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Magnetocaloric effect in (TbcR1-c)Co2 (R=Er and Ho)
Journal of Alloys and Compounds 618 (2015) 386–389
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Magnetocaloric effect in ðTbc R1c ÞCo2 (R ¼ Er and Ho) N.A. de Oliveira a,⇑, P.J. von Ranke a, A. Troper b a b
Instituto de Física Armando Dias Tavares, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro 20550-013, RJ, Brazil Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil
a r t i c l e
i n f o
Article history: Received 7 June 2014 Received in revised form 17 August 2014 Accepted 26 August 2014 Available online 9 September 2014
a b s t r a c t In this paper we theoretically discuss the magnetocaloric effect in ðTbc R1c ÞCo2 (R = Er and Ho). To this end, we use a model Hamiltonian in which are included the 4f electrons from rare earth ions and the 3d electrons from Co ions. In the model is also included an extra term to account for the crystal electric field, which plays an important role in the magnetic and caloric properties of these compounds. Ó 2014 Elsevier B.V. All rights reserved.
Keywords: Rare earth alloys and compounds Disordered systems Thermodynamic modeling
1. Introduction The magnetocaloric effect [1–5] has been intensively studied in the last years. Apart from its technological application in magnetic refrigeration, the magnetocaloric effect is worth an investigation from the point of view of the fundamental physics. For instance, the magnetocaloric effect can be used to discuss the order of the phase transitions in magnetic compounds [6]. The intermetallic compounds RCo2 (R = Er, Ho and Dy) [7–9], which undergo first order phase transitions, exhibit large values of the magnetocaloric quantity DSiso . Moreover, it is well known that the magnetization and the magnetocaloric properties of these compounds can be changed by doping [10,11] or by applying external pressure [12,13]. In addition, it has been theoretically predicted [14] that these compounds exhibit large values of the correspondbar ing barocaloric quantities DSbar iso and DT ad and sizeable anisotropic magnetocaloric effect. In this context, it is very interesting to study the magnetocaloric effect in the series of compounds ðRa1c Rgc ÞCo2 . Although the magnetocaloric effect in some compounds belonging to this series has already been studied in the literature [15–20], many aspects of their magnetocaloric properties are not yet completely understood. In this work, we theoretically discuss the magnetocaloric effect in ðTbc R1c ÞCo2 where R stands for Er and Ho. For this purpose, we use a model Hamiltonian in which are included both the 4f electrons from rare earth ions and the 3d electrons from Co ions. In the model, the magnetoelastic coupling and the effect
⇑ Corresponding author. Tel.: +55 21 23340379. E-mail address: [email protected] (N.A. de Oliveira). http://dx.doi.org/10.1016/j.jallcom.2014.08.203 0925-8388/Ó 2014 Elsevier B.V. All rights reserved.
of the crystal electric field are considered as well. This work is important because it explains very reasonable the available experimental data [10] of DSiso for ðTbc Ho1c ÞCo2 and theoretically predicts the magnetocaloric effect in the compound ðTbc Er1c ÞCo2 . 2. Formulation In order to describe the magnetocaloric effect in the doped compounds ðTbc R1c ÞCo2 , where R = Ho and Er, we use the following Hamiltonian H ¼ H4f þ H3d where
2 X X f f 1 X ~f Ji ~ J l þ J1 ~ J fi ~ J fl B ~ J fi þ H4f ¼ J0~ g fi lB~ J J 2 i df i i i;l X ~ HCF sdi þ i
ð1Þ
i
describes a subsystem of localized 4f -magnetic moments associated with the rare earth ions and H3d given by:
H3d ¼
X
e0r dþir dir þ
ir
X X 1 X ~f d þ T ilr dir dlr þ U ni" ni# þ J J ~ s 2 i df i i ir ilr
X B ~ sdi g ei lB~
ð2Þ
i
describes a subsystem of 3d-itinerant electrons from Co ions. In the Hamiltonian H4f , the first term represents the effective exchange interaction between rare earth ions and the biquadratic term somehow describes the magnetoelastic interaction in the framework of momentum operators [21], where J0 ; J1 are model parameters. The third represents the Zeeman interaction and the fourth
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N.A. de Oliveira et al. / Journal of Alloys and Compounds 618 (2015) 386–389
