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Origin of anomalous magnetocaloric effect in „ Dy 1 Àz Er z … Al 2 alloys A. L. Lima, 1 I. S. Oliveira, 1 A. M. Gomes, 1 and P. J. von Ranke 2, * 1 Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, Rio de Janeiro22290-180, Brazil 2 Universidade do Estado do Rio de Janeiro, Rua Sa ˜ o Francisco Xavier, 524, Rio de Janeiro 20550-013, Brazil ~Received 27 February 2002; published 29 April 2002! We report a theoretical description of the anomalous magnetocaloric peak in (Dy 12z Er z )Al 2 in the concen- tration range 0.15,z ,0.5 which was experimentally discovered by Gschneidner and co-workers. The anoma- lous peak was investigated using a Hamiltonian that includes the crystalline electrical field effects. DOI: 10.1103/PhysRevB.65.172411 PACS number~s!: 75.30.Sg, 65.40.2b, 65.40.Gr We investigated theoretically (Dy 1 2z Er z )Al 2 alloys that are strong candidates for magnetic refrigerant materials. One of the biggest challenges in the magnetic refrigeration is to find a proper magnetic material to work as a refrigerant ma- terial, which must present two fundamental characteristics 1 : ~a! large isothermal entropy changes upon variation of the external magnetic field and ~b! large temperature change in the adiabatic process. In simple ferromagnetic systems, for instance, DyAl 2 and ErAl 2 , only one peak in the magneto- caloric potential near or at Curie temperature is expected. Recently, it was experimentally observed that (Dy 1 2z Er z )Al 2 alloys, for concentration varying 0.15,z ,0.5, present two peaks in the adiabatic temperature variation upon the change of the applied magnetic field from zero to 7.5 T. The upper temperature peak is due to ferromagnetic ordering, while the nature of the lower one was not understood. 2 In order to investigate the origin of the second peak, we developed a microscopic model, in which the Hamiltonian is solved ex- actly, considering the exchange interaction, the crystalline electric field ~CEF! and the Zeeman interaction. We deter- mined the adiabatic temperature change vs temperature in (Dy 1 2z Er z )Al 2 for z 50.0, 0.30, 0.5, 1.0. The theoretical re- sults obtained are in good agreement with the experimental measurements and the anomalous peak was fully understood and associated with the high density of CEF levels. The in- vestigation on the microscopic mechanism that is responsible for the large range of temperature variation in a adiabatic process in which refrigeration/heating occurs, can have a high impact on the refrigerant magnetic materials research. 3 The thermodynamic properties of our interest, in the mag- netic system (Dy 1 2z Er z )Al 2 , can be calculated starting from the following Hamiltonian, H ˆ CEF 5W F X F 4 ~ O 4 0 15 O 4 4 ! 1 ~ 1 2u X u ! F 6 ~ O 6 0 221O 6 4 ! G 2g m B HJ z , ~1! where the first term describes the single-ion CEF Hamil- tonian written in the Lea, Leask, and Wolf ~LLW! notation 4 where W gives the CEF energy scale and X ( 21 ,X ,1) gives the relative contributions of the fourth and sixth degree in O n m Stevens’ equivalent operators. 5 The constants F 4 and F 6 have the values F 4 560 and F 6 5138 62. The second term is the effective Zeeman interaction in which the exchange interaction was included in molecular field approximation. Here, g is the Lande factor, m B is the Bohr magneton and H 5H 0 1l M is the external magnetic field plus the effective molecular field with the molecular field constant l, and M is the magnetization, which can be calculated from the self- consistent solution of the magnetic state equation, M 5g m B ( ^ « i u J z u « i & exp@ 2« i / KT # ( exp@ 2« i / KT # , ~2! where « i and u « i & are, respectively, the energy eigenvalues and eigenvectors of Hamiltonian ~1!. The three main contributions to the total entropy in the considered magnetic system are S ~ H , T ! 53 R H 4 S T Q D D 3 E 0 Q D/ T x 3 dx exp~ x ! 