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The influence of magnetic and electric coupling properties on the magnetocaloric effect in quantum paraelectric EuTiO3
Journal of Magnetism and Magnetic Materials 324 (2012) 1290–1295
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The influence of magnetic and electric coupling properties on the magnetocaloric effect in quantum paraelectric EuTiO3 P.J. von Ranke a,n, B.P. Alho a, E.P. No´brega a, V.S.R. de Sousa b, T.S.T. Alvarenga a, A. Magnus G. Carvalho c, N.A. de Oliveira a a
Instituto de Fı´sica, Universidade do Estado do Rio de Janeiro—UERJ, Rua Sa~ o Francisco Xavier, 524, 20550-013 RJ, Brazil Instituto de Fı´sica Gleb Wataghin, Universidade Estadual de Campinas—UNICAMP, 13083-970 Campinas, SP, Brazil c ~ de Metrologia de Materiais (DIMAT), Instituto Nacional de Metrologia, Normalizac- a~ o e Qualidade Industrial (INMETRO), Duque de Caxias, RJ 25250-020, Brazil Divisao b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 October 2011 Received in revised form 6 November 2011 Available online 2 December 2011
We report on the magnetic and magnetocaloric effect calculations in antiferromagnetic perovskite-type EuTiO3. From the isothermal magnetic entropy change calculated upon low magnetic field changes (below 1 T) several results were predicted: inverse magnetocaloric effect, latent heat associated to spin AFM-FM reorientation transition and a temperature interval (controlled by magnetic field) where the EuTiO3 does not change heat in an isothermic process. The magnetocaloric effect described through magnetic entropy change was correlated with magnetocapacitance formula. The theoretical investigation was carried out using a Heisenberg Hamiltonian considering the G-type antiferromagnetic structure with exchange interactions, in mean field approximation, between nearest-neighbor and next-nearest-neighbor magnetic Eu þ 2 ions. & 2011 Elsevier B.V. All rights reserved.
Keywords: Magnetocaloric effect Antiferromagnetic state equation Perovskite oxides Europium titanate
1. Introduction The magnetocaloric effect (MCE) was discovered by Warburg [1] in 1881 and the first potential application was in adiabatic refrigeration at low temperatures [2]. The MCE appears when a magnetic material presents temperature changes under the action of a magnetic field and is usually characterized by two thermodynamic quantities: DTad (the adiabatic temperature change) and DST (the isothermal entropy change). In the last fourteen years the interest in MCE were strongly renewed due to the discovery of the first giant magnetocaloric material reported by Pecharsky and Gschneidner [3] and to the possibility of its technological application in room temperature magnetic refrigeration. The DTad can be directly measured while DST can be only measured indirectly, usually through the heat capacity or magnetic measurements. From theoretical point of view, the total entropy can be calculated embodying several microscopic interactions parameters, which can be systematically studied and compared to the experimental data [4]. For example, in Gd1 xPrxAl2 (x¼ 0.25, 0.5 and 075) an inverse MCE was predicted and experimentally confirmed due to the antiparallel coupling between Gd and Pr moments and the crystalline electrical field interaction [5]; the influence of the magnetoelastic interaction on the onset of first order magnetic phase transition was described for ferrimagnetic systems [6]; the
n
Corresponding author. E-mail address: [email protected] (P.J. von Ranke).
0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.11.037
charge-ordering contribution [7] to the heat capacity and DST; the magnetic disorder problems [8] and spins fluctuations in 3d transition metal compounds [9] have already been addressed in the literature. Therefore, besides the technological interest, the MCE is being shown as a powerful tool to explain and to predict several properties of magnetic materials, especially those connected to the coupling between the magnetic and crystalline lattices. In what follows we discuss the use of the MCE for better understanding of the physics of magnetoelectric materials. One of the big challenges in magnetoelectric materials is to describe the microscopic mechanism, which leads the magnetic properties coupling to electrical ones. In a few magnetoelectric materials the so-called multiferroics, the electrical order (polarization) coexist with magnetic order (magnetization), in a single phase. These materials have high technological interest such as for sensitive sensors where induction of magnetization by an electrical field and vice-versa can be processed [10–11]. Katsufiji and Takagi investigated the magnetic field dependence of the dielectric constant of quantum paraelectric, type-G antiferromagnetic EuTiO3 compound and proposed a relation between the dielectric constant and spin correlation (the magnetocapacitance formula) [12]. Recently, using the Katsufiji and Takagi formulation, Kolodiazhnyi et al. [13] studied the magnetoelectric material EuZrO3 and concluded that the magnitude of the magnetodielectric coupling in EuZrO3 is much smaller than that of EuTiO3. Jiang and Wu [14] proposed a microscopic coupling mechanism between magnetism and dielectric properties in EuTiO3. The origin of the coupling interaction comes from the deformation
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dependence on the superexchange interaction between nearest Eu þ 2 ions, which is intermediate by oxygen. From the Jiang and Wu model, the Katsufiji and Takagi insight formulation of the magnetocapacitance formula was obtained analytically, in the framework of soft mode theory [15]. Investigations on the coupling between spins and phonons were also performed through first principle calculations [16] and used to explain the large infrared-active phonon splitting in ZnCr2O4 at Neel temperature [17]. Furthermore, first principle calculations confirmed that the behavior described by magnetocapacitance formula [12] in EuTiO3 is due to the coupling between the spin and infraredactive phonon [18]. However, as far as we know, the coupling between the magnetic and electric properties was never investigated from the manifestation of the MCE analysis. In this work, we focus on the magnetic and MCE properties of EuTiO3. The model Hamiltonian includes the exchange interactions between the nearest and next-nearest-neighbors Eu þ 2 ions and the Zeeman effect. The magnetocaloric potential DST was fully investigated, considering the magnetic field applied along (and no along) the antiferromagnetic axis in EuTiO3. The existence of inverse-MCE, a latent heat associated to a discontinuous antiferro-ferromagnetic phase transition and an inert magnetocaloric temperature interval (in which DST ¼ 0) were predicted and discussed on the light of temperature and field dependence of the magnetic configurations. Also, the dielectric properties of EuTiO3 were investigated, through the magnetocapacitance formula associated with the MCE.
