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3MnO3 single crystal
Author’s Accepted Manuscript Monte Carlo simulation of the magnetocaloric effect in La2/3Ca1/3MnO3 single crystal R. Zouari, A. Chehaidar
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S0304-8853(16)30937-4 http://dx.doi.org/10.1016/j.jmmm.2016.05.104 MAGMA61518
To appear in: Journal of Magnetism and Magnetic Materials Received date: 11 December 2015 Revised date: 23 May 2016 Accepted date: 31 May 2016 Cite this article as: R. Zouari and A. Chehaidar, Monte Carlo simulation of the magnetocaloric effect in La2/3Ca1/3MnO3 single crystal, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.05.104 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Monte Carlo simulation of the magnetocaloric effect in La2/3Ca1/3MnO3 single crystal R. Zouari, A. Chehaidar
Laboratory of Physics-Mathematics and Applications, University of Sfax, Faculty of Sciences, Department of Physics, P.O. Box 1171, 3000 Sfax, Tunisia.
*Corresponding author: [email protected] Abstract The present work is devoted to a theoretical simulation study of the magnetocaloric effect in magnetically homogeneous
single crystal. Using the standard Monte Carlo-
Metropolis algorithm and the classical Heisenberg model Hamiltonian, we have computed the two main magnetocaloric properties such as the isothermal entropy change and the adiabatic temperature change upon an abrupt variation of the intensity of the applied magnetic field, as function of temperature. A good qualitative agreement is observed between our simulation and experiment. We have shown that the maximum entropy change increases by increasing the intensity of the applied magnetic field. In addition, it occurs at the ferromagneticparamagnetic transition temperature regardless of the intensity of the applied magnetic field. Our simulation shows, moreover, that the adiabatic temperature change behaves as the isothermal entropy change with respect to the material temperature and the applied magnetic field variation. Quantitatively, however, the experimental data deviate more or less, depending on powder preparation conditions, from our simulation data. This demonstrates the deviation of the prepared powders with respect to an ideal magnetic structure, as expected experimentally. Our simulation expects a maximum isothermal entropy change of and a maximum adiabatic temperature change of
under a magnetic field variation
of 5T. On approaching room temperature, the magnitude of the magnetocaloric effect in single crystal decreases but remains significant under a magnetic field variation of at least 2 Tesla.
Keywords: Manganites, Magnetocaloric effect, Monte Carlo simulation, modelling.
1
1. Introduction The magnetocaloric effect (MCE) is recognized as the warming or cooling of magnetic material subjected to an abrupt variation of the intensity of applied magnetostatic field. It was first discovered in 1881 by Warburg on iron, and first quantified experimentally in 1917 by Weiss and Piccard by measuring a sizable and reversible temperature change in nickel near its Curie temperature (See Ref. [1] for history). Since then, the search of magnetocaloric effect materials never stopped. In particular, solid materials for magnetic refrigeration have aroused particular interest to replace conventional gas refrigerants, and this fits into the framework of environmental preservation, on one hand, and maximization of the performance /cost ratio, on the other hand. In selecting magnetic refrigerants, particular attention was paid to the solid materials which exhibit large magnetic entropy change under a magnetic field, especially to those that can be used at room temperature or in its vicinity. Several compounds that meet these requirements have been proposed [2]; in particular, calcium doped lanthanum-manganite of chemical formula. The latter is found to be characterized further by high chemical stability and a high electrical resistivity that promotes a low warming of the material by the eddy currents [3-6]. The measurements of the quantities characterizing the magnetocaloric effect in the bulk
, namely the maximum isothermal entropy
change, the corresponding adiabatic temperature change and the relative cooling power, are performed on polycrystalline powders. Different values were, however, reported for the same magnetocaloric parameters. This discrepancy was interpreted as being due to differences in the sample processing routes and/or to the differently chemical composition [2]. So, we can ask what values can take the magnetocaloric parameters of a perfect
bulk
material. The answer to this question can be provided by the simulation technique. This is the purpose of our present work.
2
2. Computation Method A single crystal of
, free of defects, is considered here with a simple
cubic perovskite structure [7, 8]. Thus, a unit cell of this crystal is a cube containing one ion of Ca or La in its center, one Mn ion in each corner, and an O ion in the middle of each edge. As the La and Ca ions do not have the same ionization, the Mn ions are engaged in such a structure in two valances:
and
. The electronic structure of
make them non-magnetic ions, while and
and
,
and
are paramagnetic ions with spins
, respectively. The electrostatic interaction between the Mn ion and its
ligands leads to a redistribution of its valence electrons between two separated energy levels denoted by
and
in accordance with octahedral site symmetry. Regarding
ion, its
four valence electrons fall into three electrons on the triply-degenerate level energy) and one electron on the doubly-degenerate level ion, its three valence electrons occupy the the normal vibrational modes of the
(lower
(higher energy). For the
level. The coupling between the
-orbitals and
octahedron leads to the static distortion of the
oxygen’s cage, an effect called Jahn-Teller distortion [9]. This distortion manifests itself in a tetragonal elongation of the octahedron, leading to a lowering of the site symmetry and, consequently, the lifting of the degeneracy of the
level; the latter splits into two non-
degenerate and close sublevels denoted, hereinafter,
and
[10]. In a crystal, the
cooperative Jahn-Teller distortions favor the appearance of orbital ordering [11-15], which consists in a regular alternation of the occupied
and
orbitals with lobes pointed toward
the neighboring oxygen ions. In addition to orbital order, mixed-valence manganites are found to exhibit charge as well as spin ordering [16-20]. In those compounds in which the proportions of
and
ions are rational fractions, spin-, charge- and orbital-
ordering effects are particularly pronounced [21]. The single crystal
is
ordered ferromagnetically at low temperature with spins-arrangement of B-type according to the notation of Wallan and Koehler [16]; the charge- and orbital-ordering in this compound, established previously by Hotta et al. [21] and more recently by Restrepo-Para et al. [22], are illustrated schematically in Fig. 1 showing a cluster of
manganese ions whose
periodic repetition in the three directions of space reproduces the bulk crystal. Due to computation limitation, the crystal model is supposed to be contained in a large supercell with 3D periodic boundary conditions. In our present study, a cubic supercell of
3
ions
was
thus
generated
to
crystalline
simulate
the
magnetocaloric
properties
of
the
bulk
.
