The influence of an electric field on the latent heat of the ferroelectric phase transition in KDP

This is the accepted manuscript prior to house edits. ©2005 IOP Publishing Ltd The Version of Record is at J. Phys.: Condens. Matter 17 (2005) 2645-2654 doi:10.1088/0953-8984/17/17/012 Influence of the electric field on the latent heat of ferroelectric phase transition in KDP Jose Maria Delgado-Sanchez, Jos´e Mar´ıa Mart´ın-Olalla, Mar´ıa Carmen Gallardo, and Saturio Ramos Departamento de F´ısica de la Materia Condensada. Instituto Mixto de Ciencia de Materiales CSIC-Universidad de Sevilla Ap Correos 1065, ES-41080 Sevilla, SPAIN Marceli Koralewski Institute of Physics, Adam Mickiewicz University, Umultowska 85, 61-614, Poznan, Poland Jaime del Cerro Departamento de F´ısica de la Materia Condensada. Instituto Mixto de Ciencia de Materiales. CSIC-Universidad de Sevilla Ap Correos 1065, ES-41080 Sevilla, SPAIN (Submitted: 31 January 2005; Revised: March 24 2005; Published: 15 April 2005) The specific heat, heat flux (DTA trace) and dielectric constant of KDP ferroelectric crystal have been measured simultaneously for various electric fields with a conduction calorimeter. The specific heat presents a strong anomaly but these simultaneous measurements allow us to evaluate the latent heat accurately. Latent heat decreases with field and the value of critical electric field —that where latent heat disappears— is estimated to be (0.44 ±0.03) kV cm−1. Incidentally, we have measured simultaneously the dielectric permittivity which suggests that latent heat is developed as domains are growing. I. INTRODUCTION The KDP family is one of the most extensively studied[1, 2] hydrogen-bonded ferroelectric crystals. The potassium dihydrogen phosphate KH2PO4crystal exhibits a discontinuous phase transition at T0= 121 K from a tetragonal paraelectric phase to an orthorhombic ferroelectric phase. The specific heat anomaly at transition temperature shows[3–5] a strong λ-type anomaly; perhaps, that is why the transition was initially considered continuous. However Reese[4] showed that the transition is discontinuous by measuring its latent heat, which was evaluated to be 46.1 J mol−1. It is shown that discontinuity disappear under the influence of pressure or electric field. The value of the critical electric field Ecfor which the discontinuity disappears is a subject of discussion. Reese et al.[6] also carried out a measurement of the specific heat with an applied field of 294 V cm−1finding a controversial evidence of latent heat. Above E= 785 V cm−1 they found no evidence of latent heat and suggested Ec= 300 V cm−1from the shift of the maximum of the specific heat as function of the electric field. Measurements of specific heat under electric field higher than 360 V cm−1was carried out by Sandvold and Fossheim[7] and the authors suggest that Landau theory for continuous transitions with 2 −4−6 potential was appropriate to describe the shape of specific heat curves. Other works have lead to estimations of the critical field. For instance, Strukov et al.[8] evaluated Ec= 124 V cm−1from electrocaloric experiments. Sidnenko and Gladki[9] found Ec= 370 V cm−1while Okada and Sugie[10] obtained 160 V cm−1for Ec. Vallade[11] deduced a value of 254 V cm−1from birefringence measurements. In contrast to these values, Kobayashi et al.[12] found, by X-ray measurements, 8000 V cm−1; Eberhard and Horn[13] derived a value of 6500 V cm−1from dielectric susceptibility. In this frame it would be interesting to evaluate the latent heat as a function of the electric field to determine the value of the critical electric field Ecfor which latent heat becomes null. The difficulty of an accurate determination of Ecfrom calorimetric measurements in standard equipments — such as differential thermal analysis DTA and differential scanning calorimeter DSC —is that these systems really measure changes of enthalpy which has two contributions near the transition: one due to the latent heat and other due to the variation of specific heat with temperature. In the case of phase transitions near a tricritical point or a discontinuous ferroelectric phase transition under an electric field close to the critical field, the specific heat presents a strong anomaly and latent heat becomes very small. This fact makes difficult to separate both contributions and to distinguish the temperature interval where the latent heat is present. This may explain the lack of study of the influence of electric field, smaller than Ec, on specific heat and latent heat of KDP after Reese[6]. Our group has developed a method, named square