High-resolution electrocaloric and heat capacity measurements in barium titanate

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August 2013

EPL, 103 (2013) 47001 www.epljournal.orgdoi: 10.1209/0295-5075/103/47001

High-resolution electrocaloric and heat capacity measurementsin barium titanate

Nikola Novak1, Zdravko Kutnjak

1,2 and Rasa Pirc1

1 Jozef Stefan Institute - Jamova 39, 1000 Ljubljana, Slovenia, EU2 Jozef Stefan International Postgraduate School - Jamova 39, 1000 Ljubljana, Slovenia, EU

received on 14 August 2013; accepted by J. F. Scott on 21 August 2013published online 30 August 2013

PACS 77.70.+a – Pyroelectric and electrocaloric effectsPACS 05.70.Jk – Critical point phenomenaPACS 65.40.Ba – Heat capacity

Abstract – The electrocaloric (EC) effect in a BaTiO3 single crystal oriented along the [001]direction has been studied by direct high-resolution EC measurements in the vicinity of the field-induced critical point. It is shown that the observed behavior of the EC temperature changeΔT , as well as the heat capacity anomalies are in good agreement with the predictions of theLandau theory. We also present the electric field-temperature phase diagram for the field-inducedparaelectric-to-ferroelectric phase transition line, derived from the calculated latent heat.

editor’s choice Copyright c© EPLA, 2013

Introduction. – Barium titanate BaTiO3 (BTO) isone of the most widely studied ferroelectric materials,which is also frequently used in various practical applica-tions [1]. At high temperatures in zero electric field BTOhas an average cubic perovskite structure and undergoesa first-order paralelectric-to-ferroelectric phase transitioninto a tetragonal ferroelectric phase at the Curie temper-ature near ∼400 K. This transition has long been believedto be of a purely displacive type, characterized by thesoftening of a transverse optic phonon [2]; however, it wasshown later that another order-disorder–type componentexists due to the off-center displacements of the Ti ionsalong the [111] directions in the high-temperature phase.The off-center Ti model was proposed by Comes et al. [3]and has been later confirmed by first-principles calcula-tions [4,5] as well as by X-ray [6], NMR [7], and tera-hertz dielectric spectroscopy [8]. A mesoscopic model ofBTO which includes a coupling between the soft-phononmode [9] and the symmetry-adapted occupational proba-bilities for the Ti displacements was formulated in ref. [10].

It has been pointed out by several authors [11–13] thata phenomenological description of BTO based on the Lan-dau free-energy expansion must include terms up to theeighth order in powers of the polarization components Pi

(i= 1, 2, 3) in order to reproduce the full sequence ofphase transitions in zero field: paraelectric cubic 405 K−−−→ferroelectric tetragonal 281K−−−→ orthorhombic 193K−−−→ rhom-

bohedral [14]. Porta et al. [13] studied the effects of straindisorder on the critical point in the E-T phase diagram ofBTO and calculated the heat capacity of PMN-PT nearthe critical point for several values of the electric field.

In the present study, we focus on the behavior of theelectrocaloric (EC) effect in bulk BTO along the line offirst-order transitions in the E-T phase diagram. As iswell known, a material shows a high EC effect when highreversible entropy changes can be induced as a functionof temperature and electric field [15–20]. Thus, it is ex-pected to observe the highest EC response in materialsat the first-order phase transition where a significant vari-ation in polarization occurs. However, despite the greatinterest in the EC effect, there is still a lack of direct high-resolution EC data in the vicinity of the first-order phasetransitions. Here we present new direct high-resolutionEC data obtained in a BTO [001] single crystal by mea-suring the EC temperature change near the field-inducedfirst-order phase transitions. The experimental EC tem-perature profile will be compared with the theoretical re-sults based on the Landau phenomenological model. Inaddition, the heat capacity anomalies calculated from theLandau model will be compared with recent experimentaldata [21].

Landau theory of heat capacity. – The Landau free-energy density of BTO [12] in an external electric field

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Nikola Novak et al.