After generating the structural model, we turn to the description of the Hamiltonian of the system. Our present calculations are based on the classical Heisenberg model, according to which the Hamiltonian of interacting spins in the presence of an applied magnetic field is written as follows [7]: ⃗
∑
⃗
∑ (⃗
∑ ⃗
⃗)
⃗⃗
(1)
The first term in this expression reflects the exchange interaction between manganese ions; the sum runs over all the sites i in the supercell and their nearest neighbor sites j. The second term gives the core cubic magnetocrystalline anisotropy, where
is the single ion bulk
anisotropy constant and the unit vector ⃗ indicates the easy axis direction. The last term accounts for the Zeeman interaction of spin magnetic dipole with a uniform applied magnetic induction, where
and
stand for the electron g-factor and the Bohr magneton,
respectively. As can be inferred from Fig. 1, three exchange integrals are involved in the development of exchange term in Eq. (1), namely (Mn4+ - Mn3+(
) and
(Mn4+ - Mn3+(
(Mn3+(
- Mn3+ (
),
). These integrals along with the
magnetocrystalline constant are treated as empirical parameters to be determined from experimental data. Now we come to the step of computing the magnetic and the magnetocaloric properties of our structural model for the crystal magnetization
. Specifically, we are interested in the
, the specific heat
, the isothermal entropy change
and the adiabatic temperature change
,
upon magnetic field variation. The
equilibrium spin configurations accessed by the magnetic system at a given temperature T are obtained by Monte Carlo simulations based on the standard single spin Metropolis algorithm [23]. In each MC step per spin, a site is randomly selected for a random spin reorientation. Typically,
MC steps per spin are considered for thermalization; after that,
MC
steps per spin are performed to evaluate the average of the desired thermodynamical quantity. If we denote by 〈 〉 the average of a physical quantity X, then the internal energy and the magnitude of the magnetization 〈 〉
of the system of spins are given by:
(2)
and 4
〈‖ ⃗‖〉
〈‖ ∑
⃗ ‖〉
(3)
where N is the number of magnetic sites in the supercell. The magnetic specific heat is estimated from the fluctuations of the internal energy according to the following relation: [〈 where
〉
〈 〉 ]
(4)
is the Boltzmann constant. The magnetic entropy
as a function of
temperature and applied magnetic field, can be computed by integration of the internal energy function starting at infinite temperature according to the following relation [24]: (
∫ Nevertheless, the term
)
(5)
in this equation is unknown; for a paramagnetic phase,
however, we expect a substantially magnetic field independent function. It worthwhile to note here that this assumption is supported by the simulation of the entropy function for thin films reported recently by Restrepo-Parra et al. [10]; these authors have computed the entropy directly from the partition function of the system using the random walk in the energy space method of Landau and Wang [25]. Under this assumption, the isothermal entropy change
, the measurable quantity that
interests us, in fact, can be computed from Eq. (5) by omitting this term. The adiabatic temperature change
, another interesting measurable thermodynamic quantity, is
calculated according to the following formula [26]: (6) where
is the sum of the magnetic specific heat and the
crystalline lattice specific heat expressed by [26,27]: { ( ) ∫ Here ,
and
⁄
( )
⁄
}
(7)
stand for the universal gas constant, the number of ions per formula unit
and the Debye temperature, respectively.
5
3. Results and discussion The starting point of our calculations is the determination of the exchange parameters and the magnetocrystalline anisotropy constant involved in the expression of the Hamiltonian of the system given by Eq. (1). This task has been addressed previously by Restrepo-Parra et al. [17] who simulated the phase diagram of the manganites 1. For
with x ranging from 0 to
, they have reported the numerical values ,
, and
, . The magnetocrystalline
anisotropy constant, in fact, was not treated as a free parameter; rather, it was fixed at a value derived experimentally by Christides et al. [28]. However, this value was postponed for strained epitaxial
thin films, and it is obviously much greater than that of
the bulk, as already mentioned by these authors too. Furthermore, in their same simulation study, Restrepo-Parra et al. [17] postponed to the parent compound LaMnO3 an anisotropy constant twice that obtained for
crystal, i.e.
; this value is,
unfortunately, incomparable to the experimental value of al. [29], neither to the theoretical value of et al. [30] for the parent compound
reported by Moussa et
obtained by ab initio calculation by Naji . We then readjusted the exchange parameter
values provided by Restrepo-Parra and coworkers for and to the value
compounds with
setting the basic anisotropy constant (denoted by
in Ref. [17])
i.e. half the experimental value derived experimentally by Moussa
and coworkers for
crystal. A fairly good agreement between theoretical and
experimental magnetic transition temperatures is obtained for the proposed compounds, respecting the same proportions between the exchange parameters of the different stoichiometries, established by Restrepo-Parra et al. [17]. For compound, in particular, we have obtained , and temperature
,
,
, leading to a theoretical Curie
slightly higher than the experimental value
[31].
The evolution of the macroscopic magnetization as a function of temperature for our crystal model subjected to the action of a uniform magnetic field has been simulated by cooling from
in steps of
. The results for
,
,
,
and
are displayed as curves in Fig. 2. The expected paramagnetic-ferromagnetic transition in crystal is clearly visible on these magnetization curves. It is steep in the absence of an applied magnetic field, and manifests itself at 6
; the Curie
⁄
temperature is determined here at the greatest slope in
. It becomes less and less steep
by increasing the intensity of the applied magnetic field, thus making account experimental observations [31]. The transition temperature is found to slightly upward shift by increasing the magnetic field intensity up to 2T, whereas it becomes undefined beyond. The dependence of the magnetic entropy with the applied magnetic field and the temperature,
, was simulated according to the Eq. (5) in which the integral is started at
with
; rather it is the difference between the entropy at temperature T
and that at infinite temperature,
, that is computed numerically. The results
are shown in Fig. 3 as curves corresponding to
,
,
,
and
.
The curves are virtually superimposed everywhere except around the transition temperature, as shows more clearly the inset. In fact, the same is observed on energy curves (not shown here). We found that the contribution of the Zeeman term to the total energy of the crystal for the range of magnetic field strengths considered here is relatively low, which explains the very small difference between the curves of energy and therefore of the magnetic entropy in the paramagnetic and ferromagnetic phases. However, the presence of the applied magnetic field accelerates, depending on its intensity of course, the ordering of the spins in the crystal, as is evident on the magnetization curves (see Fig. 2); thus, the transition to the ferromagnetic phase has to start earlier in the presence of the applied magnetic field, which may explain the difference between the entropy curves around the transition temperature. Theoretically, the difference
should tend to zero at infinite
temperature regardless of the value of . This is realized by our simulation that gave a very small value at
independent of the value of B. The cutoff temperature of
is thus a good approximation to the infinite temperature at least for the range of B-values considered here. The temperature dependence of the magnetic entropy change,
, for
under various magnetic field intensities was deduced from the preceding curves; the results are displayed in Fig. 4. The curves show a broad peak substantially centered about the transition temperature
regardless of the intensity of the applied magnetic field; its
magnitude as well as its full width at half maximum (FWHM) by cons, increase by increasing the intensity of the external magnetic field. A comparison of our simulation results with experiment is shown in Fig. 4 and in Table 1. The inset of Fig. 4 shows the entropy change, in absolute value, measured by Zhang et al. [3] for
bulk material over the
temperature range under applied magnetic fields up to 7
. In Table 1 we
postponed the values of maximum entropy change
, FWHM (
cooling power (RCP) defined by
) and the relative
, deduced from the preceding
experimental curves in comparison with our theoretical values. Our simulation data reproduce satisfactorily the overall aspect of the experimental curves as well as the negative sign of the entropy change along the explored temperature range. More quantitatively, the experimental data of Zhang and coworkers are in fairly good agreement with our theoretical data, although a slight difference is obvious; indeed, the measured maximum entropy change and the RCP are smaller than those simulated for a single crystal, while it is the opposite for the FWHM. One glaring difference, however, is noticed with the measurements of Guo et al. [4] who reported a value of
and
magnetic field variation of
for
and
. An even larger deviation is noted with the measurements
made by Lin et al. [6] and Sun et al. [5] who reported values of for
, under
and
, respectively, for a
and
magnetic field variations, respectively. Then we can deduce
that the polycrystalline powders prepared by Zhang and coworkers [3] seem to be the closest to an ideal crystal structure of
.