describes the coupling with the 3d-itinerant electrons. The term HCF i accounts for the crystal electric field. In the Hamiltonian H3d , the first term represents an effective atomic energy, the second describes the hoping energy and the third the two body Coulomb interaction. The fourth and the fifth terms describe the coupling with the 4f magnetic moments and the Zeeman interaction respectively. In order to simplify the calculations we take some approximations. Here, we treat the two body interaction and the chemical disorder introduced in the rare earth sublattice, through the extended mean field approximation [22]. It should be mentioned that a more rigorous treatment of the disorder should be done in the framework of Monte Carlo calculation [23–26]. However, such a calculation is much more complex and is beyond the scope of the present work. In the formalism of this extended mean field approach, we can write the Hamiltonian H4f as H4f ¼ H4f ðaÞ þ H4f ðgÞ , where
H4f ðaÞ ¼
Xh
a 3 ag a g aa Jaa 0eff ðcÞhJ i þ J0eff ðcÞhJ i þ J1eff ðcÞ J
i
SðT; B; cÞ ¼ cSa ðT; B; cÞ þ ð1 cÞSg ðT; B; cÞ a
"
3 Z Ha T D Slat ðT; cÞ ¼ Ni 3R ln 1 e T þ 12R a a
HD
Xh
ð3Þ
represents contribution from the crystalline lattice, where R is the gas constant; HaD is the Debye temperature and N i is the total numof
atoms
per
" ðaÞ S4f mag ðT; B; cÞ ¼ R ln
gg ga ga gg have defined Jgg 0eff ðcÞ ¼ J0 Z gg ðcÞ, J0eff ðcÞ ¼ J0 Z ga ðcÞ; J1eff ðcÞ ¼ ga ga Jgg 1 Z gg ðcÞ and J1eff ðcÞ ¼ J1 Z ga ðcÞ. In the definitions of these effec-
tive parameters, the term Z ag ðcÞ represents the number of first next neighbors of g-ion surrounding a site occupied by an ion of a-type. Similar descriptions hold for Z aa ðcÞ; Z gg ðcÞ and Z ga ðcÞ. CFðaÞ In the cubic symmetry, Hi is given by [27]: CFðaÞ
ð5Þ
i
where W is an energy scale and x gives the relative importance of the fourth and sixth order terms. Onm are the Steven’s operators [28]; F 4 and F 6 are numerical factors common to all matrix elements [29]. a (i ¼ x; y; z) are calculated by The mean values Ji a P P a a a a bEj bEaj hJ i ¼ ½ hE J E ie = e , where Ea ; Ea are the energy i
j
j
i
j
aÞ S3dð mag ðT; B; cÞ ¼ N Co R
XZ l r
ag and Jag 1eff ðcÞ ¼ J1 Z ag ðcÞ. Similarly, in the expression for H4f ðgÞ we
Hi
4f ðaÞ Samag ðT; B; cÞ ¼ Smag ðT; B; cÞ
unit.
P a bEa # j X bEa 1 j Ej e e j þ P bEa kB T j j je
"
ð4Þ
describes a sublattice of rare earth ions of g-type. In the present work, where we are discussing the doped compounds ðTbc R1c ÞCo2 ; a and g stand for Tb and R ions respectively. In the expression for H4f ðaÞ we have defined the effective paramag ag aa aa aa eters, Jaa 0eff ðcÞ ¼ J0 Z aa ðcÞ; J 0eff ðcÞ ¼ J0 Z ag ðcÞ; J1eff ðcÞ ¼ J1 Z aa ðcÞ
3a 2 0 O4 þ 5O44 O06 21O46 5 ¼ W 4x þ ð 1 j xj Þ F4 F6
formula
ð8Þ
represents the contribution from the 4f -magnetic moments and 3dðaÞ Smag ðT; B; cÞ given by [30]:
i
1 CFðgÞ þJ1eff ðcÞhJ g ihJ a i2 þ g f ðgÞ lB B þ Jdf hs3d i J gi þ Hi 2
0
# x3 dx ; ex 1
3dðaÞ ðT; B; cÞ is the magnetic entropy [3], where: þSmag
ga gg g a g 3 Jgg 0eff ðcÞhJ i þ J0eff ðcÞhJ i þ J1eff ðcÞhJ i
ga
HaD =T
ð7Þ
describes a sublattice of rare earth ions of a-type and
H4f ðgÞ ¼
ð6Þ
g
where S ðT; B; cÞ and S ðT; B; cÞ represent the total entropy for the effective compounds TbCo2 and RCo2 respectively. This total entropy has the form Sa ðT; B; cÞ ¼ Samag ðT; B; cÞ þ Salat ðT; cÞ þ Sael ðTÞ where Sael ¼ cT is the contribution from the conduction electrons, with c being the Sommerfeld coefficient. The term Salat ðT; cÞ which is given by:
ber
1 aÞ a g 2 f ða Þ lB B þ Jdf hs3d i Jai þ HCFð þJag i 1eff ðcÞhJ ihJ i þ g 2
the magnetization of an effective compound RCo2 . Similarly, the total entropy in the compound ðTbc R1c ÞCo2 can be written as:
j
eigenvalues and eigenvectors of the mean field Hamiltonian H4f ðaÞ and b ¼ 1=kB T, with kB being the Boltzmann constant. Similar relations hold for the mean values J gi (i ¼ x; y; z). The Hamiltonian that describes the itinerant electrons is written in the mean field approximation [3] as an effective one-body ðaÞ
