where H ′ is the maximum value of applied magnetic field. ∆ S M ( T, H ′ ) indicates the maxi-mum magnetic entropy change regardless of a protocol for applying magnetic fields when themagnetic field H ( ≤ H ′ ) is applied. Note that when H ′ ≥ H max ( T ), max { S M ( T, H ) | H ≤ H ′ } = S M ( T, H max ( T )). Figure 1 (c) shows T /T c and H ′ /J dependence of ∆ S M ( T, H ′ ).In the ferromagnet, ∆ S M ( T, H ′ ) has a large value around T c /J . Thus, the ferromagnetexhibits a large magnetic entropy change around the Curie temperature. However, in theordered phase below T c /J , the magnetic entropy change is exceedingly small. In contrast,in antiferromagnets, ∆ S M ( T, H ′ ) becomes large below T c /J . This indicates that antiferro-magnets can exhibit a large magnetic entropy change in the ordered phase below the N´eeltemperature rather than around the N´eel temperature. Moreover, the temperature regionin which ∆ S M ( T, H ′ ) has a large value moves towards lower temperature as increasing the5 erromagnet ★ C-type antiferromagnetA-type antiferromagnetG-type antiferromagnet
FACG C A F
FIG. 2. (Color online) A set of regions such that each magnetic structure exhibits a larger magneticentropy change than other three magnetic structures. The star indicates the parameter used inFig. 3. number of antiferromagnetic interactions.Based on the results of ∆ S M ( T, H ′ ), Fig. 2 shows a region in which each magnetic struc-ture is most suited for magnetic refrigeration from among the four types of magnetic struc-tures. That is, in each region, a drawn magnetic structure exhibits a larger magnetic en-tropy change than other three magnetic structures. The obtained value of magnetic entropychange in each region can be known from corresponding contour map of ∆ S M ( T, H ′ ) shownin Fig. 1 (c). Figure 2 indicates that the ferromagnet is always suited for magnetic refriger-ation at high temperatures above T c /J . In contrast, at low temperatures below T c /J , thereis a wide region where antiferromagnets are more useful for magnetic refrigeration than theferromagnet.We showed that at low temperatures, antiferromagnets exhibit a larger magnetic en-tropy change than the ferromagnet when the magnetic field is varied from a finite valueto H max ( T ) instead of zero. Below T c /J for antiferromagnets, H max ( T ) is nonzero valueshown in the inset of Fig. 1 (b), and thus H max ( T ) must be known to execute our proposedprotocol for applying magnetic fields. Here, we present a method by which H max ( T ) can beeasily obtained for antiferromagnets below T c /J . Note that H max ( T ) is always zero above T c /J as mentioned above. Suppose we calculate the difference between magnetic entropies S M ( T, H ) − S M ( T, H ′ ) as a function of H ( ≤ H ′ ) at a fixed temperature. In antiferromag-6ets, there should be a peak in the difference when H ′ ≥ H max ( T ), and the peak positionis H max ( T ). For example, H/J dependence of S M ( T, H ) − S M ( T, H ′ ) with H ′ /J = 5 at T /T c = 0 .
85 of the Ising models defined by Eq. (1) is shown in Fig. 3. The position of theparameters is indicated in Fig. 2. The value of S M ( T, H ) − S M ( T, H ′ ) at H = H max ( T )represents ∆ S M ( T, H ′ ) defined by Eq. (2) when our proposed protocol is used. In this case,it is clear that the A-type antiferromagnet is the most suitable as a magnetic refrigerationmaterial. In contrast, the value of S M ( T, H ) − S M ( T, H ′ ) at H = 0 is the magnetic entropychange using the conventional protocol where the magnetic filed is varied from H ′ to zero.If the conventional protocol is used, the ferromagnet is regarded as the most suitable asa magnetic refrigeration material. The magnetic entropy change obtained from our pro-posed protocol increases by 170 % (resp. 200 % and 1280 %) compared with that obtainedfrom the conventional protocol in the A-type antiferromagnet (resp. C-type and G-typeantiferromagnets). The method to obtain H max ( T ) can be used with the thermodynamicformula: S M ( T, H ) − S M ( T, H ′ ) = Z HH ′ (cid:18) ∂M∂T (cid:19) H ′′ dH ′′ , (3)where M is the magnetization, and the integrating interval is [ H ′ , H ]. This means that onlythe magnetization process under various temperatures is required. Thus, this method canbe performed by data which were already obtained in experimental researches on magneticrefrigeration. Moreover, the value of S M ( T, H ) − S M ( T, H ′ ) can be also estimated by thespecific heat.In conclusion, we demonstrated the microscopic features of the magnetocaloric effects inthe ferromagnetic and antiferromagnetic Ising models by Monte Carlo simulations based onthe Wang-Landau method. In antiferromagnets, the magnetic entropy reaches a maximumvalue at a finite magnetic field H max ( T ) when the temperature is fixed below the N´eeltemperature. Thereby, in order to obtain the maximum magnetic entropy change below theN´eel temperature, the magnetic field should be varied from a finite value to H max ( T ) insteadof zero. By using this protocol, we found that antiferromagnets exhibit a large magneticentropy change in the ordered phase below the N´eel temperature rather than around theN´eel temperature. We also showed that antiferromagnets are more useful for magneticrefrigeration than ferromagnets at low temperatures. In non-ferromagnetic materials, theordered state is destroyed by applying the magnetic field, and there should be a finite7 FMA-type AFMC-type AFMG-type AFM
FIG. 3. (Color online) Magnetic field
H/J dependence of the difference between magnetic entropies S M ( T, H ) − S M ( T, H ′ ) at T /T c = 0 .
85 for ferromagnet (FM) and three types of antiferromagnets(AFMs). The maximum applied magnetic field is H ′ /J = 5. magnetic field H max ( T ) at which the magnetic entropy reaches a maximum value. Thus, ourproposed protocol for applying magnetic fields can be widely applied to non-ferromagneticmaterials to achieve a maximum efficiency for magnetic refrigeration.We thank Kenjiro Miyano and Shu Tanaka for useful comments and discussions. R.T.and H. K. were partially supported by a Grand-in-Aid for Scientific Research (C) (Grant No.25420698). In addition, R. T. was partially supported by National Institute for MaterialsScience. The computations in the present work were performed on super computers at theSupercomputer Center, Institute for Solid State Physics, University of Tokyo and NationalInstitute for Materials Science. REFERENCES V. K. Pecharsky and K. A. Gschneidner Jr., J. Magn. Magn. Mater. , 44 (1999). M. A. Novotny and P. A. Rikvold, in
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