�E = (E1, E2, E3) is written in the form

F = α1

∑i

P 2i + α11

∑i

P 4i + α12

∑i�=j

P 2i P 2

j

+ α111

∑i

P 6i + α112

∑i�=j �=k

P 4i (P 2

j + P 2k )

+ α123P21 P 2

2 P 23 + α1111

∑i

P 8i

+ α1112

∑i�=j �=k

P 6i (P 2

j + P 2k ) + α1122

∑i�=j

P 4i P 4

j

+ α1123

∑i�=j �=k

P 4i (P 2

j + P 2k ) −

∑i

EiPi, (1)

where i, j, k run over the Cartesian indices 1, 2, 3 [11–13].Specifically, the coefficient α1 is given by

α1 =12a1(T − T0), (2)

where the coefficients a1 and T0 for our sample are a1 =1.696 × 106 C−2m2NK−1 and T0 = 399.3 K (see table 1),and the remaining coefficients are, as usual, assumed tobe T -independent.

For our sample in zero electric field the free energy (1)predicts a first-order paraelectric-to-ferroelectric phasetransition at TC = 405.4 K with the spontaneous polar-ization Ps = 0.1081 Cm−2. It should be noted that theLandau coefficients reported by Porta et al. [13] could notreproduce the Curie temperature TC = 405.5 K and thecritical field ECP = 10 kV/cm of our sample, which weredetermined by dielectric susceptibility and heat capacitymeasurements [21]. Therefore, in order to calculate thetemperature and field dependence of the heat capacity andelectrocaloric effect we had to determine anew the valuesof the relevant Landau coefficients.

We will be primarily interested in the field-inducedphase transitions and the critical point at temperaturesT > TC for fields along one of the cubic directions, e.g.,�E = (0, 0, E). For symmetry reasons, the polarization vec-tor will also have only one nonzero component, namely,�P = (0, 0, P ). The free energy (1) can thus be rewrittenin compact form [12],

F =12aP 2 +

14bP 4 +

16cP 6 +

18dP 8 − PE, (3)

where the coefficients a, b, c, d can be trivially expressed interms of the coefficients α1, . . . , α1123, namely,

a = 2α1; b = 4α11; c = 6α111; d = 8α1111, (4)

which are listed in table 1. The dielectric polariza-tion P (E, T ) is calculated by minimizing the free en-ergy (3) [13]. It follows that for an electric field E alongthe [001] direction, the first-order transition is shifted to ahigher temperature TC(E) and the jump in the polariza-tion P (E, T ) decreases in analogy to fig. 1 of ref. [13].

Table 1: Values of coefficients a, b, c, d in the Landau freeenergy (3) estimated from the dielectric susceptibility and heatcapacity measurement data.

Coefficient Value Units

a 1.696 × 106(T − 399.3) (C−2m2N)b −3.422 × 109 (C−4m6N)c 1.797 × 1011 (C−6m10N)d 3.214 × 1012 (C−8m14N)

Finally, at a critical field ECP = 10 kV/cm and tem-perature TCP = 414.83 K the discontinuity in P (E, T )disappears and the transition becomes of second order.The calculated temperature dependence of the dielectricsusceptibility χ(E, T ) = (∂P/∂E)T is shown in fig. 1(a)for a set of electric-field values. Clearly, for E < ECP

the susceptibility shows sharp anomalies at the first-ordertransition temperatures TC(E) and diverges at the criti-cal point ECP , TCP . The values of the standard criticalexponents at the field-induced critical point are α = 2/3,β = 1/3, γ = 2/3, and δ = 1/3, which differ from theusual mean-field exponents in zero field [18,22]. Beyondthe field-induced critical point the system is in a supercrit-ical state characterized by a smooth variation of P (E, T )and by rounded maxima of χ(E, T ), the positions of whichdefine the so-called Widom line [23].