2. Theory The EuTiO3 has a type-G antiferromagnetic structure and the magnetic moment comes from the 4f7 magnetic state in divalent Eu þ 2 ions. Fig. 1 shows the cubic perovskite spin arrangement of the EuTiO3 where each Eu þ 2 ion has six nearest-neighbors Eu þ 2 ions connected by exchange interaction J1 with opposite spins and twelve next-nearest-neighbor Eu þ 2 ions with the same spin direction, connected by exchange interaction J2. The O 2 ions are in midway between the next-nearest-neighbors Eu þ 2 ions. The magnetic Hamiltonian used to describe the EuTiO3 system is
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given by X !! X !! !X ! ! H¼ J1 J i J j J2 J i J j g mB B ð J aj þ J bj Þ: o i,j 4
½i,j
ð1Þ
j
The first and second terms describe the exchange interactions between nearest and next-nearest-neighbors, respectively. Therefore, /i, jS and [i, j] denote the summations over the nearest neighbors and next-nearest-neighbors, respectively. The last term ! represents the Zeeman interaction being B the external magnetic field applied parallel to the z axis, g is the Lande´ factor and mB the Bohr magneton. Under mean field approximation, the above Hamiltonian can be separated into two spins lattices (a) and (b) Hamiltonians, given by H ¼ Ha þHb
ð2Þ
where Ha ¼ Hxa J xa Hza J za ,
ð3Þ
Hb ¼ Hxb J xb Hzb J zb :
ð4Þ
The two magnetic lattices are coupled by the effective field components Hxa ¼ 6J 1 /J xb Sþ 12J 2 /J xa S,
ð5Þ
Hza ¼ g mB B þ6J 1 /J zb Sþ 12J 2 /J za S,
ð6Þ
Hxb ¼ 6J 1 /J xa Sþ 12J 2 /J xb S,
ð7Þ
Hzb ¼ g mB B þ6J 1 /J za Sþ 12J 2 /J zb S:
ð8Þ
For a given value of total angular momentum J the eigenvalues of Hamiltonians (3) and (4) can be analytically determined leading to the following mean values for the x and z-magnetic moment components for (a) and (b) sublattices: JHza BJ ðxa Þ /J za S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðHxa Þ2 þ ðHza Þ2
ð9Þ
JHxa BJ ðxa Þ , /J xa S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðHxa Þ2 þ ðHza Þ2
ð10Þ
JHzb BJ ðxb Þ /J zb S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðHxb Þ2 þ ðHzb Þ2
ð11Þ
JHxb BJ ðxb Þ : /J xb S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðHxb Þ2 þ ðHzb Þ2
ð12Þ
where BJ(xZ) is the Brillouin function, and, (Z ¼a, b) and k is the Boltzmann constant. The total magnetization dependence on the temperature and magnetic field can be directly calculated from the relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MðB,TÞ ¼ g mB ð/J xa S þ/J xb SÞ2 þ ð/J za S þ/J zb SÞ2 ð13Þ The total magnetic entropy is the sum of magnetic entropy in sublattices (a) and (b) and is given by " ! sin h ð2J þ 1=2JÞxa R ln xa BJ ðxa Þ Smag: ¼ 2 sin h ðxa =2JÞ # ! sin h ð2J þ1=2JÞxb ð14Þ þln xb BJ ðxb Þ , sin h ðxb =2JÞ Fig. 1. Schematic representation of the G-type antiferromagnetic crystal structure for EuTiO3, where (a) and (b) show different magnetic sites of Eu þ 2 ions. J1 and J2 represent the exchange interactions between nearest and next-nearest-neighbors, respectively.
where R¼8.314 J/mol K, is the universal gas constant. The factor ½, comes from the fact that we have half Eu þ 2 in sites (a) and the other half Eu þ 2 in sites (b), per mol of EuTiO3.
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The isothermal magnetic entropy change upon variation of external magnetic field (DST), is given by
DST ðB,TÞ ¼ Smag: ðB,TÞSmag: ðB ¼ 0,TÞ
ð15Þ
Alternatively, DST can be calculated and discussed from the integration of Maxwell relation Z B @M DST ðB,TÞ ¼ dB ð16Þ @T B 0 The spin correlation function between the nearest magnetic ions can be self-consistently calculated in mean field approximation !! / J i J j S ¼ /Jxa S/J xb Sþ /Jza S/J zb S ð17Þ This spin correlation function connects the magnetic properties to the electric ones in EuTiO3 through the magnetocapacitance relation [12]: !! eðB,TÞ ¼ eðB ¼ 0,TÞ½1 þ a/ J i J j S ð18Þ where a is the coupling constant between spin correlation function and dielectric constant, e.
Fig. 2. Diagram showing the main orientations of the magnetic moments in Eu þ 2 ions localized in magnetic sublattices (a) (up line) and (b) (down line), increasing temperature from left to right side. The initial antiferromagnetic configuration is aligned along the z-magnetic field direction. At T¼TSR the magnetic moment of (b) sublattice presents a discontinuous reorientation transitions. For T4 TSR both sublattices are aligned along the magnetic field direction.
3. Application and discussions In order to apply the above model to investigate the magnetocaloric effect in EuTiO3 we use J¼7/2 for the total angular moment in Eu þ 2 ions and the exchange parameters J1 ¼ 0.037 K and J2 ¼ 0.069 K (from Ref. 12). These exchange parameters lead to Ne el temperature TN ¼ 5.5 K. It is interesting to note that the nextnearest-neighbor interaction (J2), which is intermediated by the oxygen ions (see Fig. 1) is ferromagnetic with absolute value larger than that of the antiferromagnetic next-neighbor interaction (J1). From relations (9)–(12), we see that for a fixed temperature and magnetic field the x and z-magnetic moment components for (a) and (b) sublattices are coupled through the effective fields given by relations (5)–(8). The calculation of the four components /J x,z a,b S is performed by a numerical self-consistent procedure. Two cases are discussed in this work: 1] the magnetic moments in both (a) and (b) sublattices are aligned along the magnetic field direction (z-direction); 2] the magnetic moments are not aligned along the magnetic field direction. The first case is simulated considering an initial lattice magnetic configuration with/J xa S ¼ /J xb S ¼ 0 and the second case is simulated considering an initial lattice magnetic configuration with all components different of zero. The main configurations of the magnetic moments arrangements for the investigated cases 1] and 2] are presented in Figs. 2 and 3, respectively. Fig. 2 shows an outline of the magnetic moment configurations in sublattices (a) and (b) for different values of temperature. The column (1) is the initial magnetic moment configuration for nearest-neighbor Eu þ 2 pair at T¼0 K and B¼0 T. The column (2) shows a configuration where a magnetic field is applied along z-direction with T40 K; the 9/Jza S94 9/J zb S since the probability of occupation of magnetic moments along the z-positive magnetic field direction minimizes the energy. Increasing temperature till T¼TRS a reorientation (along magnetic field direction) in the magnetic moment of the Eu þ 2, belong to the b-sublattice, take place as shown in column (3). It is worth noticing that the intensity of magnetic moment in (a) sublattice decreases and the intensity of magnetic moment in (b) sublattice increase when the temperature increases from little below TRS, as depicted in columns (2) and (3). This behavior was ascribed to the fact stated above, where the ferromagnetic exchange interaction in EuTiO3 has absolute value larger than that of the antiferromagnetic one.
Fig. 3. Diagram showing the main orientations of the magnetic moments in Eu þ 2 ions localized in magnetic sublattices (a) (up line) and (b) (down line), increasing temperature from left to right side. The initial antiferromagnetic configuration is not aligned along the z-magnetic field direction. At T ¼T0 the magnetic moments in both sublattices are aligned along the magnetic field direction.