Hamiltonian with the renormalized energy eir ¼ e0 þ U hnr i þ0:25rJdf J a rlB B, where a ¼ a or g. The magnetization of the compound ðTbc R1c ÞCo2 per formula unit can be calculated by MðT; B; cÞ ¼ cM a ðT; B; cÞ where M a ðT; B; cÞ ¼ M 4f ðaÞ ðT; B; cÞ þ 2M 3dðaÞ þð1 cÞM g ðT; B; cÞ, ðT; B; cÞ represents the magnetization of an effective compound TbCo2 and M g ðT; B; cÞ ¼ M 4f ðgÞ ðT; B; cÞ þ 2M 3dðgÞ ðT; B; cÞ represents
1 X þ kB T r
1
Z l 1
aÞ ln 1 þ ebðelÞ q3dð ðeÞde r # 3dðaÞ
ðe lÞqr
ðeÞf ðeÞde
ð9Þ
is the contribution from the 3d-electrons, where N Co ¼ 2 represents the number of Co ions per formula unit and f ðeÞ is the Fermi distri3dðaÞ bution function. In this expression qr is a spin polarized density of states [30]. The magnetocaloric quantities DSiso and DT ad upon magnetic field variation DB ¼ B2 B1 are respectively calculated by DSiso ðT; DB; cÞ ¼ SðT; B2 ; cÞ SðT; B1 ; cÞ and by DT ad ðT; DB; cÞ ¼ T 2 T 1 under the adiabatic condition SðT 2 ; B2 ; cÞ ¼ SðT 1 ; B1 ; cÞ. 3. Results and discussion In order to calculate the magnetocaloric effect in ðTbc Ho1c ÞCo2 , we used the parameters J Tb ¼ 6; g Tb ¼ 1:5; J Ho ¼ 8; g Ho ¼ 1:25 taken from Hund rule. The bare values of the exchange interaction gg parameters (Jaa 0 ; J0 ) and the magnetoelastic coupling ones gg aa (J1 ; J1 ) (a ¼ Tb and g ¼ Ho) were adjusted to fit the experimental data of the pure compounds HoCo2 and TbCo2 . The effective gg ag parameters [Jaa 0eff ðcÞ; J0eff ðcÞ and J 0eff ðcÞ] for intermediate concentrations, gg
were gg
calculated ag
ag
aa Jaa 0eff ðcÞ ¼ J0 Z aa ðcÞ;
by ga
ga
J0eff ðcÞ ¼ J0 Z gg ðcÞ; J 0eff ðcÞ ¼ J0 Z ag ðcÞ and J 0eff ðcÞ ¼ J0 Z ga ðcÞ considering the following parameterization Z aa ðcÞ ¼ cZ aa ð0Þ; Z ag ðcÞ ¼ cð1 cÞZ aa ð0Þ; Z gg ðcÞ ¼ ð1 cÞZ gg ð0Þ and Z ga ðcÞ ¼ cð1 cÞZ gg ð0Þ, where Z aa ð0Þ ¼ Z gg ð0Þ ¼ 8. Such a parameterization has been independently proposed by Tang and Nolting [31] and by de Oliveira [22]. The effective magnetoelastic coupling parameters gg ag ga [Jaa 1eff ðcÞ; J1eff ðcÞ; J1eff ðcÞ and J1eff ðcÞ] were estimated in the very same way. The Debye temperature and the Sommerfeld coefficient Ho 2 were taken as HTb D ¼ HD ¼ 200 K and c ¼ 5 (mJ/mol K ), which are in the physical range of the values commonly used in the literature. In order to discuss the order of the phase transition in this doped compound, we self-consistently calculated the magnetization by increasing (open symbols plus dashed lines) and decreasing
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N.A. de Oliveira et al. / Journal of Alloys and Compounds 618 (2015) 386–389
(solid lines) temperature. In Fig. 1 we plot the magnetization in ðTbc Ho1c ÞCo2 for some values of Tb concentration. From this figure, we can observe that the magnetization curves for the pure compound HoCo2 (c ¼ 0) go abruptly to zero around the critical temperature (T C ). Besides, the values of the magnetization around T C , calculated in the heating process are different from the ones calculated in the cooling process. As a result there is a thermal hysteresis. These both features are clear indications that the phase transition in HoCo2 is of first order. As the magnetization curves for c ¼ 0:2, shown in Fig. 1, exhibit the same behavior as the ones for HoCo2 , we can conclude that the phase transition in ðTb0:2 Ho0:8 ÞCo2 is also of first order. On the other hand, for c ¼ 0:4 the magnetization curves calculated by increasing and decreasing temperature go smoothly to zero without thermal hysteresis. So, we can assure that phase transition in ðTb0:4 Ho0:6 ÞCo2 is of second order. Therefore, according to our calculations, the phase transition in ðTbc Ho1c ÞCo2 is of first order for c < 0:4 and second order for c P 0:4. This change of the order of the phase transition as a function of Tb concentration is due to the increase of the 3d energy bandwidth and the reduction of the magnetoelastic coupling. In Fig. 2, we present the isothermal entropy changes (DSiso ) for a magnetic field variation from 0 to 5 T. From this figure we can observe a good agreement between our calculations and the available experimental data [10]. For the corresponding adiabatic temperature changes (DT ad ), which are shown in Fig. 3, further experimental data are necessary to compare with our theoretical
Fig. 3. DT ad in ðTbc Ho1c ÞCo2 calculated upon magnetic field variation from 0 to 5 T.