21 2ln F 1 2exp S 2 Q D T D G J 1g T 1R F ln F ( exp S 2 « i KT D G 1 ^ E & KT G , ~3! where R is the universal gas constant, Q D is the Debye tem- perature, g is the electronic heat capacity coefficient and ^E& is the mean energy. The first and second terms in relation ~3! represent the lattice ~Debye term! and electronic contribution that are very easy to be calculated since they depend only on temperature. The last term comes from magnetic interaction and besides the temperature, it depends also on H 5H 0 1l M . Therefore, the magnetic entropy term must be calcu- lated in self-consistent way. For a given temperature T and for an external magnetic field H 0 , the transcendental equa- tion ~2! must be solved in order to obtain the exchange field to update the magnetic entropy. The adiabatic increase in the sample temperature, 2D T ad 5T 2 2T 1 ~the magnetocaloric effect! is theoretically calculated considering the variation of the external magnetic field, from zero to H 0 for instance, and solving the following equation S ( H 0 50, T 1 ) 5S ( H 0 0, T 2 ). The numerical procedures to treat the model Hamiltonian applied to the magnetic system (Dy 1 2z Er z )Al 2 is simplified since both Dy and Er have the same total angular momen- tum, J 5 15 2 , leading to the same matrix order for all z con- centration. The other magnetic parameters $ g , l , W, X % were considered to be dependent on concentration. In this PHYSICAL REVIEW B, VOLUME 65, 172411 0163-1829/2002/65~17!/172411~3!/$20.00 ©2002 The American Physical Society 65 172411-1
Transcript of alloys
PHYSICAL REVIEW B, VOLUME 65, 172411
Origin of anomalous magnetocaloric effect in„Dy1ÀzEr z…Al2 alloys
A. L. Lima,1 I. S. Oliveira,1 A. M. Gomes,1 and P. J. von Ranke2,*1Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, Rio de Janeiro 22290-180, Brazil
2Universidade do Estado do Rio de Janeiro, Rua Sa˜o Francisco Xavier, 524, Rio de Janeiro 20550-013, Brazil~Received 27 February 2002; published 29 April 2002!
We report a theoretical description of the anomalous magnetocaloric peak in (Dy12zErz)Al2 in the concen-tration range 0.15,z,0.5 which was experimentally discovered by Gschneidner and co-workers. The anoma-lous peak was investigated using a Hamiltonian that includes the crystalline electrical field effects.
DOI: 10.1103/PhysRevB.65.172411 PACS number~s!: 75.30.Sg, 65.40.2b, 65.40.Gr
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We investigated theoretically (Dy12zErz)Al2 alloys thatare strong candidates for magnetic refrigerant materials.of the biggest challenges in the magnetic refrigeration isfind a proper magnetic material to work as a refrigerant mterial, which must present two fundamental characteristi1:~a! large isothermal entropy changes upon variation ofexternal magnetic field and~b! large temperature changethe adiabatic process. In simple ferromagnetic systemsinstance, DyAl2 and ErAl2 , only one peak in the magnetocaloric potential near or at Curie temperature is expecRecently, it was experimentally observed that (Dy12zErz)Al2alloys, for concentration varying 0.15,z,0.5, present twopeaks in the adiabatic temperature variation upon the chaof the applied magnetic field from zero to 7.5 T. The upptemperature peak is due to ferromagnetic ordering, whilenature of the lower one was not understood.2 In order toinvestigate the origin of the second peak, we developemicroscopic model, in which the Hamiltonian is solved eactly, considering the exchange interaction, the crystalelectric field ~CEF! and the Zeeman interaction. We detemined the adiabatic temperature change vs temperatur(Dy12zErz)Al2 for z50.0, 0.30, 0.5, 1.0. The theoretical rsults obtained are in good agreement with the experimemeasurements and the anomalous peak was fully undersand associated with the high density of CEF levels. Thevestigation on the microscopic mechanism that is responsfor the large range of temperature variation in a adiabprocess in which refrigeration/heating occurs, can havhigh impact on the refrigerant magnetic materials resear3
The thermodynamic properties of our interest, in the mnetic system (Dy12zErz)Al2 , can be calculated starting fromthe following Hamiltonian,
HCEF5WF X
F4~O4
015O44!1
~12uXu!F6
~O60221O6
4!G2gmBHJz, ~1!