modulated differential thermal analysis SMDTA[14–16], based on conduction calorimetry, which is able to measure absolute values of specific heat and the heat flux exchanged by the sample when its temperature is changed at a rate as low as 0.1 K h−1. The comparison of the data allows us to separate the above two contributions to the total enthalpy and to evaluate the latent heat, in case there were any. Typeset by REVT EX 2 This technique has been successfully applied to the study of the almost tricritical phase transition of KMnF3[14], whose latent heat was firstly measured with this technique. Furthermore, the effect of the substitution of Mn by Ca was also investigated measuring the latent heat[15], which showed that the doping makes the transition become continuous[17]. The method of SMDTA has been also applied to show that the phase transition in CoO[18], whose character was also controversial, is continuous. In this paper we have applied this method to study a KDP single crystal. We have measured the specific heat and the heat flux exchanged by the sample in the neighborhood of the ferroelectric phase transition. The measurements have been carried out at four values of E: 0 V cm−1, 100 V cm−1, 400 V cm−1, 1000 V cm−1. Simultaneously to these measurements, the dielectric susceptibility of the sample has also been measured and we have related its behaviour around transition temperature with the temperature interval where the latent heat is produced. Dielectric measurements provide information about the mechanism of ferroelectric phase transition of KDP crystal[19, 20] and the simultaneous measurement of thermal and dielectric properties would be worthy. II. EXPERIMENTAL The measurements were performed in a high resolution conduction calorimeter which has been described previously in details[21–23]. The sensor is formed by two identical heat fluxmeters, each one having 48 chromelconstantan thermocouples connected electrically in series but thermally in parallel. The sample is pressed between both fluxmeters whose signal is measured by a Keithley 182 nanovoltmeter. Two electrodes and two heaters are placed between sample and fluxmeters. The sensor is placed inside a calorimeter block which is suspended within two cylindrical radiation shields. The whole assembly is then placed in a hermetic outer case at a high vacuum. The device is then surrounded by a coiled tube and placed in an alcohol bath. Liquid N2circulates through the coil and regulates the temperature of the bath with a good thermal stabilization. As a result, it is possible to change smoothly the temperature of the sample (at a rate of about 0.1 K h−1) without observing significant temperature fluctuations (always less than 10−6K) in the block temperature. The specific heat is measured using the method previously described[16]. The same constant power Wis dissipated in both heaters (dissipation branch) for twelve minutes and a steady state characterized by a constant temperature difference between the sample and the calorimeter block is reached. The power is then cut off until a new steady state is reached twelve minutes later (relaxation branch). Then, the power is again switched on and the sequence is continuously repeated while the temperature of the assembly is changed at a low constant rate. That is a long-periodic serial of square thermal pulses is superposed to a heating or cooling ramp. The temperature increase of the sample due to the thermal pulse of twelve minutes (circa 50 ×10−3K) is higher than the temperature variation of the sample produced by the rate of change of temperature in these twelve minutes (circa ±20 ×10−3K). Hence, the sample is being cooled and heated alternatively during a run. The integration of the electromotive force given by the fluxmeter between every pair of steady states allows us to determine sample thermal capacity. Hence, the method is able to determine two data of heat capacity in each cycle. The first one is calculated from the dissipation branch Cd, and the second one from the relaxation branch Cr. Heat capacity obtained in either branch show a regular behaviour if there is no phase transition or if it is continuous. When a discontinuous phase transition occurs, both data become different showing an anomalous behaviour in the temperature interval where the latent heat is produced as a result of the thermal hysteresis and transition kinetics. That behaviour is an evidence of the discontinuous character of the transition[16]. Incidentally, we must point out that specific heat data are not reliable when this behaviour is observed. On the other hand, the