Fig. 4. Magnetization in ðTbc Er1c ÞCo2 , calculated by increasing (symbols plus dashed lines) and decreasing (solid lines) temperature.
Fig. 1. Magnetization in ðTbc Ho1c ÞCo2 , calculated by increasing (symbols plus dashed lines) and decreasing (solid lines) temperature.
Fig. 5. DSiso upon magnetic field variation from 0 to 5 T in ðTbc Er1c ÞCo2 .
Fig. 2. DSiso upon magnetic field variation from 0 to 5 T in ðTbc Ho1c ÞCo2 . Solid lines represent our calculations and symbols are experimental data [10].
predictions. Notice that the values of the peaks of the magnetocaloric quantities decrease with increasing Tb concentration, due to the change of the magnetic phase transition from first to second order. In order to calculate the magnetocaloric effect in ðTbc Er1c ÞCo2 , we took J Tb ¼ 6; g Tb ¼ 1:5; J Er ¼ 7:5; g Er ¼ 1:2 and adjusted the gg exchange interaction parameters (Jaa 0 ; J0 ) and the magnetoelasgg tic coupling ones (Jaa ; J ) ( a ¼ Tb and g ¼ Er) to reproduce the 1 1 experimental data of the pure compounds ErCo2 and TbCo2 . For
N.A. de Oliveira et al. / Journal of Alloys and Compounds 618 (2015) 386–389
389
not only shifts the magnetic ordering temperature but also changes the type of the phase transition from first to second order. This observation is in very good agreement with the available experimental data for ðTbc Ho1c ÞCo2 [10]. Our theoretical calculations for ðTbc Er1c ÞCo2 need experimental data to be confirmed. It is also worth mentioning that it would be very interesting to grow single crystals in order to better investigate the effect of anisotropy on the caloric properties of these compounds. Acknowledgment This work has been done under the auspices of the Brazilian agencies CNPq and FAPERJ. References Fig. 6. DT ad in ðTbc Er1c ÞCo2 calculated upon magnetic field variation from 0 to 5 T.
intermediate concentrations, the effective parameters gg ag [Jaa 0eff ðcÞ; J0eff ðcÞ and J 0eff ðcÞ] were estimated as described in the previous case. We plot in Fig. 4 the magnetization in ðTbc Er1c ÞCo2 calculated by increasing (open symbols plus dashed lines) and decreasing (solid lines) temperature. From this figure, we can observe that the magnetization curves for the pure compound ErCo2 and for ðTb0:2 Er0:8 ÞCo2 go abruptly to zero around the critical temperature T C . Besides that, there are also thermal hysteresis, characterizing first order phase transitions. On the contrary, the magnetization curves for c ¼ 0:4 and c ¼ 1:0 go smoothly to zero without thermal hysteresis. Therefore, the phase transitions in ðTb0:4 Er0:6 ÞCo2 and TbCo2 are of second order. As a result from this analysis, we can say that the phase transition in ðTbc Er1c ÞCo2 is of first order for c < 0:4 and second order for c P 0:4. In Figs. 5 and 6 we respectively plot the isothermal entropy changes and the adiabatic temperature changes calculated for DB ¼ 5 T. From these figures, we can observe that the maximum values of the magnetocaloric quantities in ðTbc Er1c ÞCo2 , decrease with increasing Tb concentration. This is due to the change in the type of the phase transition from first to second order. In conclusion, we have calculated the magnetocaloric effect in the doped compounds ðTbc Ho1c ÞCo2 and ðTbc Er1c ÞCo2 . Our theoretical calculations show that the increase of Tb concentration
[1] A.M. Tishin, Y.I. Spichkin, The Magnetocaloric Effect and its Applications, first ed., Institute of Physics, Bristol and Philadelphia, 2003. [2] K.A. Gschneidner Jr., V.K. Pecharsky, A.O. Tsokol, Rep. Prog. Phys. 68 (2005) 1479–1539. [3] N.A. de Oliveira, P.J. von Ranke, Phys. Rep. 489 (2010) 89–159. [4] N.A. de Oliveira, P.J. von Ranke, A. Troper, Int. J. Refrig. 37 (2014) 237–248. [5] S.B. Roy, Handb. Magn. Mater. 22 (2014) 203–316. [6] C.M. Bonilla et al., Phys. Rev. B 81 (2010) 224424. [7] H. Wada et al., J. Magn. Magn. Mater. 196/197 (1999) 689–690. [8] H. Wada et al., J. Alloys Comp. 316 (2001) 245–249. [9] J. Herero-Albillos et al., Phys. Rev. B 73 (2006) 134410. [10] M. Balli, D. Fruchart, D. Gignoux, J. Magn. Magn. Mater. 314 (2007) 16–20. [11] J. Cwick, T. Palewski, K. Nenkov, O. Gutfleisch, J. Klamut, Intermetallics 19 (2011) 1656–1660. [12] O. Syschenko et al., J. Magn. Magn. Mater. 