where the first term describes the single-ion CEF Hamtonian written in the Lea, Leask, and Wolf~LLW ! notation4
where W gives the CEF energy scale andX (21,X,1)gives the relative contributions of the fourth and sixth degin On
m Stevens’ equivalent operators.5 The constantsF4 andF6 have the valuesF4560 andF65138 62. The second termis the effective Zeeman interaction in which the exchaninteraction was included in molecular field approximatio
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Here, g is the Lande factor,mB is the Bohr magneton andH5H01lM is the external magnetic field plus the effectivmolecular field with the molecular field constantl, andM isthe magnetization, which can be calculated from the sconsistent solution of the magnetic state equation,
M5gmB
(^« i uJzu« i&exp@2« i /KT#
( exp@2« i /KT#, ~2!
where« i and u« i& are, respectively, the energy eigenvaluand eigenvectors of Hamiltonian~1!.
The three main contributions to the total entropy in tconsidered magnetic system are
S~H,T!53RH 4S T
QDD 3E
0
QD/T x3dx
exp~x!21
2 lnF12expS 2QD
T D G J1gT1RF lnF( expS 2
« i
KTD G1^E&KT G , ~3!
whereR is the universal gas constant,QD is the Debye tem-perature,g is the electronic heat capacity coefficient and^E&is the mean energy. The first and second terms in relation~3!represent the lattice~Debye term! and electronic contributionthat are very easy to be calculated since they depend onltemperature. The last term comes from magnetic interacand besides the temperature, it depends also onH5H01lM . Therefore, the magnetic entropy term must be callated in self-consistent way. For a given temperatureT andfor an external magnetic fieldH0 , the transcendental equation ~2! must be solved in order to obtain the exchange fito update the magnetic entropy.
The adiabatic increase in the sample temperatu2DTad5T22T1 ~the magnetocaloric effect! is theoreticallycalculated considering the variation of the external magnfield, from zero toH0 for instance, and solving the followingequationS(H050, T1)5S(H0Þ0, T2).
The numerical procedures to treat the model Hamiltonapplied to the magnetic system (Dy12zErz)Al2 is simplifiedsince both Dy and Er have the same total angular momtum, J5 15
2 , leading to the same matrix order for allz con-centration. The other magnetic parameters$g, l, W, X%were considered to be dependent on concentration. In
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BRIEF REPORTS PHYSICAL REVIEW B 65 172411
approximation, for a givenz (0<z<1) value, the set parameter used in (Dy12zErz)Al2 was $g, l, W, X%Dy(12z)Er(z)Al2
5z$g, l, W, X%ErAl21(12z)$g, l, W, X%DyAl2, wherethe set parameters for the extreme concentrations$6/5, 13.3 I2/meV, 20.0252 meV,20.262%ErAl2 and$4/3, 44.0I2/meV, 20.011 meV ,0.3%DyAl2, taken from Ref.6. The effective Debye temperature was taken from the nmagnetic and isostructural systems LaAl2 and LuAl2 usingthe assumptions considered in Ref. 7;~4! The electronic heacapacity coefficient,g55.5 mJ mol21 K22, was assumed tobe equal to that of the nonmagnetic compound LuAl2 .7
Figure 1 shows the 2DTad vs temperature in(Dy12zErz)Al2 for ~z50, 0.3, 0.5, and 1.0! for magneticfield change from zero to 7.5 T. The symbols representexperimental data taken from Ref. 2 and the solid curwere calculated from our theoretical model described aboThe agreements between theoretical and experimental dagood since no fitting procedure was performed in the moparameters. The main goal of this paper is not to fit a stdard model to experimental data, but to understand theture of the lower anomalous peaks observed experimen~see in Fig. 1 the curves forz50.3 andz50.5!. These lower
FIG. 2. The Gg3-DOS vs X-CEF parameter calculated forz
50.3 andz50.5. The lower inset gives the LLW diagrams and tupper inset shows the splitting of theGg