It should be noted that the existence of magnetic moment orientation, at least as a first order transition process, depend on the magnetic field intensity. Above TRS both sublattices present the same magnetic moment which decrease with the same rate increasing temperature, as shown in the column (4). Fig. 3 shows an outline of the magnetic moment configurations in sublattices (a) and (b) when the magnetic moments do not have the initial z-direction of the applied magnetic field, so the evolution of the spin-flop phase occurs. The column (1) shows the anti-parallel moments in nearest-neighbor Eu þ 2 ions at T¼0 K and B ¼0 T. The column (2) shows the angular displacement of the magnetic moment in both sublattices when the magnetic field is applied along the z-direction (in this situation we still have T¼0 K). It should be noted (in column (2) of the diagram) that applied magnetic field produces a torque in such way to increases /J xa S, i.e., the x-component of the magnetic moment of Eu þ 2 in sublattice (a). Also we can observe that /Jxb S ¼ /J xa S and /J za S ¼ /J zb S. The columns (3), (4), (5) and (6) of the diagram show the effect of temperature increases on the magnetic moments (spin-flop phase and the antiferro-ferromagnetic transition). In (3) both absolute values of the components /J xb Sand /J xa Sdecrease with the same rate as temperature increases, and the components /Jza S ¼ /Jzb S remain fixed. As the temperature continues increasing, the x-components of magnetic moments go continually to zero. At T¼T0 (see column (5)) both magnetic moments are aligned along the z-magnetic field direction. It is
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interesting to note that in presence of applied magnetic field, the z-components are equals /Jza S ¼ /Jzb S and do not decrease till T¼T0. Increasing the temperature for T4T0, the z-components /Jza S and /J zb S decrease continuously to zero, as shown in column (6) of the diagram. Fig. 4 shows the detailed temperature dependence at B ¼0.8 T on the x, and z-components of magnetic moments of Eu þ 2 in sublattices (a) and (b). The symbols represent the numerical results when the antiferromagnetic moments are aligned along z-magnetic field direction (case 1) and the solid curves represent the results when the antiferromagnetic moments are initially not aligned along z-magnetic field direction (case 2). In case 1, only the components /J za S (circles) and /J zb S(squares) exist. The absolute values of /Jzb Sdecreases faster than that of /J za Sas expected and at T¼TRS (TRS 2 K, for B ¼0.8 T) the magnetic moment of Eu þ 2 in sublattice (b) presents a discontinuous spin reorientation transition from down to up z-magnetic field direction (see the column 3, in Fig. 2). In case 2, /J za S ¼ /J zb S 2 and not changes till T¼T0 (T0 4.6 K, for B¼0.8 T). The components /Jxa S and /J xb S present the same absolute values with opposite sign from T¼0 to T¼T0 (for T4T0, /J xa S ¼ /Jxb S ¼ 0, see columns (5) and (6), in Fig. 3). It worth noticing that, at T¼ 0 K, /Jza S0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /Jxa S20 þ /Jza S20 7=2. Also we and /Jxa S0 2:89 leading to should mention that the values /Jza S0 , /Jxa S0 , as well as, T0 and TSR depend on the applied magnetic field intensity. Fig. 5 shows the temperature dependence of the magnetization in EuTiO3 for applied magnetic fields B¼0.5, 1.0 and 2.5 T. The circles and squares curves represent M vs. T when the antiferromagnetic axis is aligned along the z-magnetic field direction with B¼0.5 and B¼1.0 T, respectively. For B¼0.5 T, the spin reorientation transition occurs at TSR 4.2 K and, for B¼ 1.0 T the two sublattices (a) and (b) present a ferromagnetic configuration, i.e., B¼1.0 T, is high enough to produce spin reorientation at T¼0 K. It worth noticing that TSR decreases when the magnetic field increases. The solid curves for B¼0.5 T and B¼1 T show the M vs. T when the applied magnetic field along z-direction is not initially aligned with the magnetic moments as displayed in the diagram of Fig. 3, column (2). As the temperature increases the magnetization remains constant; for B¼0.5 T, M 2.5 mB from T¼0 K to T0 ¼5.2 K and for B¼ 1.0 T, M 4.9 mB from T¼ 0 K to T0 ¼3.9 K. The temperature T0 is magnetic field dependent and
Fig. 4. Magnetic moment components vs. T at B¼ 0.8 T. For antiferromagnetic axis along the magnetic field direction /J za S (circles ) and /J zb S (squares). For antiferromagnetic axis not along the magnetic field direction (solid lines) with the components indicated by the arrows.
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Fig. 5. Magnetization vs. T for B ¼0.5, 1.0 and 2.5 T in EuTiO3. The solid curves were calculated for antiferromagnetic axis not aligned along the magnetic field direction and the curves represented by symbols were calculated considering the antiferromagnetic axis along the magnetic field direction.
decreases as B increases. The constant value of M in the temperature interval from T¼ 0 K to T¼T0 can be verified from relation (13). As shown in the diagram, columns (2), (3), (4) and (5), in Fig. 3, the x-components of magnetic moments in sublattices (a) and (b) cancel, i.e. /Jxb S þ /J xa S ¼ 0 and the total z-components remain constant, so from relation (13), M¼constant. This magnetic behavior has important consequence in the magnetocaloric effect, as will be discussed below. Finally, for B¼2.5 T nor TSR neither T0 exist more since this magnetic field is strong enough to induces a ferromagnetic arrangement in EuTiO3. So increasing temperature only a smoothly decreasing in M is observed. Fig. 6a and (b) shows M vs. B calculated for different values of temperature in EuTiO3, considering the antiferromagnetic axis not along and (along) the z-applied magnetic field direction. In Fig. 6a the squares and circles represent the experimental data [19] measured at T¼1.3 and 4.2 K, respectively. We can observe that for temperatures below 5 K an unusual behavior occurs, i.e., the magnetization does not change with temperature in a certain temperature-field interval. For example, all isothermic curves from T¼0 K till T¼4 K, have the same values in the magnetic field interval from B ¼0 till B ¼1 T (see the dotted line in Fig. 6a). For higher values of temperature and magnetic field the usual behavior is restored, where the M vs. B isothermal curves have M values decreasing with temperature for all magnetic field intervals. The unusual behavior will produces an anomalous magnetocaloric effect since, for example, the area between two isothermic curves dA¼ [M(T¼0 K, B)–M(T¼4 K, B)]dB ¼0 in the magnetic field interval from B ¼0 to B ¼1 T, as will be discussed below. In Fig. 6b, the isothermic curves M vs. B, were calculated applying the magnetic field along the antiferromagnetic axis and in this case no x-components of magnetic moments exist as discussed above. For isothermic curves below T¼6 K we can observe jumps in M vs. B at B¼ 0.9, 0.82, 0.67, 0.53 and 0.3 T for T¼1, 2, 3, 4 and 5 K, respectively. These jumps indicate the spin reorientation of the magnetic sublattice (b) from down to up z-magnetic field direction. For temperatures higher than T¼6 K the usual behavior can be noted where the magnetization always decreases as temperature increases. For temperatures lower than T¼5 K an unusual behavior can be observed which will lead to an inverse magnetocaloric effect as will be discussed below. Fig. 7 shows the calculated magnetocaloric effect in EuTiO3, i.e., DST vs. T under magnetic field changes from zero to B ¼0.5,
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Fig. 6. Magnetization vs. B for T¼1 to 7 K in EuTiO3. The magnetic field is not applied along the magnetic axis of the antiferromagnetic order (Fig. 6a), the squares and circles represent experimental data at T¼1.3 and 4.2 K, respectively. The magnetic field is applied along the magnetic axis (Fig. 6b).
Fig. 7. Temperature dependence of DST for EuTiO3 for a magnetic-field change from 0 to 0.5 T, 0 to 0.8 T, 0 to 1.0 T and 0 to 2.5 T. The solid and symbols lines represent the theoretical results for magnetic field applied not along and along the magnetic axis, respectively.