Finally, we show in Fig. 5 the magnetocaloric effect as the adiabatic temperature change computed according to Eq. (6) as a function of temperature and applied magnetic field. For comparison, we have reported in the inset experimental data from Lin et al. [6] for polycrystalline perovskite applied magnetic field change of
over the and
temperature range under
. It is worth mentioning here that, to our
knowledge, this is the only measurement of adiabatic temperature change available in the literature for such compound. Our simulation accounts for the overall aspect of the experimental curves. We show a broad peak centered practically on the Curie temperature regardless of the intensity of the applied magnetic field; its magnitude as well as its FWHM by cons, increase by increasing the intensity of the external magnetic field. Quantitative analysis of these curves was made by noting the peak maximum and its FWHM; the former are listed in the last column of Table 1, whereas the latter are found to be practically identical to their counterparts in the
curves (see the third column in Table 1).
Comparison with experiment is limited to the work of Lin et al. [6]; these authors have reported
values of 1.1, 1.7, 2.1 and
for magnetic field variations of 0.75, 1.33,
1.74 and 2.02T, respectively. These values are smaller than those provided by our simulation, but the deviation is not so large. From these experimental data, Lin and coworkers have deduced a linear law of variation of the maximum adiabatic temperature change as a function 8
of magnetic field variation. Our simulation shows that this is not the case, however, for larger magnetic field variation range, as can be noted from Fig. 5 showing a direct comparison between the results of our simulation and the result of a second-order polynomial fitting. We conclude this discussion by noting that the adiabatic temperature change decreases on approaching the room temperature, but remains significant for a magnetic field jump of at least 2T. Indeed, at
,
takes the values
,
,
and
under a
magnetic field variation of 1, 2, 3 and 4T, respectively. These findings are therefore encouraging for the use of the
compound in the magnetic refrigeration even
at room temperature.
4. Conclusion: In this work we have presented a theoretical study of the magnetocaloric effect of a magnetically homogeneous
crystal. The main purpose of this study is to
complement the tremendous experimental efforts that have been reported on this material for a better characterization of its MCE. Our Monte Carlo simulation was carried out using the conventional Metropolis importance sampling algorithm, where the energy of interaction of the spins-system is described by the three-dimensional vector model of Heisenberg. Our simulation reproduces reasonably the shape of the experimentally derived curves of
and
as functions of temperature and applied magnetic field. We have found in addition a good qualitative agreement with experiment. Indeed, the maximum entropy change as well as the maximum adiabatic temperature change are found to increase by increasing the applied magnetic field intensity, whereas they occur at the same temperature, the ferromagneticparamagnetic transition temperature of the considered crystal, regardless of the intensity of the applied magnetic field. In quantitative terms, we noted a slight disagreement between experiment and simulation. As expected, this is due to the fact that the prepared polycrystalline powders deviate more or less, depending on the preparation conditions, from an ideal homogeneous magnetic structure of
single crystal. The
effectiveness of this manganite in magnetic refrigeration, near room temperature, is thus approved by our simulation. A single crystal is found to be more efficient than a polycrystalline powder.
9
References [1] A. Smith, “Who discovered the magnetocaloric effect? Warburg, Weiss, and the connection between magnetism and heat”, Eur. Phys. J. H 38 (2013) 507–517. doi:10.1140/epjh/e2013-40001-9 [2] M.-H. Phana, and S.–C. Yu, “Review of the magnetocaloric effect in manganite materials”, J. Mag. Mag. Mater. 308 (2007) 325–340. doi:10.1016/j.jmmm.2006.07.025 [3] X. X. Zhang, J. Tejada, Y. Xin, G. F. Sun, K. W. Wong, and X. Bohigas, “Magnetocaloric effect in La0.67Ca0.33MnOõ and La0.60Y0.07Ca0.33MnOõ bulk materials”, Appl. Phys. Lett. 69 (1996) 3596-3598. doi: 10.1063/1.117218 [4] Z.B. Guo, Y.W. Du, J.S. Zhu, H. Huang, W.P. Ding, and D. Feng, “Large magnetic entropy change in perovskite-type manganese oxides”, Phys. Rev. Lett. 78 (1997) 1142-1145. doi:10.1103/PhysRevLett.78.1142 [5] Y. Sun, X. Xu, and Y. Zhang, “Large magnetic entropy change in the colossal magnetoresistance material La2/3Ca1/3MnO3”, J. Mag. Mag. Mater. 219 (2000) 183-185. [6] G.C. Lin, Q. Wei, and J.X. Zhang, “Direct measurement of the magnetocaloric effect in La0.67Ca0.33MnO3”, J. Mag. Mag. Mater. 300 (2006) 392-396. [7] O. Pavlukhina, V. Buchelnikov, V. Sokolovskiy, and M. Zaghrebin, “Monte Carlo Study of the Magnetic and Magnetocaloric Properties of La1-xCaxMnO3 (x = 0.33 and 0.5)”, Solid State Phenomena 190 (2012) 347-350. [8] S. Valencia, A. Gaupp, W. Gudat, Ll. Abad, L1. Balcells, and B. Martinez, “Impact of microstructure on the Mn valence of La2/3Ca1/3MnO3 thin films”, Phys. Rev. B 75 (2007) 184431-184442. doi: 10.1103/PhysRevB.75.184431 [9] H. A. Jahn and E. Teller, “Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy”, Proc. R. Soc. London A. 161 (1937) 220-235. doi: 10.1098/rspa.1937.0142. [10] E. Restrepo-Parra, L. Ramos-Rivera, and J. Londoño-Navarro, “Magnetocaloric properties in La1-xCaxMnO3 thin films: Monte-Carlo simulations", J. Magn. Mag. Mater. 351 (2014) 65-69. doi: 10.1016/j.jmmm.2013.09.011 [11] J. Kanamori, “Crystal distortion in magnetic compounds”, J. Appl. Phys. 31 (1960) 1423. doi: 10.1063/1.1984590 [12] R. Maezono, S. Ishihara, and N. Nagaosa, “Phase diagram of manganese oxides”, Phys. Rev. B 58 (1998) 11583-11596. doi: 10.1103/PhysRevB.58.11583 10
[13] J. van den Brink, and D. I. Khomskii, “Orbital ordering of complex orbitals in doped Mott
insulators”,
Phys.
Rev.