The heat capacity at constant field ΔCE = T (∂S/∂T )E

is derived from the entropy S = (∂F/∂T )E. From eq. (3)we obtain

S = −12a1P

2, (5)

and thusΔCE = Tχ(E, T )[a1P (E, T )]2. (6)

Here we have applied the equilibrium condition (∂F/∂P )E = 0, and the relation (∂P/∂T )E = a1Pχ which fol-lows from it. Since χ(E, T ) shows sharp anomalies at first-order transitions for each field value E and a divergence atthe critical point, a similar behavior can be expected forthe heat capacity ΔCE(T ). The calculated heat capacityis shown in fig. 1(b) for the same set of electric-field valuesE as in fig. 1(a). The experimental data for ΔCE(T ) inBTO oriented along the [001] direction obtained by high-resolution measurements are shown in fig. 2(a) for threediscrete values of the electric field E. For comparison, thetheoretical prediction for ΔCE(T ), calculated from eq. (6)at the same field values, is plotted in fig. 2(b). The pre-dicted temperature profile of ΔCE(T ) qualitatively agreeswith the experimentally observed behavior; however, thereis a difference of ∼ 2 K between the experimental and theo-retical values of the transition temperature for finite fields.The reason for this difference is that the field dependenceof the measured polarization P (E, T ) and susceptibilityχ(E, T ) cannot be described with sufficient accuracy bythe Landau free energy (3) with temperature-independentcoefficients b, c, d. An analogous problem occurred in

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High-resolution electrocaloric and heat capacity measurements

Fig. 1: (Colour on-line) (a) Calculated temperature depen-dence of the dielectric susceptibility for a set of field valuesE/ECP , as indicated. (b) Calculated temperature dependenceof the heat capacity anomaly for the same set of field values.

Fig. 2: (Colour on-line) (a) Experimental data show-ing the specific-heat anomalies occurring at paraelectric-to-ferroelectric phase transition in BTO [001] for three selectedvalues of the electric field, obtained by the ac high-resolutioncalorimetric measurement. (b) Calculated temperature depen-dence of the excess specific heat using the coefficients listed intable 1 and the same field values as in experiments.

the case of ferroelectric polymers, where the temperature-dependent Landau coefficients b(T ) and c(T ) had to beintroduced [24]. Unfortunately, our data on P (E, T ) didnot allow us to determine the temperature dependence ofthe coefficients b, c, d.

The latent heat released at the first-order phase transi-tion is given by L = TC(E)ΔS, where ΔS is the entropychange between the two phases. Denoting by P+ and P−the values of the polarization P (E, T ) in the upper andlower phase at T = TC(E), respectively, we obtain fromeq. (5)

L =12a1(P 2

+ − P 2−). (7)

Fig. 3: (Colour on-line) Phase diagram for BTO obtainedfrom the calculations of the latent heat at the paraelectric-to-ferroelectric phase transition. The first-order transition line ofTC(E) terminates in a critical point CP at ECP = 10 kV/cmand TCP = 414.83 K, where the latent heat vanishes. Thedashed line represents the Widom line. The inset shows thecalculated latent heat as a function of temperature.

P+ and P− can be extracted from the calculated val-ues of P (E, T ) by a simple numerical procedure. Infig. 3 we show the E-T phase diagram established fromthe calculated temperature dependence of the latent heatL(E, T ) for several fixed values of the electric field E (seethe inset of fig. 3). As expected, the line of the first-order phase transition terminates in a critical point whereL(E, T ) goes to zero. The theoretical coordinates of thecritical point are ECP = 10 kV/cm, TCP = 414.83 K,and are in good agreement with the experimental valuesECP = 10 ± 2 kV/cm, TCP = 412 ± 3 K [21].

Electrocaloric response. – Recently, the EC stud-ies [15] near the paraelectric-to-ferroelectric phase tran-sition in a BTO single crystal were reported by Bai etal. [25] and Moya et al. [19]. In ref. [19] the EC responseof the BTO single crystal was studied via direct mea-surements and calculated by an indirect method based onthe Maxwell relation. These studies showed a large ECstrength near the first-order ferroelectric phase transition.