0.8, 1.0 and 2.5 T. This calculation was performed using the relation (15). The solid curves represent the calculation considering the case 2, where the magnetic field is not initially aligned
with the magnetic Eu þ 2 moments. Therefore, as discussed above, the magnetization M is constant from T¼0 to T¼T0 and from Maxwell relation (16) no magnetocaloric effect is expected (DST ¼0). In this way, we observe DST ¼0 below T0 ¼5.1, 4.6 and 4.0 K for DB ¼0.5, 0.8 and 1.0 T, respectively (solid curves). The arrows point out the same T0 ¼4.6 K in Fig. 4 and in Fig. 7 calculated for B ¼0.8 T. For DB¼1.0 T and T4T0 ¼4.0 K an increase occurs in DST curve since M is no longer constant. Alternatively, it can be confirmed analyzing Fig. 6a where for T44 K, for example, T¼5 K we have an area no null between two consecutives curves, i.e., dA¼[M(T¼4 K, B)–M(T¼5 K, B)]dB40, which leads DST 40 in accordance with Maxwell relation (16). The maximum values of DST vs. T occur at Ne´el temperature TN ¼5.5 K. For case 1, where the applied magnetic field is along the magnetic moments an inverse magnetocaloric effect (DST o0) is predicted to occurs in EuTiO3. This inverse magnetocaloric effect is expected to occur in this situation, since the magnetization (the order parameter) increases with temperature at low temperature (see Fig. 5, circles-curve, for B¼0.5 T). However, only with the M vs. T curve we cannot explain the inverse magnetocaloric effect that occurs for B ¼1 T, as shown in Fig. 7. This case should be investigated carefully considering the M vs. B – isothermic curves (see Fig. 6b). At low temperatures the area between two consecutive isothermic curves is negative, for example, dA¼[M(T¼1 K, B)–M(T¼2 K, B)]dBo 0 when integrated from B¼ 0 to B ¼1 T (the vertical dotted line in Fig. 6b). In general way, for all temperature interval (Ti þ 1 4Ti) where dA¼[M(Ti, B)– M(Ti þ 1, B)]dB o0 an inverse magnetocaloric occurs, and if dA40 a normal magnetocaloric effect (DST 40) occur. In our investigated case, dA¼[M(T¼5 K, B)–M(T¼6 K, B)]dB40, so DST 40 is expected in this temperature interval (see DST vs. T for DB¼ 1.0, in Fig. 7). For DB¼2.5 T we predict a normal magnetocaloric effect in EuTiO3 for all temperature range. It can be verified displacing the vertical dot line (in Fig. 6b) from B¼1 T to B ¼2.5 T and noticing that dA40 for all temperature interval. Therefore, in antiferromagnetic materials is necessary a complete analyze of the spin reorientation using the isothermic-M vs. B curves to predict the possible existence of an inverse magnetocaloric effect. Besides, the spin reorientation can also be determined from the DST vs. T curves. The arrow in Fig. 7 indicate the temperature (TSR ¼2 K) where the spin reorientation transition is predicted to occur in EuTiO3 for B ¼0.8 T (see also this transition in Fig. 4, dotted line). The discontinuity of the magnetization also appears for B¼ 0.5 T at TSR 4.2 K (see transition in Fig. 5, dotted line). Associated with this discontinuity in the magnetization DM 2 mB a latent heat can be calculated L¼TSR. DST 3.54 J/mol (DST is the discontinuity in magnetic entropy at TSR, see Fig. 7, dotted line in circle-curve). When the magnetic field increases the latent heat decrease, for example, L¼ 2.2 J/mol for B ¼0.8 T (see in Fig. 7 the discontinuity DST at TSR ¼2 K, square-curve). !! Fig. 8 shows in left axis the spin correlation function, / J i J j S vs. T (circle curves) and in right axis the DST vs. T (solid curves) for different values of applied magnetic field B ¼0.5, 0.8 and 1.0 T. It worth noticing the systematic magnetic field behavior of !! / J i J j S, where it reaches the maximum at T0, the arrow in Fig. 8 indicates the value of T0 for B ¼1 T. Therefore, considering the magnetocapacitance (relation 18) we can predict two anomalous physical behavior: 1] in temperature interval where the dielectric constant increases (for a given magnetic field intensity B0), the EuTiO3 does not change heat under isothermal magnetic field change from zero to B0, and 2] the temperature of EuTiO3 does not changes upon magnetic field change from zero to B0 in an adiabatic process. These two model predictions are valid under the following condition: 1] the coupling constant a is positive (Katsufuji and Takagi [12] determined by adjusting a ¼2.74 10 3 for EuTiO3); 2] the lattice entropy is magnetic
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temperature (where the spin pair correlation between nearest (a)–(b) sites peaks12), may have an impact on experimental investigations in order to design new materials with anisotropic magnetocaloric effects. As far as we know, no magnetocaloric measurements have been carried out on EuTiO3 compound and single crystal experiments showing the anisotropy would be particularly important to compare with our theoretical predictions.
Acknowledgments We acknowledge financial support from CNPq – Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico – Brazil, FAPERJ – Fundac- a~ o de Amparo a Pesquisa do Estado do Rio de Janeiro, CAPES – Coordenac- a~ o de Aperfeic- oamento do Pessoal de Nı´vel Superior, and FAPESP – Fundac- a~ o de Amparo a Pesquisa do Estado de Sa~ o Paulo. Fig. 8. Temperature dependence of /Ji JjS for EuTiO3 for a magnetic-field B¼0.5, 0.8 and 1.0 T (circles)- left axis . The solid lines-(right axis) represent DST, as in Fig. 7. The inset shows the T0 dependence on B.
field independent (in this way, the total entropy change is equal to the magnetic entropy change in an isothermal process). Another important aspect is that the temperature T0 can be tuned by the magnetic field intensity. The inset in Fig. 8 shows the T0 dependence on magnetic field, increasing B the T0 decreases smoothly.
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[6] [7]
4. Final comments The thermal and magnetic properties in EuTiO3 were theoretically investigated focusing on the magnetocaloric effect. Two case of magnetic antiferromagnetic anisotropy were explored: when the magnetic field is applied along (and not along) the antiferromagnetic axis. The magnetocaloric effect, studied by the DST vs. T, reveled to be a powerful tool to reveal several behaviors about the antiferromagnetic EuTiO3. From DST vs. T curves the inverse magnetocaloric effect and the latent heat associated to the spin reorientation transition were predicted for low magnetic field changes. Also, the special possibility where the magnetic system EuTiO3 does not change heat below the described T0
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
E. Warburg, Annals of Physics 13 (1881) 141. P. Debye Ann., Physik 81 (1926) 1154. V.K. Pecharsky, K.A. Gschneidner Jr., Physical Review Letters 78 (1997) 4494. N.A. de Oliveira, P.J. von Ranke, Physics Report 489 (2010) 89. V.S.R. de Souza, A. Magnus, G. Carvalho, E.J.R. Plaza, B.P. Alho, J.C.G. Tedesco, A.A. Coelho, N.A. de Oliveira, P.J. von Ranke, Journal of Magnetism and Magnetic Materials 323 (2011) 794. B.P. Alho, N.A. de Oliveira, V.S.R. de Sousa, E.J.R. Plaza, A. Magnus, G. Carvalho, P.J. von Ranke, Journal of Physics Condensed Matter 22 (2010) 486008. P. Sande, L.E. Hueso, D.R. Miguens, J. Rivas, F. Rivadulla, M.A. Lopez-Quintela, Applied Physics Letters 79 (2001) 2040. E.P. Nobrega, N.A. de Oliveira, P.J. von Ranke, A. Troper, Physical Review B 74 (2006) 144429. H. Yamada, T. Goto, Physica B 346 (2004) 104. Nicola A. Spaldin, Manfred Fiebig, Science 309 (2005) 391. Manfred Fiebig, Journal of Physics D: Applied Physics 38 (2005) R123–R152. T. Katsufuji, H. Takagi, Physical Review B 64 (2001) 054415. T. Kolodiazhnyi, F. Fujita, L. Wang, Y. Zong, K. Tanaka, Y. Sakka, E. TakayamaMuromachi, Applied Physics Letters 96 (2010) 252901. Jiang Qing, Wu Hua, Chinese Physics 11 (2002) 1303. M.E. Lines, A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Oxford, Clarendon, 1977. Craig J. Fennie, Karin M. Rabe, Physical Review Letters 96 (2006) 205505. A.B. Sushkov, O. Tchernyshyov, W. Ratcliff II, S.W. Cheong, H.D. Drew, Physical Review Letters 94 (2005) 137202. Craig J. Fennie, Karin M. Rabe, Physical Review Letters 97 (2006) 267602. T.R. McGuire, M.W. Shafer, R.J. Joenk, H.A. Alperin, S.J. Pickart, Journal of Applied Physics 37 (1966) 981.