B
63
(2001)
140416
-140419.
doi:
10.1103/PhysRevB.63.140416. [14] T. Hotta, S. Yunoki, M. Mayr, and E. Dagotto, “A-type antiferromagnetic and C-type orbital-ordered states in LaMnO3 using cooperative Jahn-Teller phonons”, Phys. Rev. B 60 (1999) R15009 –R150012. doi: doi: 10.1103/PhysRevB.60.R15009 [15] J. van den Bink, and D. Khomskii, “A-type antiferromagnetic and C-type orbital-ordered states in LaMnO3 using cooperative Jahn-Teller phonons”, Phys. Rev. B 60 (1999) R15009 – R150012. doi: doi: 10.1103/PhysRevB.60.R15009 [16] E.O. Wallan, and W. C. Koehler, “Neutron diffraction study of the magnetic properties of the series of perovskite-type compounds [(1-x)La, xCa]MnO3”, Phys. Rev. 100 (1955) 545563. doi:10.1103/PhysRev.100.545 [17] E. Restrepo-Parra, C.D. Salazar-Enríquez, J. Londonõ-Navarro, J.F. Jurado, J. Restrepo, “Magnetic phase diagram simulation of La1-xCaxMnO3 system by using Monte Carlo, Metropolis algorithm and Heisenberg model", J. Magn. Mag. Mater. 323 (2011) 1477–1483. doi: 10.1016/j.jmmm.2011.01.003 [18] C.H. Chen and S.-W. Cheong, “Commensurate to incommensurate charge ordering and its real-space images in La0.5Ca0.5MnO3”, Phys. Rev. Lett. 76 (1996) 4042-4045. doi: 10.1103/PhysRevLett.76.4042 [19] C. P. Adams, J. W. Lynn, Y. M. Mukovskii, A. A. Arsenov, and D. A. Shulyatev, “Charge ordering and polaron formation in the magnetoresistive oxide La0.7Ca0.3MnO3”, Phys. Rev. Lett. 85 (2000) 3954-3957. doi: 10.1103/PhysRevLett.85.3954 [20] T. Hotta, A.L. Malvezzi, and E. Dagotto, “Charge-orbital ordering and phase separation in the two-orbital model for manganites: Roles of Jahn-Teller phononic and Coulombic interactions”, Phys. Rev. B. 62 (2000) 9432-9452. doi: 10.1103/PhysRevB.62.9432 [21] T. Hotta, A. Feiguin, and E. Dagotto, “Stripes induced by orbital ordering in layered manganites”, Phys. Rev. Lett. 86 (2001) 4922-4925. doi: 10.1103/PhysRevLett.86.4922 [22] E. Restrepo-Parra, C.M. Bedoya-Hincapié, F.J. Jurado, J.C. Riano-Rojas, and J. Restrepo, “Monte Carlo study of the critical behavior and magnetic properties of La2/3Ca1/3MnO3 thin films", J. Magn. Magn. Mater. 322 (2010) 3514-3518. doi: doi:10.1016/j.jmmm.2010.06.055 [23] D.P. Landau, and K. Binder, “A Guide to Monte Carlo Simulations in Statistical Physics”, 3rd edition, Cambridge University Press, UK (2009).
11
[24] K. Binder, “Monte Carlo study of entropy for face-centered cubic Ising antiferromagnets”, Z. Phys. B - Condensed Matter 45 (1981) 61-69. [25] D.P. Landau, and F. Wang, “A New Approach to Monte Carlo Simulations in Statistical Physics”, Braz. J. Phys. 34 (2004) 354-362. doi: 10.1590/S0103-97332004000300004 [26] O. Pavlukhina, V. Buchelnikov, and V. Sokolovskiy, “Modeling of magnetic and magnetocaloric properties of La0.7Ba0.3MnO3 manganites by Monte Carlo method”, Functional Mater. 19 (2012) 97-100. [27] N.A. de Oliveira, and P.J. von Ranke, “Theoretical aspects of the magnetocaloric effect”, Phys. Rep. 489 (2010) 89-159. doi:10.1016/j.physrep.2009.12.006 [28] C. Christides, N. Moutis, Ph. Komninou, Th. Kehagias, and G. Nouet, “Dependence of exchange bias energy on spin projections at (La,Ca)MnO3 ferromagnetic/antiferromagnetic interfaces”, J. Appl. Phys. 92 (2002) 397-405. doi:10.1063/1.1484230 [29] F. Moussa, M. Hennion, J. Rodriguez-Carvajal, H. Moudden, L. Pinsard, A. Revcolevschi, “ Spin waves in the antiferromagnet perovskite LaMnO3: A neutron-scattering study”, Phys. Rev. B. 54 (1996) 15149-15152. doi:10.1103/PhysRevB.54.15149 [30] S. Naji, A. Benyoussef, A. El Kenz, H. Ez-Zahraouy, and M. Loulidi, “Monte Carlo study of phase transitions and manetic properties of LaMnO3: Heisenberg model”, Physica A 391 (2012) 3885-3894. doi: 10.1016/j.physa.2012.03.003 [31] P. Schiffer, A.P. Ramirez, W. Bao, and S.-W. Cheong, “Low temperature magnetoresistance and the magnetic phase diagram of La1-xCaxMnO3”, Phys. Rev. Lett. 75 (1995) 3336-3339. doi: 10.1103/ PhysRevLett.75.3336
Fig. 1: 3D schematic representation of the charge and orbital ordering in La2/3Ca1/3MnO3 single crystal used in the simulation. Fig. 2: Temperature dependence of the magnetization per manganese ion under different magnetic fields. Fig. 3: Temperature dependence of the entropy per manganese ion under different magnetic fields. The inset: a zoom on the
temperature range.
12
Fig. 4: Temperature dependence of the isothermal magnetic entropy change per manganese ion under different applied magnetic fields. The inset shows experimental data reproduced from the publication of Guo et al. [4]. Fig. 5: Temperature dependence of the adiabatic temperature change in La2/3Ca1/3MnO3 under different applied magnetic fields. The inset shows experimental data reproduced from the publication of Lin et al. [6]. Fig. 6: Magnetic field change dependence of the maximum adiabatic temperature change in La2/3Ca1/3MnO3 single crystal: the points are the simulated values, and the solid line is the result of their second-order polynomial fit.
Table 1: The magnetocaloric effect parameters computed for La2/3Ca1/3MnO3 single crystal subjected to the action of a magnetic field. The experimental values are those reported by Guo et al. [4] for a polycrystalline powder. B (T)
(J/kg.K)
(K)
(K)
(J/kg)
Theor.
Exp.
Theor.
Exp.
Theor.
Exp.
Theor.
1
-1.77
-1.14
25.56
29
45.24
33.06
1.7
2
-2.81
-1.95
33.33
37
93.65
72.15
3.0
3
-3.79
-2.59
36.67
44
139.00
114.00
4.1
4
-4.54
-
43.33
-
196.72
-
5.0
Highlights:
We present a Monte Carlo simulation of the MCE in
A good qualitative agreement is observed with available experimental data.
Quantitatively, a slight disagreement between experiment and simulation is observed.
This manganite is found to be effective in room temperature magnetic refrigeration.
A single crystal is found to be more efficient than a polycrystalline powder.
13
single crystal.