In the present paper we present new direct high-resolution EC measurements. The EC temperaturechange ΔT in bulk BTO [001] was measured via a directelectrocaloric method. Each EC measurement consisted oftwo parts, i.e., a heating and a cooling run [17,21,26,27].In the heating run the applied electric field was linearlyincreased to the final value, and in the cooling run thefield was step-like decreased. The actual EC temperaturechange ΔT was determined from the cooling run. Thetemperature step above the field-induced transition was0.5 K, whereas in the vicinity and below the transition thestep was 0.2 K, providing the highest resolution of the ECdata in BTO so far.

In figs. 4(a) and (b) we compare the experimental andcalculated EC response, respectively, for three discrete val-ues of the field E. The EC temperature change in fig. 4(b)

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Nikola Novak et al.

Fig. 4: (Colour on-line) (a) The EC response obtained via di-rect electrocaloric measurements on BTO [001] at the samevalues of the applied electric field change as in theoretical cal-culations. (b) EC response of BTO calculated for three valuesof the electric field magnitude below and above the value of thecritical field ECP .

was calculated by the Landau method, which enabled us todetermine the polarization P (E, T ), and hence the dipo-lar entropy Sdip(E, T ) at an arbitrary point in the phasediagram, without the use of thermodynamic Maxwell re-lations. In general, the resulting EC temperature changecan be written as ΔT = T2 − T , where T2 satisfies theself-consistent equation [18],

T2 = T exp{

a1

2Clatt[P 2(E, T2) − P 2(0, T )]

}, (8)

and Clatt is the lattice contribution to the total heat capac-ity. This equation was solved for the same set of electricfields and temperature range as used in experiments. Thecalculated EC response shows a discontinuous jump at theCurie temperature, where the second term in the exponentof eq. (8) is zero for T > TC . Below E < ECP a steep dropof the EC response at the field-induced transition temper-ature is obtained as a result of a discontinuous change ofthe first term in the exponent in eq. (8). Above ECP , how-ever, the EC temperature change decreases continuouslydue to a continuous evolution of polarization in the super-critical regime. The experimental data qualitatively agreewith the theoretical predictions. However, the calculatedtemperature profile of the EC response at temperaturesTC < T < TCP is broader than the one observed in exper-iments for the same reason as discussed above in the caseof heat capacity anomalies.

Conclusions. – We have carried out a series of di-rect high-resolution electrocaloric (EC) and heat capac-ity measurements at the paraelectric-to-ferroelectric phasetransition in a BTO [001] single crystal. The experimen-tal data obtained previously [21] were used to determine

the coefficients in the Landau free energy. By extendingthe Landau approach up to the eighth order in powersof polarization components, the temperature dependenceof the dielectric susceptibility and heat capacity at sev-eral constant electric fields has been calculated. Bothquantities show a divergent behavior when approachingthe field-induced critical point, which was found to be atECP = 10 kV/cm and TCP = 414.83 K. The calculatedexcess heat capacity and the experimentally observed heatcapacity anomalies are in good qualitative agreement. Byevaluating the latent heat as a function of temperature andthe applied electric field, a phase diagram of BTO [001]has been constructed. The calculated temperature depen-dence of the EC temperature change ΔT shows a steepincrease at the Curie temperature TC(E = 0) and thendrops sharply at the field-induced transition temperatureTC(E) for electric fields lower then ECP . For E > ECP ,however, the EC response decreases continuously due tothe smooth evolution of polarization in the supercriticalregime [28]. The calculated temperature profile of the ECresponse in the BTO [001] single crystal agrees quantita-tively with the high-resolution experimental data, clearlyshowing that the maximum EC effect is reached preciselyat TC(E = 0) for all field amplitudes.

∗ ∗ ∗

This research was supported by the Slovenian ResearchAgency under programs P1-0125 and, P1-0044, and theNAMASTE Centre of Excellence.

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