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The influence of magnetic and electric coupling properties on the magnetocaloric effect in quantum paraelectric EuTiO3 P.J. von Ranke a,n, B.P. Alho a, E.P. No´brega a, V.S.R. de Sousa b, T.S.T. Alvarenga a, A. Magnus G. Carvalho c, N.A. de Oliveira a a
Instituto de Fı´sica, Universidade do Estado do Rio de Janeiro—UERJ, Rua Sa~ o Francisco Xavier, 524, 20550-013 RJ, Brazil Instituto de Fı´sica Gleb Wataghin, Universidade Estadual de Campinas—UNICAMP, 13083-970 Campinas, SP, Brazil c ~ de Metrologia de Materiais (DIMAT), Instituto Nacional de Metrologia, Normalizac- a~ o e Qualidade Industrial (INMETRO), Duque de Caxias, RJ 25250-020, Brazil Divisao b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 October 2011 Received in revised form 6 November 2011 Available online 2 December 2011
We report on the magnetic and magnetocaloric effect calculations in antiferromagnetic perovskite-type EuTiO3. From the isothermal magnetic entropy change calculated upon low magnetic field changes (below 1 T) several results were predicted: inverse magnetocaloric effect, latent heat associated to spin AFM-FM reorientation transition and a temperature interval (controlled by magnetic field) where the EuTiO3 does not change heat in an isothermic process. The magnetocaloric effect described through magnetic entropy change was correlated with magnetocapacitance formula. The theoretical investigation was carried out using a Heisenberg Hamiltonian considering the G-type antiferromagnetic structure with exchange interactions, in mean field approximation, between nearest-neighbor and next-nearest-neighbor magnetic Eu þ 2 ions. & 2011 Elsevier B.V. All rights reserved.
Keywords: Magnetocaloric effect Antiferromagnetic state equation Perovskite oxides Europium titanate
1. Introduction The magnetocaloric effect (MCE) was discovered by Warburg [1] in 1881 and the first potential application was in adiabatic refrigeration at low temperatures [2]. The MCE appears when a magnetic material presents temperature changes under the action of a magnetic field and is usually characterized by two thermodynamic quantities: DTad (the adiabatic temperature change) and DST (the isothermal entropy change). In the last fourteen years the interest in MCE were strongly renewed due to the discovery of the first giant magnetocaloric material reported by Pecharsky and Gschneidner [3] and to the possibility of its technological application in room temperature magnetic refrigeration. The DTad can be directly measured while DST can be only measured indirectly, usually through the heat capacity or magnetic measurements. From theoretical point of view, the total entropy can be calculated embodying several microscopic interactions parameters, which can be systematically studied and compared to the experimental data [4]. For example, in Gd1 xPrxAl2 (x¼ 0.25, 0.5 and 075) an inverse MCE was predicted and experimentally confirmed due to the antiparallel coupling between Gd and Pr moments and the crystalline electrical field interaction [5]; the influence of the magnetoelastic interaction on the onset of first order magnetic phase transition was described for ferrimagnetic systems [6]; the
n
Corresponding author. E-mail address: [email protected] (P.J. von Ranke).
0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.11.037
charge-ordering contribution [7] to the heat capacity and DST; the magnetic disorder problems [8] and spins fluctuations in 3d transition metal compounds [9] have already been addressed in the literature. Therefore, besides the technological interest, the MCE is being shown as a powerful tool to explain and to predict several properties of magnetic materials, especially those connected to the coupling between the magnetic and crystalline lattices. In what follows we discuss the use of the MCE for better understanding of the physics of magnetoelectric materials. One of the big challenges in magnetoelectric materials is to describe the microscopic mechanism, which leads the magnetic properties coupling to electrical ones. In a few magnetoelectric materials the so-called multiferroics, the electrical order (polarization) coexist with magnetic order (magnetization), in a single phase. These materials have high technological interest such as for sensitive sensors where induction of magnetization by an electrical field and vice-versa can be processed [10–11]. Katsufiji and Takagi investigated the magnetic field dependence of the dielectric constant of quantum paraelectric, type-G antiferromagnetic EuTiO3 compound and proposed a relation between the dielectric constant and spin correlation (the magnetocapacitance formula) [12]. Recently, using the Katsufiji and Takagi formulation, Kolodiazhnyi et al. [13] studied the magnetoelectric material EuZrO3 and concluded that the magnitude of the magnetodielectric coupling in EuZrO3 is much smaller than that of EuTiO3. Jiang and Wu [14] proposed a microscopic coupling mechanism between magnetism and dielectric properties in EuTiO3. The origin of the coupling interaction comes from the deformation
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dependence on the superexchange interaction between nearest Eu þ 2 ions, which is intermediate by oxygen. From the Jiang and Wu model, the Katsufiji and Takagi insight formulation of the magnetocapacitance formula was obtained analytically, in the framework of soft mode theory [15]. Investigations on the coupling between spins and phonons were also performed through first principle calculations [16] and used to explain the large infrared-active phonon splitting in ZnCr2O4 at Neel temperature [17]. Furthermore, first principle calculations confirmed that the behavior described by magnetocapacitance formula [12] in EuTiO3 is due to the coupling between the spin and infraredactive phonon [18]. However, as far as we know, the coupling between the magnetic and electric properties was never investigated from the manifestation of the MCE analysis. In this work, we focus on the magnetic and MCE properties of EuTiO3. The model Hamiltonian includes the exchange interactions between the nearest and next-nearest-neighbors Eu þ 2 ions and the Zeeman effect. The magnetocaloric potential DST was fully investigated, considering the magnetic field applied along (and no along) the antiferromagnetic axis in EuTiO3. The existence of inverse-MCE, a latent heat associated to a discontinuous antiferro-ferromagnetic phase transition and an inert magnetocaloric temperature interval (in which DST ¼ 0) were predicted and discussed on the light of temperature and field dependence of the magnetic configurations. Also, the dielectric properties of EuTiO3 were investigated, through the magnetocapacitance formula associated with the MCE.