226–230 (2001) 1062–1067. [13] N.A. de Oliveira, J. Phys.: Condens. Matter 20 (2008) 175209. [14] N.A. de Oliveira, Appl. Phys. Lett. 90 (2007) 052501. [15] N.A. de Oliveira, P.J. von Ranke, A. Troper, Phys. Rev. B 66 (2002) 094402. [16] N.A. de Oliveira, P.J. von Ranke, J. Magn. Magn. Mater. 264 (2003) 55–61. [17] N.A. de Oliveira, Eur. Phys. J.B. 65 (2008) 207–212. [18] Y. Zhu et al., Cryogenics 51 (2011) 494–498. [19] M. Balli, D. Fruchart, D. Gignoux, J. Alloys Comp. 509 (2011) 3907–3912. [20] J. Cwick, J. Alloys Comp. 580 (2013) 341–347. [21] Th. Strässle et al., Phys. Rev. B 92 (2004) 257202. [22] N.A. de Oliveira, J. Alloys Comp. 455 (2008) 81–86. [23] E.P. Nóbrega et al., Phys. Rev. B 72 (2005) 134426. [24] E.P. Nóbrega et al., Phys. Rev. B 74 (2006) 144429. [25] E.P. Nóbrega et al., J. Magn. Magn. Mater. 310 (2007) 2805–2807. [26] V.D. Buchelnikov et al., Phys. Rev. B 78 (2008) 184427. [27] K.R. Lea, M.J.M. Leask, W.P. Wolf, J. Phys. Chem. Sol. 23 (1962) 1381–1405. [28] K.W.H. Stevens, Proc. Phys. Soc., London, Sect. A 65 (1952) 209–215. [29] M.T. Hutchings, Sol. Stat. Phys. 16 (1964) 227. [30] N.A. de Oliveira, Eur. Phys. J.B. 40 (2004) 259–264. [31] G.X. Tang, W. Nolting, Phys. Rev. B 73 (2006) 024415.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Magnetocaloric effect in ðTbc R1c ÞCo2 (R ¼ Er and Ho) N.A. de Oliveira a,⇑, P.J. von Ranke a, A. Troper b a b
Instituto de Física Armando Dias Tavares, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Rio de Janeiro 20550-013, RJ, Brazil Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil
a r t i c l e
i n f o
Article history: Received 7 June 2014 Received in revised form 17 August 2014 Accepted 26 August 2014 Available online 9 September 2014
a b s t r a c t In this paper we theoretically discuss the magnetocaloric effect in ðTbc R1c ÞCo2 (R = Er and Ho). To this end, we use a model Hamiltonian in which are included the 4f electrons from rare earth ions and the 3d electrons from Co ions. In the model is also included an extra term to account for the crystal electric field, which plays an important role in the magnetic and caloric properties of these compounds. Ó 2014 Elsevier B.V. All rights reserved.
Keywords: Rare earth alloys and compounds Disordered systems Thermodynamic modeling
1. Introduction The magnetocaloric effect [1–5] has been intensively studied in the last years. Apart from its technological application in magnetic refrigeration, the magnetocaloric effect is worth an investigation from the point of view of the fundamental physics. For instance, the magnetocaloric effect can be used to discuss the order of the phase transitions in magnetic compounds [6]. The intermetallic compounds RCo2 (R = Er, Ho and Dy) [7–9], which undergo first order phase transitions, exhibit large values of the magnetocaloric quantity DSiso . Moreover, it is well known that the magnetization and the magnetocaloric properties of these compounds can be changed by doping [10,11] or by applying external pressure [12,13]. In addition, it has been theoretically predicted [14] that these compounds exhibit large values of the correspondbar ing barocaloric quantities DSbar iso and DT ad and sizeable anisotropic magnetocaloric effect. In this context, it is very interesting to study the magnetocaloric effect in the series of compounds ðRa1c Rgc ÞCo2 . Although the magnetocaloric effect in some compounds belonging to this series has already been studied in the literature [15–20], many aspects of their magnetocaloric properties are not yet completely understood. In this work, we theoretically discuss the magnetocaloric effect in ðTbc R1c ÞCo2 where R stands for Er and Ho. For this purpose, we use a model Hamiltonian in which are included both the 4f electrons from rare earth ions and the 3d electrons from Co ions. In the model, the magnetoelastic coupling and the effect
⇑ Corresponding author. Tel.: +55 21 23340379. E-mail address: [email protected] (N.A. de Oliveira). http://dx.doi.org/10.1016/j.jallcom.2014.08.203 0925-8388/Ó 2014 Elsevier B.V. All rights reserved.