Figure 1
x z y
Figure 2
14
Figure 3
15
Figure 4
16
Figure 5
17
Figure 6
18
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PII: DOI: Reference:
S0304-8853(16)30937-4 http://dx.doi.org/10.1016/j.jmmm.2016.05.104 MAGMA61518
To appear in: Journal of Magnetism and Magnetic Materials Received date: 11 December 2015 Revised date: 23 May 2016 Accepted date: 31 May 2016 Cite this article as: R. Zouari and A. Chehaidar, Monte Carlo simulation of the magnetocaloric effect in La2/3Ca1/3MnO3 single crystal, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.05.104 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Monte Carlo simulation of the magnetocaloric effect in La2/3Ca1/3MnO3 single crystal R. Zouari, A. Chehaidar
Laboratory of Physics-Mathematics and Applications, University of Sfax, Faculty of Sciences, Department of Physics, P.O. Box 1171, 3000 Sfax, Tunisia.
*Corresponding author: [email protected] Abstract The present work is devoted to a theoretical simulation study of the magnetocaloric effect in magnetically homogeneous
single crystal. Using the standard Monte Carlo-
Metropolis algorithm and the classical Heisenberg model Hamiltonian, we have computed the two main magnetocaloric properties such as the isothermal entropy change and the adiabatic temperature change upon an abrupt variation of the intensity of the applied magnetic field, as function of temperature. A good qualitative agreement is observed between our simulation and experiment. We have shown that the maximum entropy change increases by increasing the intensity of the applied magnetic field. In addition, it occurs at the ferromagneticparamagnetic transition temperature regardless of the intensity of the applied magnetic field. Our simulation shows, moreover, that the adiabatic temperature change behaves as the isothermal entropy change with respect to the material temperature and the applied magnetic field variation. Quantitatively, however, the experimental data deviate more or less, depending on powder preparation conditions, from our simulation data. This demonstrates the deviation of the prepared powders with respect to an ideal magnetic structure, as expected experimentally. Our simulation expects a maximum isothermal entropy change of and a maximum adiabatic temperature change of
under a magnetic field variation
of 5T. On approaching room temperature, the magnitude of the magnetocaloric effect in single crystal decreases but remains significant under a magnetic field variation of at least 2 Tesla.
Keywords: Manganites, Magnetocaloric effect, Monte Carlo simulation, modelling.
1
1. Introduction The magnetocaloric effect (MCE) is recognized as the warming or cooling of magnetic material subjected to an abrupt variation of the intensity of applied magnetostatic field. It was first discovered in 1881 by Warburg on iron, and first quantified experimentally in 1917 by Weiss and Piccard by measuring a sizable and reversible temperature change in nickel near its Curie temperature (See Ref. [1] for history). Since then, the search of magnetocaloric effect materials never stopped. In particular, solid materials for magnetic refrigeration have aroused particular interest to replace conventional gas refrigerants, and this fits into the framework of environmental preservation, on one hand, and maximization of the performance /cost ratio, on the other hand. In selecting magnetic refrigerants, particular attention was paid to the solid materials which exhibit large magnetic entropy change under a magnetic field, especially to those that can be used at room temperature or in its vicinity. Several compounds that meet these requirements have been proposed [2]; in particular, calcium doped lanthanum-manganite of chemical formula. The latter is found to be characterized further by high chemical stability and a high electrical resistivity that promotes a low warming of the material by the eddy currents [3-6]. The measurements of the quantities characterizing the magnetocaloric effect in the bulk
, namely the maximum isothermal entropy
change, the corresponding adiabatic temperature change and the relative cooling power, are performed on polycrystalline powders. Different values were, however, reported for the same magnetocaloric parameters. This discrepancy was interpreted as being due to differences in the sample processing routes and/or to the differently chemical composition [2]. So, we can ask what values can take the magnetocaloric parameters of a perfect
bulk
material. The answer to this question can be provided by the simulation technique. This is the purpose of our present work.
2
2. Computation Method A single crystal of
, free of defects, is considered here with a simple
cubic perovskite structure [7, 8]. Thus, a unit cell of this crystal is a cube containing one ion of Ca or La in its center, one Mn ion in each corner, and an O ion in the middle of each edge. As the La and Ca ions do not have the same ionization, the Mn ions are engaged in such a structure in two valances:
and
. The electronic structure of
make them non-magnetic ions, while and
and
,
and
are paramagnetic ions with spins
, respectively. The electrostatic interaction between the Mn ion and its
ligands leads to a redistribution of its valence electrons between two separated energy levels denoted by
and
in accordance with octahedral site symmetry. Regarding
ion, its
four valence electrons fall into three electrons on the triply-degenerate level energy) and one electron on the doubly-degenerate level ion, its three valence electrons occupy the the normal vibrational modes of the
(lower
(higher energy). For the
level. The coupling between the
-orbitals and
octahedron leads to the static distortion of the
oxygen’s cage, an effect called Jahn-Teller distortion [9]. This distortion manifests itself in a tetragonal elongation of the octahedron, leading to a lowering of the site symmetry and, consequently, the lifting of the degeneracy of the
level; the latter splits into two non-
degenerate and close sublevels denoted, hereinafter,
and
[10]. In a crystal, the
cooperative Jahn-Teller distortions favor the appearance of orbital ordering [11-15], which consists in a regular alternation of the occupied
and
orbitals with lobes pointed toward
the neighboring oxygen ions. In addition to orbital order, mixed-valence manganites are found to exhibit charge as well as spin ordering [16-20]. In those compounds in which the proportions of
and
ions are rational fractions, spin-, charge- and orbital-
ordering effects are particularly pronounced [21]. The single crystal
is
ordered ferromagnetically at low temperature with spins-arrangement of B-type according to the notation of Wallan and Koehler [16]; the charge- and orbital-ordering in this compound, established previously by Hotta et al. [21] and more recently by Restrepo-Para et al. [22], are illustrated schematically in Fig. 1 showing a cluster of
manganese ions whose
periodic repetition in the three directions of space reproduces the bulk crystal. Due to computation limitation, the crystal model is supposed to be contained in a large supercell with 3D periodic boundary conditions. In our present study, a cubic supercell of
3
ions
was
thus
generated
to
crystalline
simulate
the
magnetocaloric
properties
of
the
bulk
.