2. Theory The EuTiO3 has a type-G antiferromagnetic structure and the magnetic moment comes from the 4f7 magnetic state in divalent Eu þ 2 ions. Fig. 1 shows the cubic perovskite spin arrangement of the EuTiO3 where each Eu þ 2 ion has six nearest-neighbors Eu þ 2 ions connected by exchange interaction J1 with opposite spins and twelve next-nearest-neighbor Eu þ 2 ions with the same spin direction, connected by exchange interaction J2. The O 2 ions are in midway between the next-nearest-neighbors Eu þ 2 ions. The magnetic Hamiltonian used to describe the EuTiO3 system is
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given by X !! X !! !X ! ! H¼ J1 J i J j J2 J i J j g mB B ð J aj þ J bj Þ: o i,j 4
½i,j
ð1Þ
j
The first and second terms describe the exchange interactions between nearest and next-nearest-neighbors, respectively. Therefore, /i, jS and [i, j] denote the summations over the nearest neighbors and next-nearest-neighbors, respectively. The last term ! represents the Zeeman interaction being B the external magnetic field applied parallel to the z axis, g is the Lande´ factor and mB the Bohr magneton. Under mean field approximation, the above Hamiltonian can be separated into two spins lattices (a) and (b) Hamiltonians, given by H ¼ Ha þHb
ð2Þ
where Ha ¼ Hxa J xa Hza J za ,
ð3Þ
Hb ¼ Hxb J xb Hzb J zb :
ð4Þ
The two magnetic lattices are coupled by the effective field components Hxa ¼ 6J 1 /J xb Sþ 12J 2 /J xa S,
ð5Þ
Hza ¼ g mB B þ6J 1 /J zb Sþ 12J 2 /J za S,
ð6Þ
Hxb ¼ 6J 1 /J xa Sþ 12J 2 /J xb S,
ð7Þ
Hzb ¼ g mB B þ6J 1 /J za Sþ 12J 2 /J zb S:
ð8Þ
For a given value of total angular momentum J the eigenvalues of Hamiltonians (3) and (4) can be analytically determined leading to the following mean values for the x and z-magnetic moment components for (a) and (b) sublattices: JHza BJ ðxa Þ /J za S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðHxa Þ2 þ ðHza Þ2
ð9Þ
JHxa BJ ðxa Þ , /J xa S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðHxa Þ2 þ ðHza Þ2
ð10Þ
JHzb BJ ðxb Þ /J zb S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðHxb Þ2 þ ðHzb Þ2
ð11Þ
JHxb BJ ðxb Þ : /J xb S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðHxb Þ2 þ ðHzb Þ2
ð12Þ
where BJ(xZ) is the Brillouin function, and, (Z ¼a, b) and k is the Boltzmann constant. The total magnetization dependence on the temperature and magnetic field can be directly calculated from the relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MðB,TÞ ¼ g mB ð/J xa S þ/J xb SÞ2 þ ð/J za S þ/J zb SÞ2 ð13Þ The total magnetic entropy is the sum of magnetic entropy in sublattices (a) and (b) and is given by " ! sin h ð2J þ 1=2JÞxa R ln xa BJ ðxa Þ Smag: ¼ 2 sin h ðxa =2JÞ # ! sin h ð2J þ1=2JÞxb ð14Þ þln xb BJ ðxb Þ , sin h ðxb =2JÞ Fig. 1. Schematic representation of the G-type antiferromagnetic crystal structure for EuTiO3, where (a) and (b) show different magnetic sites of Eu þ 2 ions. J1 and J2 represent the exchange interactions between nearest and next-nearest-neighbors, respectively.
where R¼8.314 J/mol K, is the universal gas constant. The factor ½, comes from the fact that we have half Eu þ 2 in sites (a) and the other half Eu þ 2 in sites (b), per mol of EuTiO3.
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The isothermal magnetic entropy change upon variation of external magnetic field (DST), is given by
DST ðB,TÞ ¼ Smag: ðB,TÞSmag: ðB ¼ 0,TÞ
ð15Þ
Alternatively, DST can be calculated and discussed from the integration of Maxwell relation Z B @M DST ðB,TÞ ¼ dB ð16Þ @T B 0 The spin correlation function between the nearest magnetic ions can be self-consistently calculated in mean field approximation !! / J i J j S ¼ /Jxa S/J xb Sþ /Jza S/J zb S ð17Þ This spin correlation function connects the magnetic properties to the electric ones in EuTiO3 through the magnetocapacitance relation [12]: !! eðB,TÞ ¼ eðB ¼ 0,TÞ½1 þ a/ J i J j S ð18Þ where a is the coupling constant between spin correlation function and dielectric constant, e.
Fig. 2. Diagram showing the main orientations of the magnetic moments in Eu þ 2 ions localized in magnetic sublattices (a) (up line) and (b) (down line), increasing temperature from left to right side. The initial antiferromagnetic configuration is aligned along the z-magnetic field direction. At T¼TSR the magnetic moment of (b) sublattice presents a discontinuous reorientation transitions. For T4 TSR both sublattices are aligned along the magnetic field direction.
3. Application and discussions In order to apply the above model to investigate the magnetocaloric effect in EuTiO3 we use J¼7/2 for the total angular moment in Eu þ 2 ions and the exchange parameters J1 ¼ 0.037 K and J2 ¼ 0.069 K (from Ref. 12). These exchange parameters lead to Ne el temperature TN ¼ 5.5 K. It is interesting to note that the nextnearest-neighbor interaction (J2), which is intermediated by the oxygen ions (see Fig. 1) is ferromagnetic with absolute value larger than that of the antiferromagnetic next-neighbor interaction (J1). From relations (9)–(12), we see that for a fixed temperature and magnetic field the x and z-magnetic moment components for (a) and (b) sublattices are coupled through the effective fields given by relations (5)–(8). The calculation of the four components /J x,z a,b S is performed by a numerical self-consistent procedure. Two cases are discussed in this work: 1] the magnetic moments in both (a) and (b) sublattices are aligned along the magnetic field direction (z-direction); 2] the magnetic moments are not aligned along the magnetic field direction. The first case is simulated considering an initial lattice magnetic configuration with/J xa S ¼ /J xb S ¼ 0 and the second case is simulated considering an initial lattice magnetic configuration with all components different of zero. The main configurations of the magnetic moments arrangements for the investigated cases 1] and 2] are presented in Figs. 2 and 3, respectively. Fig. 2 shows an outline of the magnetic moment configurations in sublattices (a) and (b) for different values of temperature. The column (1) is the initial magnetic moment configuration for nearest-neighbor Eu þ 2 pair at T¼0 K and B¼0 T. The column (2) shows a configuration where a magnetic field is applied along z-direction with T40 K; the 9/Jza S94 9/J zb S since the probability of occupation of magnetic moments along the z-positive magnetic field direction minimizes the energy. Increasing temperature till T¼TRS a reorientation (along magnetic field direction) in the magnetic moment of the Eu þ 2, belong to the b-sublattice, take place as shown in column (3). It is worth noticing that the intensity of magnetic moment in (a) sublattice decreases and the intensity of magnetic moment in (b) sublattice increase when the temperature increases from little below TRS, as depicted in columns (2) and (3). This behavior was ascribed to the fact stated above, where the ferromagnetic exchange interaction in EuTiO3 has absolute value larger than that of the antiferromagnetic one.