of the crystal electric field are considered as well. This work is important because it explains very reasonable the available experimental data [10] of DSiso for ðTbc Ho1c ÞCo2 and theoretically predicts the magnetocaloric effect in the compound ðTbc Er1c ÞCo2 . 2. Formulation In order to describe the magnetocaloric effect in the doped compounds ðTbc R1c ÞCo2 , where R = Ho and Er, we use the following Hamiltonian H ¼ H4f þ H3d where
2 X X f f 1 X ~f Ji ~ J l þ J1 ~ J fi ~ J fl B ~ J fi þ H4f ¼ J0~ g fi lB~ J J 2 i df i i i;l X ~ HCF sdi þ i
ð1Þ
i
describes a subsystem of localized 4f -magnetic moments associated with the rare earth ions and H3d given by:
H3d ¼
X
e0r dþir dir þ
ir
X X 1 X ~f d þ T ilr dir dlr þ U ni" ni# þ J J ~ s 2 i df i i ir ilr
X B ~ sdi g ei lB~
ð2Þ
i
describes a subsystem of 3d-itinerant electrons from Co ions. In the Hamiltonian H4f , the first term represents the effective exchange interaction between rare earth ions and the biquadratic term somehow describes the magnetoelastic interaction in the framework of momentum operators [21], where J0 ; J1 are model parameters. The third represents the Zeeman interaction and the fourth
387
N.A. de Oliveira et al. / Journal of Alloys and Compounds 618 (2015) 386–389
describes the coupling with the 3d-itinerant electrons. The term HCF i accounts for the crystal electric field. In the Hamiltonian H3d , the first term represents an effective atomic energy, the second describes the hoping energy and the third the two body Coulomb interaction. The fourth and the fifth terms describe the coupling with the 4f magnetic moments and the Zeeman interaction respectively. In order to simplify the calculations we take some approximations. Here, we treat the two body interaction and the chemical disorder introduced in the rare earth sublattice, through the extended mean field approximation [22]. It should be mentioned that a more rigorous treatment of the disorder should be done in the framework of Monte Carlo calculation [23–26]. However, such a calculation is much more complex and is beyond the scope of the present work. In the formalism of this extended mean field approach, we can write the Hamiltonian H4f as H4f ¼ H4f ðaÞ þ H4f ðgÞ , where
H4f ðaÞ ¼
Xh
a 3 ag a g aa Jaa 0eff ðcÞhJ i þ J0eff ðcÞhJ i þ J1eff ðcÞ J
i
SðT; B; cÞ ¼ cSa ðT; B; cÞ þ ð1 cÞSg ðT; B; cÞ a
"
3 Z Ha T D Slat ðT; cÞ ¼ Ni 3R ln 1 e T þ 12R a a
HD
Xh
ð3Þ
represents contribution from the crystalline lattice, where R is the gas constant; HaD is the Debye temperature and N i is the total numof
atoms
per
" ðaÞ S4f mag ðT; B; cÞ ¼ R ln
gg ga ga gg have defined Jgg 0eff ðcÞ ¼ J0 Z gg ðcÞ, J0eff ðcÞ ¼ J0 Z ga ðcÞ; J1eff ðcÞ ¼ ga ga Jgg 1 Z gg ðcÞ and J1eff ðcÞ ¼ J1 Z ga ðcÞ. In the definitions of these effec-
tive parameters, the term Z ag ðcÞ represents the number of first next neighbors of g-ion surrounding a site occupied by an ion of a-type. Similar descriptions hold for Z aa ðcÞ; Z gg ðcÞ and Z ga ðcÞ. CFðaÞ In the cubic symmetry, Hi is given by [27]: CFðaÞ
ð5Þ
i
where W is an energy scale and x gives the relative importance of the fourth and sixth order terms. Onm are the Steven’s operators [28]; F 4 and F 6 are numerical factors common to all matrix elements [29]. a (i ¼ x; y; z) are calculated by The mean values Ji a P P a a a a bEj bEaj hJ i ¼ ½ hE J E ie = e , where Ea ; Ea are the energy i
j
j
i
j
aÞ S3dð mag ðT; B; cÞ ¼ N Co R
XZ l r
ag and Jag 1eff ðcÞ ¼ J1 Z ag ðcÞ. Similarly, in the expression for H4f ðgÞ we
Hi
4f ðaÞ Samag ðT; B; cÞ ¼ Smag ðT; B; cÞ
unit.