DTA trace is continuously measured in a second run without dissipation in the sample and using the same temperature scanning rate used to measure the specific heat. Due to the high number of thermocouples and their good thermal stability of the sample, the equipment works like a very sensitive DTA device. The electromotive force given by the fluxmeters is proportional to the heat flux, φdexchanged between sample and calorimeter block. From the specific heat data obtained in the first running and using a method previously described[14–16], we calculate the heat flux φcwhich would have been due exclusively to the behaviour of the thermal capacity of the sample around the transition temperature. Comparing the measured φdand the calculated φcwe deduce that only in the temperature range (Tf, Tp) where both data do not coincide there is an effect from the latent heat. Its value is determined by integrating φd/v, where vis the rate of temperature change, between (Tf, Tp) and using the straight line φd/v(Tf)−φd/v(Tp) as baseline[15]. The sensitivity of the method is estimated to be better than 5 mJ. The single crystal of KH2PO4was grown at the Institute of Physics of Poznan University (Poland). The sample has 0.3857 g mass, with a thickness of 2.16 mm along the ferroelectric axis and electrodes circular faces with 78.5 mm2in surface. The sample was placed out in the calorimeter. Gold electrodes were evaporated of the surface of the sample; those electrodes were connected to a capacitance bridge ESI −SP5400 which has allowed us to measure the dielectric permittivity of the sample, simultaneously to the heat flux, with an imposed external bias field of 0 V cm−1, 100 V cm−1, 400 V cm−1and 3 1000 V cm−1. III. RESULTS III.1. Calorimetric measurements The temperature dependence of the specific heat cpof a sample of KDP was measured on cooling, on quasiestatic conditions at a scanning temperature rate of ca. v∼ 0.1 Kh−1using the method described in Sec. II for different applied electric fields. In Figure 1, the specific heat data in a wide temperature interval for different applied electric field (a) E= 0 V cm−1, (b) E= 100 V cm−1, (c) E= 400 V cm−1and (d) E= 1000 V cm−1is shown. These specific heat data show a linear temperature dependence in the paraelectric phase and for 0 V cm−1, 100 V cm−1and 400 V cm−1a sharp λ-type anomaly in a narrow temperature interval is observed. For 1000 V cm−1the maximum is more rounded and the transition is smeared. The shape of specific heat curve for high electric field tends to be almost symmetric, such behaviour was also suggested by Reese[6]. The maximum value of cinitially increases with increasing field but decreases for sufficiently high values. As expected, the specific heat tail in the paraelectric phase increases with field as a consequence of the coupling of the order parameter to the field. In Figure 2, we have plotted the specific heat excess obtained in the dissipation branch (cd) and in the relaxation branch (cr) in a narrow temperature interval, 1 K, for each electric field. We observed that for 0 V cm−1, 100 V cm−1and 400 V cm−1,cdand crdata do not coincide around the transition temperature while for 1000 V cm−1both series of data almost coincide in the whole range of temperature. As we have stated above, the temperature variation of the sample due to the thermal pulses is slightly higher than the variation due to the temperature ramp. This means that in every period the temperature of the sample increases and decreases consecutively. Due to thermal hysteresis, to kinetics of the phase transition etc, the process of heating and cooling when two phases coexist is different and consequently data obtained in the dissipation branch and relaxation branch become different under these conditions. We have reported previously that the difference is very notorious even in systems near the tricritical point, where the latent heat is very small ( L = 0.13 J g−1for KMnF3, L = 0.010 J g−1for KMn0.997Ca0.003F3)[16]. Hence, in the case of KDP, it is clear from Figure 2 that for 0 V cm−1, 100 V cm−1and 400 V cm−1the phase transition is discontinuous. On the contrary, the similar behaviour of cdand crfor 1000 V cm−1, Figure 2d, indicate that no trace of latent heat is present, so we can deduce that for 1000 V cm−1the phase transition is continuous. Hence the critical field lies between 400 V cm−1 and 1000 V cm−1. On the other hand, the baseline respect which the fluxmeter emf is integrated in order to calculate the specific heat is the underline signal due to the temperature ramp imposed on the calorimeter. This signal changes very slowly