Fig. 3. Diagram showing the main orientations of the magnetic moments in Eu þ 2 ions localized in magnetic sublattices (a) (up line) and (b) (down line), increasing temperature from left to right side. The initial antiferromagnetic configuration is not aligned along the z-magnetic field direction. At T ¼T0 the magnetic moments in both sublattices are aligned along the magnetic field direction.
It should be noted that the existence of magnetic moment orientation, at least as a first order transition process, depend on the magnetic field intensity. Above TRS both sublattices present the same magnetic moment which decrease with the same rate increasing temperature, as shown in the column (4). Fig. 3 shows an outline of the magnetic moment configurations in sublattices (a) and (b) when the magnetic moments do not have the initial z-direction of the applied magnetic field, so the evolution of the spin-flop phase occurs. The column (1) shows the anti-parallel moments in nearest-neighbor Eu þ 2 ions at T¼0 K and B ¼0 T. The column (2) shows the angular displacement of the magnetic moment in both sublattices when the magnetic field is applied along the z-direction (in this situation we still have T¼0 K). It should be noted (in column (2) of the diagram) that applied magnetic field produces a torque in such way to increases /J xa S, i.e., the x-component of the magnetic moment of Eu þ 2 in sublattice (a). Also we can observe that /Jxb S ¼ /J xa S and /J za S ¼ /J zb S. The columns (3), (4), (5) and (6) of the diagram show the effect of temperature increases on the magnetic moments (spin-flop phase and the antiferro-ferromagnetic transition). In (3) both absolute values of the components /J xb Sand /J xa Sdecrease with the same rate as temperature increases, and the components /Jza S ¼ /Jzb S remain fixed. As the temperature continues increasing, the x-components of magnetic moments go continually to zero. At T¼T0 (see column (5)) both magnetic moments are aligned along the z-magnetic field direction. It is
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interesting to note that in presence of applied magnetic field, the z-components are equals /Jza S ¼ /Jzb S and do not decrease till T¼T0. Increasing the temperature for T4T0, the z-components /Jza S and /J zb S decrease continuously to zero, as shown in column (6) of the diagram. Fig. 4 shows the detailed temperature dependence at B ¼0.8 T on the x, and z-components of magnetic moments of Eu þ 2 in sublattices (a) and (b). The symbols represent the numerical results when the antiferromagnetic moments are aligned along z-magnetic field direction (case 1) and the solid curves represent the results when the antiferromagnetic moments are initially not aligned along z-magnetic field direction (case 2). In case 1, only the components /J za S (circles) and /J zb S(squares) exist. The absolute values of /Jzb Sdecreases faster than that of /J za Sas expected and at T¼TRS (TRS 2 K, for B ¼0.8 T) the magnetic moment of Eu þ 2 in sublattice (b) presents a discontinuous spin reorientation transition from down to up z-magnetic field direction (see the column 3, in Fig. 2). In case 2, /J za S ¼ /J zb S 2 and not changes till T¼T0 (T0 4.6 K, for B¼0.8 T). The components /Jxa S and /J xb S present the same absolute values with opposite sign from T¼0 to T¼T0 (for T4T0, /J xa S ¼ /Jxb S ¼ 0, see columns (5) and (6), in Fig. 3). It worth noticing that, at T¼ 0 K, /Jza S0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /Jxa S20 þ /Jza S20 7=2. Also we and /Jxa S0 2:89 leading to should mention that the values /Jza S0 , /Jxa S0 , as well as, T0 and TSR depend on the applied magnetic field intensity. Fig. 5 shows the temperature dependence of the magnetization in EuTiO3 for applied magnetic fields B¼0.5, 1.0 and 2.5 T. The circles and squares curves represent M vs. T when the antiferromagnetic axis is aligned along the z-magnetic field direction with B¼0.5 and B¼1.0 T, respectively. For B¼0.5 T, the spin reorientation transition occurs at TSR 4.2 K and, for B¼ 1.0 T the two sublattices (a) and (b) present a ferromagnetic configuration, i.e., B¼1.0 T, is high enough to produce spin reorientation at T¼0 K. It worth noticing that TSR decreases when the magnetic field increases. The solid curves for B¼0.5 T and B¼1 T show the M vs. T when the applied magnetic field along z-direction is not initially aligned with the magnetic moments as displayed in the diagram of Fig. 3, column (2). As the temperature increases the magnetization remains constant; for B¼0.5 T, M 2.5 mB from T¼0 K to T0 ¼5.2 K and for B¼ 1.0 T, M 4.9 mB from T¼ 0 K to T0 ¼3.9 K. The temperature T0 is magnetic field dependent and
Fig. 4. Magnetic moment components vs. T at B¼ 0.8 T. For antiferromagnetic axis along the magnetic field direction /J za S (circles ) and /J zb S (squares). For antiferromagnetic axis not along the magnetic field direction (solid lines) with the components indicated by the arrows.
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Fig. 5. Magnetization vs. T for B ¼0.5, 1.0 and 2.5 T in EuTiO3. The solid curves were calculated for antiferromagnetic axis not aligned along the magnetic field direction and the curves represented by symbols were calculated considering the antiferromagnetic axis along the magnetic field direction.
decreases as B increases. The constant value of M in the temperature interval from T¼ 0 K to T¼T0 can be verified from relation (13). As shown in the diagram, columns (2), (3), (4) and (5), in Fig. 3, the x-components of magnetic moments in sublattices (a) and (b) cancel, i.e. /Jxb S þ /J xa S ¼ 0 and the total z-components remain constant, so from relation (13), M¼constant. This magnetic behavior has important consequence in the magnetocaloric effect, as will be discussed below. Finally, for B¼2.5 T nor TSR neither T0 exist more since this magnetic field is strong enough to induces a ferromagnetic arrangement in EuTiO3. So increasing temperature only a smoothly decreasing in M is observed. Fig. 6a and (b) shows M vs. B calculated for different values of temperature in EuTiO3, considering the antiferromagnetic axis not along and (along) the z-applied magnetic field direction. In Fig. 6a the squares and circles represent the experimental data [19] measured at T¼1.3 and 4.2 K, respectively. We can observe that for temperatures below 5 K an unusual behavior occurs, i.e., the magnetization does not change with temperature in a certain temperature-field interval. For example, all isothermic curves from T¼0 K till T¼4 K, have the same values in the magnetic field interval from B ¼0 till B ¼1 T (see the dotted line in Fig. 6a). For higher values of temperature and magnetic field the usual behavior is restored, where the M vs. B isothermal curves have M values decreasing with temperature for all magnetic field intervals. The unusual behavior will produces an anomalous magnetocaloric effect since, for example, the area between two isothermic curves dA¼ [M(T¼0 K, B)–M(T¼4 K, B)]dB ¼0 in the magnetic field interval from B ¼0 to B ¼1 T, as will be discussed below. In Fig. 6b, the isothermic curves M vs. B, were calculated applying the magnetic field along the antiferromagnetic axis and in this case no x-components of magnetic moments exist as discussed above. For isothermic curves below T¼6 K we can observe jumps in M vs. B at B¼ 0.9, 0.82, 0.67, 0.53 and 0.3 T for T¼1, 2, 3, 4 and 5 K, respectively. These jumps indicate the spin reorientation of the magnetic sublattice (b) from down to up z-magnetic field direction. For temperatures higher than T¼6 K the usual behavior can be noted where the magnetization always decreases as temperature increases. For temperatures lower than T¼5 K an unusual behavior can be observed which will lead to an inverse magnetocaloric effect as will be discussed below. Fig. 7 shows the calculated magnetocaloric effect in EuTiO3, i.e., DST vs. T under magnetic field changes from zero to B ¼0.5,
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Fig. 6. Magnetization vs. B for T¼1 to 7 K in EuTiO3. The magnetic field is not applied along the magnetic axis of the antiferromagnetic order (Fig. 6a), the squares and circles represent experimental data at T¼1.3 and 4.2 K, respectively. The magnetic field is applied along the magnetic axis (Fig. 6b).