After generating the structural model, we turn to the description of the Hamiltonian of the system. Our present calculations are based on the classical Heisenberg model, according to which the Hamiltonian of interacting spins in the presence of an applied magnetic field is written as follows [7]: ⃗
∑
⃗
∑ (⃗
∑ ⃗
⃗)
⃗⃗
(1)
The first term in this expression reflects the exchange interaction between manganese ions; the sum runs over all the sites i in the supercell and their nearest neighbor sites j. The second term gives the core cubic magnetocrystalline anisotropy, where
is the single ion bulk
anisotropy constant and the unit vector ⃗ indicates the easy axis direction. The last term accounts for the Zeeman interaction of spin magnetic dipole with a uniform applied magnetic induction, where
and
stand for the electron g-factor and the Bohr magneton,
respectively. As can be inferred from Fig. 1, three exchange integrals are involved in the development of exchange term in Eq. (1), namely (Mn4+ - Mn3+(
) and
(Mn4+ - Mn3+(
(Mn3+(
- Mn3+ (
),
). These integrals along with the
magnetocrystalline constant are treated as empirical parameters to be determined from experimental data. Now we come to the step of computing the magnetic and the magnetocaloric properties of our structural model for the crystal magnetization
. Specifically, we are interested in the
, the specific heat
, the isothermal entropy change
and the adiabatic temperature change
,
upon magnetic field variation. The
equilibrium spin configurations accessed by the magnetic system at a given temperature T are obtained by Monte Carlo simulations based on the standard single spin Metropolis algorithm [23]. In each MC step per spin, a site is randomly selected for a random spin reorientation. Typically,
MC steps per spin are considered for thermalization; after that,
MC
steps per spin are performed to evaluate the average of the desired thermodynamical quantity. If we denote by 〈 〉 the average of a physical quantity X, then the internal energy and the magnitude of the magnetization 〈 〉
of the system of spins are given by:
(2)
and 4
〈‖ ⃗‖〉
〈‖ ∑
⃗ ‖〉
(3)
where N is the number of magnetic sites in the supercell. The magnetic specific heat is estimated from the fluctuations of the internal energy according to the following relation: [〈 where
〉
〈 〉 ]
(4)
is the Boltzmann constant. The magnetic entropy
as a function of
temperature and applied magnetic field, can be computed by integration of the internal energy function starting at infinite temperature according to the following relation [24]: (
∫ Nevertheless, the term
)
(5)
in this equation is unknown; for a paramagnetic phase,
however, we expect a substantially magnetic field independent function. It worthwhile to note here that this assumption is supported by the simulation of the entropy function for thin films reported recently by Restrepo-Parra et al. [10]; these authors have computed the entropy directly from the partition function of the system using the random walk in the energy space method of Landau and Wang [25]. Under this assumption, the isothermal entropy change
, the measurable quantity that
interests us, in fact, can be computed from Eq. (5) by omitting this term. The adiabatic temperature change
, another interesting measurable thermodynamic quantity, is
calculated according to the following formula [26]: (6) where
is the sum of the magnetic specific heat and the
crystalline lattice specific heat expressed by [26,27]: { ( ) ∫ Here ,
and
⁄
( )
⁄
}
(7)
stand for the universal gas constant, the number of ions per formula unit
and the Debye temperature, respectively.
5
3. Results and discussion The starting point of our calculations is the determination of the exchange parameters and the magnetocrystalline anisotropy constant involved in the expression of the Hamiltonian of the system given by Eq. (1). This task has been addressed previously by Restrepo-Parra et al. [17] who simulated the phase diagram of the manganites 1. For
with x ranging from 0 to
, they have reported the numerical values ,
, and
, . The magnetocrystalline
anisotropy constant, in fact, was not treated as a free parameter; rather, it was fixed at a value derived experimentally by Christides et al. [28]. However, this value was postponed for strained epitaxial
thin films, and it is obviously much greater than that of
the bulk, as already mentioned by these authors too. Furthermore, in their same simulation study, Restrepo-Parra et al. [17] postponed to the parent compound LaMnO3 an anisotropy constant twice that obtained for
crystal, i.e.
; this value is,
unfortunately, incomparable to the experimental value of al. [29], neither to the theoretical value of et al. [30] for the parent compound
reported by Moussa et
obtained by ab initio calculation by Naji . We then readjusted the exchange parameter
values provided by Restrepo-Parra and coworkers for and to the value
compounds with
setting the basic anisotropy constant (denoted by
in Ref. [17])
i.e. half the experimental value derived experimentally by Moussa
and coworkers for
crystal. A fairly good agreement between theoretical and
experimental magnetic transition temperatures is obtained for the proposed compounds, respecting the same proportions between the exchange parameters of the different stoichiometries, established by Restrepo-Parra et al. [17]. For compound, in particular, we have obtained , and temperature
,
,
, leading to a theoretical Curie
slightly higher than the experimental value
[31].
The evolution of the macroscopic magnetization as a function of temperature for our crystal model subjected to the action of a uniform magnetic field has been simulated by cooling from
in steps of
. The results for
,
,
,
and
are displayed as curves in Fig. 2. The expected paramagnetic-ferromagnetic transition in crystal is clearly visible on these magnetization curves. It is steep in the absence of an applied magnetic field, and manifests itself at 6
; the Curie
⁄
temperature is determined here at the greatest slope in
. It becomes less and less steep
by increasing the intensity of the applied magnetic field, thus making account experimental observations [31]. The transition temperature is found to slightly upward shift by increasing the magnetic field intensity up to 2T, whereas it becomes undefined beyond. The dependence of the magnetic entropy with the applied magnetic field and the temperature,
, was simulated according to the Eq. (5) in which the integral is started at
with
; rather it is the difference between the entropy at temperature T
and that at infinite temperature,
, that is computed numerically. The results
are shown in Fig. 3 as curves corresponding to
,
,
,
and
.
The curves are virtually superimposed everywhere except around the transition temperature, as shows more clearly the inset. In fact, the same is observed on energy curves (not shown here). We found that the contribution of the Zeeman term to the total energy of the crystal for the range of magnetic field strengths considered here is relatively low, which explains the very small difference between the curves of energy and therefore of the magnetic entropy in the paramagnetic and ferromagnetic phases. However, the presence of the applied magnetic field accelerates, depending on its intensity of course, the ordering of the spins in the crystal, as is evident on the magnetization curves (see Fig. 2); thus, the transition to the ferromagnetic phase has to start earlier in the presence of the applied magnetic field, which may explain the difference between the entropy curves around the transition temperature. Theoretically, the difference
should tend to zero at infinite
temperature regardless of the value of . This is realized by our simulation that gave a very small value at
independent of the value of B. The cutoff temperature of
is thus a good approximation to the infinite temperature at least for the range of B-values considered here. The temperature dependence of the magnetic entropy change,
, for
under various magnetic field intensities was deduced from the preceding curves; the results are displayed in Fig. 4. The curves show a broad peak substantially centered about the transition temperature
regardless of the intensity of the applied magnetic field; its
magnitude as well as its full width at half maximum (FWHM) by cons, increase by increasing the intensity of the external magnetic field. A comparison of our simulation results with experiment is shown in Fig. 4 and in Table 1. The inset of Fig. 4 shows the entropy change, in absolute value, measured by Zhang et al. [3] for
bulk material over the
temperature range under applied magnetic fields up to 7
. In Table 1 we
postponed the values of maximum entropy change
, FWHM (
cooling power (RCP) defined by
) and the relative
, deduced from the preceding
experimental curves in comparison with our theoretical values. Our simulation data reproduce satisfactorily the overall aspect of the experimental curves as well as the negative sign of the entropy change along the explored temperature range. More quantitatively, the experimental data of Zhang and coworkers are in fairly good agreement with our theoretical data, although a slight difference is obvious; indeed, the measured maximum entropy change and the RCP are smaller than those simulated for a single crystal, while it is the opposite for the FWHM. One glaring difference, however, is noticed with the measurements of Guo et al. [4] who reported a value of
and
magnetic field variation of
for
and
. An even larger deviation is noted with the measurements
made by Lin et al. [6] and Sun et al. [5] who reported values of for
, under
and
, respectively, for a
and
magnetic field variations, respectively. Then we can deduce
that the polycrystalline powders prepared by Zhang and coworkers [3] seem to be the closest to an ideal crystal structure of
.