P a bEa # j X bEa 1 j Ej e e j þ P bEa kB T j j je
"
ð4Þ
describes a sublattice of rare earth ions of g-type. In the present work, where we are discussing the doped compounds ðTbc R1c ÞCo2 ; a and g stand for Tb and R ions respectively. In the expression for H4f ðaÞ we have defined the effective paramag ag aa aa aa eters, Jaa 0eff ðcÞ ¼ J0 Z aa ðcÞ; J 0eff ðcÞ ¼ J0 Z ag ðcÞ; J1eff ðcÞ ¼ J1 Z aa ðcÞ
3a 2 0 O4 þ 5O44 O06 21O46 5 ¼ W 4x þ ð 1 j xj Þ F4 F6
formula
ð8Þ
represents the contribution from the 4f -magnetic moments and 3dðaÞ Smag ðT; B; cÞ given by [30]:
i
1 CFðgÞ þJ1eff ðcÞhJ g ihJ a i2 þ g f ðgÞ lB B þ Jdf hs3d i J gi þ Hi 2
0
# x3 dx ; ex 1
3dðaÞ ðT; B; cÞ is the magnetic entropy [3], where: þSmag
ga gg g a g 3 Jgg 0eff ðcÞhJ i þ J0eff ðcÞhJ i þ J1eff ðcÞhJ i
ga
HaD =T
ð7Þ
describes a sublattice of rare earth ions of a-type and
H4f ðgÞ ¼
ð6Þ
g
where S ðT; B; cÞ and S ðT; B; cÞ represent the total entropy for the effective compounds TbCo2 and RCo2 respectively. This total entropy has the form Sa ðT; B; cÞ ¼ Samag ðT; B; cÞ þ Salat ðT; cÞ þ Sael ðTÞ where Sael ¼ cT is the contribution from the conduction electrons, with c being the Sommerfeld coefficient. The term Salat ðT; cÞ which is given by:
ber
1 aÞ a g 2 f ða Þ lB B þ Jdf hs3d i Jai þ HCFð þJag i 1eff ðcÞhJ ihJ i þ g 2
the magnetization of an effective compound RCo2 . Similarly, the total entropy in the compound ðTbc R1c ÞCo2 can be written as:
j
eigenvalues and eigenvectors of the mean field Hamiltonian H4f ðaÞ and b ¼ 1=kB T, with kB being the Boltzmann constant. Similar relations hold for the mean values J gi (i ¼ x; y; z). The Hamiltonian that describes the itinerant electrons is written in the mean field approximation [3] as an effective one-body ðaÞ
Hamiltonian with the renormalized energy eir ¼ e0 þ U hnr i þ0:25rJdf J a rlB B, where a ¼ a or g. The magnetization of the compound ðTbc R1c ÞCo2 per formula unit can be calculated by MðT; B; cÞ ¼ cM a ðT; B; cÞ where M a ðT; B; cÞ ¼ M 4f ðaÞ ðT; B; cÞ þ 2M 3dðaÞ þð1 cÞM g ðT; B; cÞ, ðT; B; cÞ represents the magnetization of an effective compound TbCo2 and M g ðT; B; cÞ ¼ M 4f ðgÞ ðT; B; cÞ þ 2M 3dðgÞ ðT; B; cÞ represents
1 X þ kB T r
1
Z l 1
aÞ ln 1 þ ebðelÞ q3dð ðeÞde r # 3dðaÞ
ðe lÞqr
ðeÞf ðeÞde
ð9Þ
is the contribution from the 3d-electrons, where N Co ¼ 2 represents the number of Co ions per formula unit and f ðeÞ is the Fermi distri3dðaÞ bution function. In this expression qr is a spin polarized density of states [30]. The magnetocaloric quantities DSiso and DT ad upon magnetic field variation DB ¼ B2 B1 are respectively calculated by DSiso ðT; DB; cÞ ¼ SðT; B2 ; cÞ SðT; B1 ; cÞ and by DT ad ðT; DB; cÞ ¼ T 2 T 1 under the adiabatic condition SðT 2 ; B2 ; cÞ ¼ SðT 1 ; B1 ; cÞ. 3. Results and discussion In order to calculate the magnetocaloric effect in ðTbc Ho1c ÞCo2 , we used the parameters J Tb ¼ 6; g Tb ¼ 1:5; J Ho ¼ 8; g Ho ¼ 1:25 taken from Hund rule. The bare values of the exchange interaction gg parameters (Jaa 0 ; J0 ) and the magnetoelastic coupling ones gg aa (J1 ; J1 ) (a ¼ Tb and g ¼ Ho) were adjusted to fit the experimental data of the pure compounds HoCo2 and TbCo2 . The effective gg ag parameters [Jaa 0eff ðcÞ; J0eff ðcÞ and J 0eff ðcÞ] for intermediate concentrations, gg
were gg
calculated ag
ag
aa Jaa 0eff ðcÞ ¼ J0 Z aa ðcÞ;
by ga
ga
J0eff ðcÞ ¼ J0 Z gg ðcÞ; J 0eff ðcÞ ¼ J0 Z ag ðcÞ and J 0eff ðcÞ ¼ J0 Z ga ðcÞ considering the following parameterization Z aa ðcÞ ¼ cZ aa ð0Þ; Z ag ðcÞ ¼ cð1 cÞZ aa ð0Þ; Z gg ðcÞ ¼ ð1 cÞZ gg ð0Þ and Z ga ðcÞ ¼ cð1 cÞZ gg ð0Þ, where Z aa ð0Þ ¼ Z gg ð0Þ ¼ 8. Such a parameterization has been independently proposed by Tang and Nolting [31] and by de Oliveira [22]. The effective magnetoelastic coupling parameters gg ag ga [Jaa 1eff ðcÞ; J1eff ðcÞ; J1eff ðcÞ and J1eff ðcÞ] were estimated in the very same way. The Debye temperature and the Sommerfeld coefficient Ho 2 were taken as HTb D ¼ HD ¼ 200 K and c ¼ 5 (mJ/mol K ), which are in the physical range of the values commonly used in the literature. In order to discuss the order of the phase transition in this doped compound, we self-consistently calculated the magnetization by increasing (open symbols plus dashed lines) and decreasing