Fig. 7. Temperature dependence of DST for EuTiO3 for a magnetic-field change from 0 to 0.5 T, 0 to 0.8 T, 0 to 1.0 T and 0 to 2.5 T. The solid and symbols lines represent the theoretical results for magnetic field applied not along and along the magnetic axis, respectively.
0.8, 1.0 and 2.5 T. This calculation was performed using the relation (15). The solid curves represent the calculation considering the case 2, where the magnetic field is not initially aligned
with the magnetic Eu þ 2 moments. Therefore, as discussed above, the magnetization M is constant from T¼0 to T¼T0 and from Maxwell relation (16) no magnetocaloric effect is expected (DST ¼0). In this way, we observe DST ¼0 below T0 ¼5.1, 4.6 and 4.0 K for DB ¼0.5, 0.8 and 1.0 T, respectively (solid curves). The arrows point out the same T0 ¼4.6 K in Fig. 4 and in Fig. 7 calculated for B ¼0.8 T. For DB¼1.0 T and T4T0 ¼4.0 K an increase occurs in DST curve since M is no longer constant. Alternatively, it can be confirmed analyzing Fig. 6a where for T44 K, for example, T¼5 K we have an area no null between two consecutives curves, i.e., dA¼[M(T¼4 K, B)–M(T¼5 K, B)]dB40, which leads DST 40 in accordance with Maxwell relation (16). The maximum values of DST vs. T occur at Ne´el temperature TN ¼5.5 K. For case 1, where the applied magnetic field is along the magnetic moments an inverse magnetocaloric effect (DST o0) is predicted to occurs in EuTiO3. This inverse magnetocaloric effect is expected to occur in this situation, since the magnetization (the order parameter) increases with temperature at low temperature (see Fig. 5, circles-curve, for B¼0.5 T). However, only with the M vs. T curve we cannot explain the inverse magnetocaloric effect that occurs for B ¼1 T, as shown in Fig. 7. This case should be investigated carefully considering the M vs. B – isothermic curves (see Fig. 6b). At low temperatures the area between two consecutive isothermic curves is negative, for example, dA¼[M(T¼1 K, B)–M(T¼2 K, B)]dBo 0 when integrated from B¼ 0 to B ¼1 T (the vertical dotted line in Fig. 6b). In general way, for all temperature interval (Ti þ 1 4Ti) where dA¼[M(Ti, B)– M(Ti þ 1, B)]dB o0 an inverse magnetocaloric occurs, and if dA40 a normal magnetocaloric effect (DST 40) occur. In our investigated case, dA¼[M(T¼5 K, B)–M(T¼6 K, B)]dB40, so DST 40 is expected in this temperature interval (see DST vs. T for DB¼ 1.0, in Fig. 7). For DB¼2.5 T we predict a normal magnetocaloric effect in EuTiO3 for all temperature range. It can be verified displacing the vertical dot line (in Fig. 6b) from B¼1 T to B ¼2.5 T and noticing that dA40 for all temperature interval. Therefore, in antiferromagnetic materials is necessary a complete analyze of the spin reorientation using the isothermic-M vs. B curves to predict the possible existence of an inverse magnetocaloric effect. Besides, the spin reorientation can also be determined from the DST vs. T curves. The arrow in Fig. 7 indicate the temperature (TSR ¼2 K) where the spin reorientation transition is predicted to occur in EuTiO3 for B ¼0.8 T (see also this transition in Fig. 4, dotted line). The discontinuity of the magnetization also appears for B¼ 0.5 T at TSR 4.2 K (see transition in Fig. 5, dotted line). Associated with this discontinuity in the magnetization DM 2 mB a latent heat can be calculated L¼TSR. DST 3.54 J/mol (DST is the discontinuity in magnetic entropy at TSR, see Fig. 7, dotted line in circle-curve). When the magnetic field increases the latent heat decrease, for example, L¼ 2.2 J/mol for B ¼0.8 T (see in Fig. 7 the discontinuity DST at TSR ¼2 K, square-curve). !! Fig. 8 shows in left axis the spin correlation function, / J i J j S vs. T (circle curves) and in right axis the DST vs. T (solid curves) for different values of applied magnetic field B ¼0.5, 0.8 and 1.0 T. It worth noticing the systematic magnetic field behavior of !! / J i J j S, where it reaches the maximum at T0, the arrow in Fig. 8 indicates the value of T0 for B ¼1 T. Therefore, considering the magnetocapacitance (relation 18) we can predict two anomalous physical behavior: 1] in temperature interval where the dielectric constant increases (for a given magnetic field intensity B0), the EuTiO3 does not change heat under isothermal magnetic field change from zero to B0, and 2] the temperature of EuTiO3 does not changes upon magnetic field change from zero to B0 in an adiabatic process. These two model predictions are valid under the following condition: 1] the coupling constant a is positive (Katsufuji and Takagi [12] determined by adjusting a ¼2.74 10 3 for EuTiO3); 2] the lattice entropy is magnetic
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temperature (where the spin pair correlation between nearest (a)–(b) sites peaks12), may have an impact on experimental investigations in order to design new materials with anisotropic magnetocaloric effects. As far as we know, no magnetocaloric measurements have been carried out on EuTiO3 compound and single crystal experiments showing the anisotropy would be particularly important to compare with our theoretical predictions.
Acknowledgments We acknowledge financial support from CNPq – Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico – Brazil, FAPERJ – Fundac- a~ o de Amparo a Pesquisa do Estado do Rio de Janeiro, CAPES – Coordenac- a~ o de Aperfeic- oamento do Pessoal de Nı´vel Superior, and FAPESP – Fundac- a~ o de Amparo a Pesquisa do Estado de Sa~ o Paulo. Fig. 8. Temperature dependence of /Ji JjS for EuTiO3 for a magnetic-field B¼0.5, 0.8 and 1.0 T (circles)- left axis . The solid lines-(right axis) represent DST, as in Fig. 7. The inset shows the T0 dependence on B.
field independent (in this way, the total entropy change is equal to the magnetic entropy change in an isothermal process). Another important aspect is that the temperature T0 can be tuned by the magnetic field intensity. The inset in Fig. 8 shows the T0 dependence on magnetic field, increasing B the T0 decreases smoothly.
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4. Final comments The thermal and magnetic properties in EuTiO3 were theoretically investigated focusing on the magnetocaloric effect. Two case of magnetic antiferromagnetic anisotropy were explored: when the magnetic field is applied along (and not along) the antiferromagnetic axis. The magnetocaloric effect, studied by the DST vs. T, reveled to be a powerful tool to reveal several behaviors about the antiferromagnetic EuTiO3. From DST vs. T curves the inverse magnetocaloric effect and the latent heat associated to the spin reorientation transition were predicted for low magnetic field changes. Also, the special possibility where the magnetic system EuTiO3 does not change heat below the described T0
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