Finally, we show in Fig. 5 the magnetocaloric effect as the adiabatic temperature change computed according to Eq. (6) as a function of temperature and applied magnetic field. For comparison, we have reported in the inset experimental data from Lin et al. [6] for polycrystalline perovskite applied magnetic field change of
over the and
temperature range under
. It is worth mentioning here that, to our
knowledge, this is the only measurement of adiabatic temperature change available in the literature for such compound. Our simulation accounts for the overall aspect of the experimental curves. We show a broad peak centered practically on the Curie temperature regardless of the intensity of the applied magnetic field; its magnitude as well as its FWHM by cons, increase by increasing the intensity of the external magnetic field. Quantitative analysis of these curves was made by noting the peak maximum and its FWHM; the former are listed in the last column of Table 1, whereas the latter are found to be practically identical to their counterparts in the
curves (see the third column in Table 1).
Comparison with experiment is limited to the work of Lin et al. [6]; these authors have reported
values of 1.1, 1.7, 2.1 and
for magnetic field variations of 0.75, 1.33,
1.74 and 2.02T, respectively. These values are smaller than those provided by our simulation, but the deviation is not so large. From these experimental data, Lin and coworkers have deduced a linear law of variation of the maximum adiabatic temperature change as a function 8
of magnetic field variation. Our simulation shows that this is not the case, however, for larger magnetic field variation range, as can be noted from Fig. 5 showing a direct comparison between the results of our simulation and the result of a second-order polynomial fitting. We conclude this discussion by noting that the adiabatic temperature change decreases on approaching the room temperature, but remains significant for a magnetic field jump of at least 2T. Indeed, at
,
takes the values
,
,
and
under a
magnetic field variation of 1, 2, 3 and 4T, respectively. These findings are therefore encouraging for the use of the
compound in the magnetic refrigeration even
at room temperature.
4. Conclusion: In this work we have presented a theoretical study of the magnetocaloric effect of a magnetically homogeneous
crystal. The main purpose of this study is to
complement the tremendous experimental efforts that have been reported on this material for a better characterization of its MCE. Our Monte Carlo simulation was carried out using the conventional Metropolis importance sampling algorithm, where the energy of interaction of the spins-system is described by the three-dimensional vector model of Heisenberg. Our simulation reproduces reasonably the shape of the experimentally derived curves of
and
as functions of temperature and applied magnetic field. We have found in addition a good qualitative agreement with experiment. Indeed, the maximum entropy change as well as the maximum adiabatic temperature change are found to increase by increasing the applied magnetic field intensity, whereas they occur at the same temperature, the ferromagneticparamagnetic transition temperature of the considered crystal, regardless of the intensity of the applied magnetic field. In quantitative terms, we noted a slight disagreement between experiment and simulation. As expected, this is due to the fact that the prepared polycrystalline powders deviate more or less, depending on the preparation conditions, from an ideal homogeneous magnetic structure of
single crystal. The
effectiveness of this manganite in magnetic refrigeration, near room temperature, is thus approved by our simulation. A single crystal is found to be more efficient than a polycrystalline powder.
9
References [1] A. Smith, “Who discovered the magnetocaloric effect? Warburg, Weiss, and the connection between magnetism and heat”, Eur. Phys. J. H 38 (2013) 507–517. doi:10.1140/epjh/e2013-40001-9 [2] M.-H. Phana, and S.–C. Yu, “Review of the magnetocaloric effect in manganite materials”, J. Mag. Mag. Mater. 308 (2007) 325–340. doi:10.1016/j.jmmm.2006.07.025 [3] X. X. Zhang, J. Tejada, Y. Xin, G. F. Sun, K. W. Wong, and X. Bohigas, “Magnetocaloric effect in La0.67Ca0.33MnOõ and La0.60Y0.07Ca0.33MnOõ bulk materials”, Appl. Phys. Lett. 69 (1996) 3596-3598. doi: 10.1063/1.117218 [4] Z.B. Guo, Y.W. Du, J.S. Zhu, H. Huang, W.P. Ding, and D. Feng, “Large magnetic entropy change in perovskite-type manganese oxides”, Phys. Rev. Lett. 78 (1997) 1142-1145. doi:10.1103/PhysRevLett.78.1142 [5] Y. Sun, X. Xu, and Y. Zhang, “Large magnetic entropy change in the colossal magnetoresistance material La2/3Ca1/3MnO3”, J. Mag. Mag. Mater. 219 (2000) 183-185. [6] G.C. Lin, Q. Wei, and J.X. Zhang, “Direct measurement of the magnetocaloric effect in La0.67Ca0.33MnO3”, J. Mag. Mag. Mater. 300 (2006) 392-396. [7] O. Pavlukhina, V. Buchelnikov, V. Sokolovskiy, and M. Zaghrebin, “Monte Carlo Study of the Magnetic and Magnetocaloric Properties of La1-xCaxMnO3 (x = 0.33 and 0.5)”, Solid State Phenomena 190 (2012) 347-350. [8] S. Valencia, A. Gaupp, W. Gudat, Ll. Abad, L1. Balcells, and B. Martinez, “Impact of microstructure on the Mn valence of La2/3Ca1/3MnO3 thin films”, Phys. Rev. B 75 (2007) 184431-184442. doi: 10.1103/PhysRevB.75.184431 [9] H. A. Jahn and E. Teller, “Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy”, Proc. R. Soc. London A. 161 (1937) 220-235. doi: 10.1098/rspa.1937.0142. [10] E. Restrepo-Parra, L. Ramos-Rivera, and J. Londoño-Navarro, “Magnetocaloric properties in La1-xCaxMnO3 thin films: Monte-Carlo simulations", J. Magn. Mag. Mater. 351 (2014) 65-69. doi: 10.1016/j.jmmm.2013.09.011 [11] J. Kanamori, “Crystal distortion in magnetic compounds”, J. Appl. Phys. 31 (1960) 1423. doi: 10.1063/1.1984590 [12] R. Maezono, S. Ishihara, and N. Nagaosa, “Phase diagram of manganese oxides”, Phys. Rev. B 58 (1998) 11583-11596. doi: 10.1103/PhysRevB.58.11583 10
[13] J. van den Brink, and D. I. Khomskii, “Orbital ordering of complex orbitals in doped Mott
insulators”,
Phys.
Rev.