3 CEF level calculated forz50.5 atT56 K.
FIG. 1. The2DTad vs temperature in (Dy12zErz)Al2 (z50,0.3, 0.5, 1.0! for a magnetic field change from 0 to 7.5 T. Thsymbols represent the experimental data and the full curves cfrom theoretical calculations.
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anomalous peaks are not connected to the phase transitiis the case of the higher peaks, which come from ferparamagnetic phase transition. The nature of anomalouswas investigated taking into account the CEF levels scheusing a LLW diagram. This diagram, displayed in the dowinset of Fig. 2, can be obtained considering only the eneeigenvalues of the CEF Hamiltonian vs theX-CEF parameterthat ranges fromX521 to X51, for a fixed value ofWscale. The following set of CEF levels appears:G8
3 ~quadru-plet!, G8
2 ~quadruplet!, G81 ~quadruplet!, G7 ~doublet!, andG6
~doublet!. The upper inset of Fig. 2 shows only theG83
ground state in presence of the molecular field that comfrom magnetization for concentrationz50.5 at T56 K ~inthis temperature the anomalous peak appears in2DTad vs T,for z50.5, see Fig. 1!. The value of the magnetization, responsible for the splitting of theG8
3 ground levels isM'6.16mB and is shown by the left arrow in Fig. 3 that givethe magnetization vs temperature forZ50.5 and Z50.3.Figure 2 shows the density of states~DOS! of theG8
3 level vsX-CEF parameter defined here as DOS54/(E42E1), where(E42E1) is the difference between the fourth and groumagnetic energy states. The DOS calculated forz50.5 pre-sents higher value compared with the one forz50.3. It wasexpected since the experimental data and theoretical pre
FIG. 3. Theoretical magnetization curves vs temperature(Dy12zErz)Al2 (z50.3,0.5).
FIG. 4. The influence of theX-CEF parameter on the behavioof the lower peak of the2DTad vs temperature curve forz50.3.
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tion show the higher anomalous peak in the2DTad vs Tcurves forz50.5 ~see Fig. 1!. In order to confirm our resultswe have performed a theoretical investigation in influencetheX parameter in the2DTad vs T curve forz50.3 ~see Fig.4!. As we change the value ofX parameter, the intensity othe second peak atT;6 K decreases. The highest intensiof the second peak occurs for the value ofX parameter forwhich the DOS of theG8
3 level presents maximum value.It can be observed, in Fig. 3, the CEF quenching in t
magnetization curves. Note that asz increases, the modulusof W-CEF parameter increases and so does the CEF queing. Also, the critical Curie temperature, calculated using o
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model, varies linearly withz-concentration and is in excelent agreement with the experimental results.
In general, the best materials to work as a magneticfrigerant in a magnetic refrigerator are those that supplymaximum amount of cooling over the widest temperatrange~tablelike characteristic!. In this way, the full theoreti-cal comprehension of the origin of the second peak2DTad vs T, that was experimentally observed, can havhigh impact on designing new magnetic materials to be uin magnetic refrigeration.
This study was supported by the following Braziliaagencies: CNPq, CAPES, FAPERJ and UERJ.
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*Corresponding author. Present address: Universidade do Esdo Rio de Janeiro, Instituto de Fı´sica, Rua Sa˜o Francisco Xavier,524, Rio de Janeiro 20550-013, Brazil. Email [email protected]
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