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(solid lines) temperature. In Fig. 1 we plot the magnetization in ðTbc Ho1c ÞCo2 for some values of Tb concentration. From this figure, we can observe that the magnetization curves for the pure compound HoCo2 (c ¼ 0) go abruptly to zero around the critical temperature (T C ). Besides, the values of the magnetization around T C , calculated in the heating process are different from the ones calculated in the cooling process. As a result there is a thermal hysteresis. These both features are clear indications that the phase transition in HoCo2 is of first order. As the magnetization curves for c ¼ 0:2, shown in Fig. 1, exhibit the same behavior as the ones for HoCo2 , we can conclude that the phase transition in ðTb0:2 Ho0:8 ÞCo2 is also of first order. On the other hand, for c ¼ 0:4 the magnetization curves calculated by increasing and decreasing temperature go smoothly to zero without thermal hysteresis. So, we can assure that phase transition in ðTb0:4 Ho0:6 ÞCo2 is of second order. Therefore, according to our calculations, the phase transition in ðTbc Ho1c ÞCo2 is of first order for c < 0:4 and second order for c P 0:4. This change of the order of the phase transition as a function of Tb concentration is due to the increase of the 3d energy bandwidth and the reduction of the magnetoelastic coupling. In Fig. 2, we present the isothermal entropy changes (DSiso ) for a magnetic field variation from 0 to 5 T. From this figure we can observe a good agreement between our calculations and the available experimental data [10]. For the corresponding adiabatic temperature changes (DT ad ), which are shown in Fig. 3, further experimental data are necessary to compare with our theoretical
Fig. 3. DT ad in ðTbc Ho1c ÞCo2 calculated upon magnetic field variation from 0 to 5 T.
Fig. 4. Magnetization in ðTbc Er1c ÞCo2 , calculated by increasing (symbols plus dashed lines) and decreasing (solid lines) temperature.
Fig. 1. Magnetization in ðTbc Ho1c ÞCo2 , calculated by increasing (symbols plus dashed lines) and decreasing (solid lines) temperature.
Fig. 5. DSiso upon magnetic field variation from 0 to 5 T in ðTbc Er1c ÞCo2 .
Fig. 2. DSiso upon magnetic field variation from 0 to 5 T in ðTbc Ho1c ÞCo2 . Solid lines represent our calculations and symbols are experimental data [10].
predictions. Notice that the values of the peaks of the magnetocaloric quantities decrease with increasing Tb concentration, due to the change of the magnetic phase transition from first to second order. In order to calculate the magnetocaloric effect in ðTbc Er1c ÞCo2 , we took J Tb ¼ 6; g Tb ¼ 1:5; J Er ¼ 7:5; g Er ¼ 1:2 and adjusted the gg exchange interaction parameters (Jaa 0 ; J0 ) and the magnetoelasgg tic coupling ones (Jaa ; J ) ( a ¼ Tb and g ¼ Er) to reproduce the 1 1 experimental data of the pure compounds ErCo2 and TbCo2 . For
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389
not only shifts the magnetic ordering temperature but also changes the type of the phase transition from first to second order. This observation is in very good agreement with the available experimental data for ðTbc Ho1c ÞCo2 [10]. Our theoretical calculations for ðTbc Er1c ÞCo2 need experimental data to be confirmed. It is also worth mentioning that it would be very interesting to grow single crystals in order to better investigate the effect of anisotropy on the caloric properties of these compounds. Acknowledgment This work has been done under the auspices of the Brazilian agencies CNPq and FAPERJ. References Fig. 6. DT ad in ðTbc Er1c ÞCo2 calculated upon magnetic field variation from 0 to 5 T.
intermediate concentrations, the effective parameters gg ag [Jaa 0eff ðcÞ; J0eff ðcÞ and J 0eff ðcÞ] were estimated as described in the previous case. We plot in Fig. 4 the magnetization in ðTbc Er1c ÞCo2 calculated by increasing (open symbols plus dashed lines) and decreasing (solid lines) temperature. From this figure, we can observe that the magnetization curves for the pure compound ErCo2 and for ðTb0:2 Er0:8 ÞCo2 go abruptly to zero around the critical temperature T C . Besides that, there are also thermal hysteresis, characterizing first order phase transitions. On the contrary, the magnetization curves for c ¼ 0:4 and c ¼ 1:0 go smoothly to zero without thermal hysteresis. Therefore, the phase transitions in ðTb0:4 Er0:6 ÞCo2 and TbCo2 are of second order. As a result from this analysis, we can say that the phase transition in ðTbc Er1c ÞCo2 is of first order for c < 0:4 and second order for c P 0:4. In Figs. 5 and 6 we respectively plot the isothermal entropy changes and the adiabatic temperature changes calculated for DB ¼ 5 T. From these figures, we can observe that the maximum values of the magnetocaloric quantities in ðTbc Er1c ÞCo2 , decrease with increasing Tb concentration. This is due to the change in the type of the phase transition from first to second order. In conclusion, we have calculated the magnetocaloric effect in the doped compounds ðTbc Ho1c ÞCo2 and ðTbc Er1c ÞCo2 . Our theoretical calculations show that the increase of Tb concentration
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