B
63
(2001)
140416
-140419.
doi:
10.1103/PhysRevB.63.140416. [14] T. Hotta, S. Yunoki, M. Mayr, and E. Dagotto, “A-type antiferromagnetic and C-type orbital-ordered states in LaMnO3 using cooperative Jahn-Teller phonons”, Phys. Rev. B 60 (1999) R15009 –R150012. doi: doi: 10.1103/PhysRevB.60.R15009 [15] J. van den Bink, and D. Khomskii, “A-type antiferromagnetic and C-type orbital-ordered states in LaMnO3 using cooperative Jahn-Teller phonons”, Phys. Rev. B 60 (1999) R15009 – R150012. doi: doi: 10.1103/PhysRevB.60.R15009 [16] E.O. Wallan, and W. C. Koehler, “Neutron diffraction study of the magnetic properties of the series of perovskite-type compounds [(1-x)La, xCa]MnO3”, Phys. Rev. 100 (1955) 545563. doi:10.1103/PhysRev.100.545 [17] E. Restrepo-Parra, C.D. Salazar-Enríquez, J. Londonõ-Navarro, J.F. Jurado, J. Restrepo, “Magnetic phase diagram simulation of La1-xCaxMnO3 system by using Monte Carlo, Metropolis algorithm and Heisenberg model", J. Magn. Mag. Mater. 323 (2011) 1477–1483. doi: 10.1016/j.jmmm.2011.01.003 [18] C.H. Chen and S.-W. Cheong, “Commensurate to incommensurate charge ordering and its real-space images in La0.5Ca0.5MnO3”, Phys. Rev. Lett. 76 (1996) 4042-4045. doi: 10.1103/PhysRevLett.76.4042 [19] C. P. Adams, J. W. Lynn, Y. M. Mukovskii, A. A. Arsenov, and D. A. Shulyatev, “Charge ordering and polaron formation in the magnetoresistive oxide La0.7Ca0.3MnO3”, Phys. Rev. Lett. 85 (2000) 3954-3957. doi: 10.1103/PhysRevLett.85.3954 [20] T. Hotta, A.L. Malvezzi, and E. Dagotto, “Charge-orbital ordering and phase separation in the two-orbital model for manganites: Roles of Jahn-Teller phononic and Coulombic interactions”, Phys. Rev. B. 62 (2000) 9432-9452. doi: 10.1103/PhysRevB.62.9432 [21] T. Hotta, A. Feiguin, and E. Dagotto, “Stripes induced by orbital ordering in layered manganites”, Phys. Rev. Lett. 86 (2001) 4922-4925. doi: 10.1103/PhysRevLett.86.4922 [22] E. Restrepo-Parra, C.M. Bedoya-Hincapié, F.J. Jurado, J.C. Riano-Rojas, and J. Restrepo, “Monte Carlo study of the critical behavior and magnetic properties of La2/3Ca1/3MnO3 thin films", J. Magn. Magn. Mater. 322 (2010) 3514-3518. doi: doi:10.1016/j.jmmm.2010.06.055 [23] D.P. Landau, and K. Binder, “A Guide to Monte Carlo Simulations in Statistical Physics”, 3rd edition, Cambridge University Press, UK (2009).
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[24] K. Binder, “Monte Carlo study of entropy for face-centered cubic Ising antiferromagnets”, Z. Phys. B - Condensed Matter 45 (1981) 61-69. [25] D.P. Landau, and F. Wang, “A New Approach to Monte Carlo Simulations in Statistical Physics”, Braz. J. Phys. 34 (2004) 354-362. doi: 10.1590/S0103-97332004000300004 [26] O. Pavlukhina, V. Buchelnikov, and V. Sokolovskiy, “Modeling of magnetic and magnetocaloric properties of La0.7Ba0.3MnO3 manganites by Monte Carlo method”, Functional Mater. 19 (2012) 97-100. [27] N.A. de Oliveira, and P.J. von Ranke, “Theoretical aspects of the magnetocaloric effect”, Phys. Rep. 489 (2010) 89-159. doi:10.1016/j.physrep.2009.12.006 [28] C. Christides, N. Moutis, Ph. Komninou, Th. Kehagias, and G. Nouet, “Dependence of exchange bias energy on spin projections at (La,Ca)MnO3 ferromagnetic/antiferromagnetic interfaces”, J. Appl. Phys. 92 (2002) 397-405. doi:10.1063/1.1484230 [29] F. Moussa, M. Hennion, J. Rodriguez-Carvajal, H. Moudden, L. Pinsard, A. Revcolevschi, “ Spin waves in the antiferromagnet perovskite LaMnO3: A neutron-scattering study”, Phys. Rev. B. 54 (1996) 15149-15152. doi:10.1103/PhysRevB.54.15149 [30] S. Naji, A. Benyoussef, A. El Kenz, H. Ez-Zahraouy, and M. Loulidi, “Monte Carlo study of phase transitions and manetic properties of LaMnO3: Heisenberg model”, Physica A 391 (2012) 3885-3894. doi: 10.1016/j.physa.2012.03.003 [31] P. Schiffer, A.P. Ramirez, W. Bao, and S.-W. Cheong, “Low temperature magnetoresistance and the magnetic phase diagram of La1-xCaxMnO3”, Phys. Rev. Lett. 75 (1995) 3336-3339. doi: 10.1103/ PhysRevLett.75.3336
Fig. 1: 3D schematic representation of the charge and orbital ordering in La2/3Ca1/3MnO3 single crystal used in the simulation. Fig. 2: Temperature dependence of the magnetization per manganese ion under different magnetic fields. Fig. 3: Temperature dependence of the entropy per manganese ion under different magnetic fields. The inset: a zoom on the
temperature range.
12
Fig. 4: Temperature dependence of the isothermal magnetic entropy change per manganese ion under different applied magnetic fields. The inset shows experimental data reproduced from the publication of Guo et al. [4]. Fig. 5: Temperature dependence of the adiabatic temperature change in La2/3Ca1/3MnO3 under different applied magnetic fields. The inset shows experimental data reproduced from the publication of Lin et al. [6]. Fig. 6: Magnetic field change dependence of the maximum adiabatic temperature change in La2/3Ca1/3MnO3 single crystal: the points are the simulated values, and the solid line is the result of their second-order polynomial fit.
Table 1: The magnetocaloric effect parameters computed for La2/3Ca1/3MnO3 single crystal subjected to the action of a magnetic field. The experimental values are those reported by Guo et al. [4] for a polycrystalline powder. B (T)
(J/kg.K)
(K)
(K)
(J/kg)
Theor.
Exp.
Theor.
Exp.
Theor.
Exp.
Theor.
1
-1.77
-1.14
25.56
29
45.24
33.06
1.7
2
-2.81
-1.95
33.33
37
93.65
72.15
3.0
3
-3.79
-2.59
36.67
44
139.00
114.00
4.1
4
-4.54
-
43.33
-
196.72
-
5.0
Highlights:
We present a Monte Carlo simulation of the MCE in
A good qualitative agreement is observed with available experimental data.
Quantitatively, a slight disagreement between experiment and simulation is observed.
This manganite is found to be effective in room temperature magnetic refrigeration.
A single crystal is found to be more efficient than a polycrystalline powder.
13
single crystal.
Figure 1
x z y
Figure 2
14
Figure 3
15
Figure 4
16
Figure 5
17
Figure 6
18