and it does not affect to the specific heat measurement except when a latent heat effect happens. It produces a non linear variation of the baseline and consequently erroneous data of specific heat may appear. Keeping in mind that the measurements have been carried out in a cooling ramp, in Figure 2a the first crat the beginning of the transition which does not coincide with the corresponding cddata is lower than the regular contribution to the specific heat. In Figure 2b we can also observe such a similar point but this effect is smeared. Finally, in Figure 2c we cannot observe any decrease in cr. Nevertheless, crdeviates from cdin a small range of temperature as in the two previous figures. Hence, we can deduce that latent heat decreases with field and should be very small for 400 V cm−1. To confirm this suggestion and to calculate the latent heat, we measured the DTA trace in a second run, changing the temperature of the sample at the same constant rate used in the specific heat measurements to make both sets of data comparable. We must point out that the rate of change of temperature is about two orders of magnitude lower than the minimum value achieved in conventional DTA equipments. In Figure 3, we represent the heat flux given by the fluxmeters φd/v (DTA trace) and the heat flux φc/v calculated from the specific heat data, using the method previously described, for the fields 0 V cm−1(a), 100 V cm−1(b), 400 V cm−1(c). For 0 V cm−1and 100 V cm−1φd/v is higher than φc/v in a very small temperature interval of about 0.05 K, showing the effect of latent heat. For 400 V cm−1φd/v is also higher than φc/v, but the difference between them is smaller, indicating a very small latent heat. The choosing of the baseline to determine the latent heat value for 0 V cm−1in graph Figure 3a(i) is difficult due to the very small temperature interval where the latent heat is present and, consequently, the few data recorded inside that interval. Anyway, by considering as baseline the straight line between the extreme temperatures where φd/v and φc/v coincide, we obtain a value of 43 J mol−1. This line is also represented in Figure 3a). The value obtained for KDP deuterated at 80% using the same set-up and procedure was 317 J mol−1[24]. To confirm the validity of that baseline election we again measured the heat flux exchanged by the sample in identical conditions. Those heat flux data are also plotted in the Figure 3a(ii). The figure shows that both heat flux data coincide with φc(stars) in the same range of temperature thus supporting the baseline used for the determination of the latent heat. In fact, the latent heat for the second run is 45 J mol−1in good agreement with that of the first one. On the other hand, these values of the latent heat are in agreement with Reese[6]. It is noteworthy that peak area —i.e. latent heat— is quite reproducible despite the kinetic effects that showed each experiment. On the contrary, the difference observed if 4 119 120 121 122 123 124 125 0 200 400 600 800 (b) c/J K−1mol−1 T/K 119 120 121 122 123 124 125 0 200 400 600 800 (d) c/J K−1mol−1 T/K 0 200 400 600 800 119 120 121 122 123 124 125 (a) c/J K−1mol−1 T/K 0 200 400 600 800 119 120 121 122 123 124 125 (c) c/J K−1mol−1 T/K FIG. 1. The specific heat of KDP for different applied electric fields in a wide temperature interval. From left to right and top to bottom, (a) 0 V cm−1and (b) 100 V cm−1, (c) 400 V cm−1and (d) 1000 V cm−1. φ/v during the phase transitions suggests the existence of non-equilibrium kinetic process as those expected for a discontinuous phase transition —for instance phase front generation— which, as a general fact, are nonreproducible. For E= 100 V cm−1and E= 400 V cm−1we decreased the scanning rate of temperature when measuring specific heat so as to get a higher number of data points in the neighborhood of the phase transition. Hence, the determination of the baseline becomes easier as shown in Figure 3. The integration of the peaks gives 35 J mol−1 and 4.2 J mol−1respectively. The obtained result confirm that latent heat diminishes with electric field (see Figure 4). At 400 V cm−1 latent heat is reduced by and order of magnitude with respect to that of zero field and it is close to the critical field. Assuming a linear behaviour of the latent heat with the field we deduce that the critical field lies on 5 121 121.2 121.4 121.6 121.8 0 200 400 600 800 (b) c/J K−1mol−1 T/K 120.8 121 121.2 121.4 121.6 121.8 0 200 400 600 800 (d) c/J K−1mol−1 T/K 0 200 400 600 800 120.6 120.8 121 121.2 121.4 121.6 (a) c/J K−1mol−1 T/K 0 200 400 600 800 120.8 121 121.2 121.4 121.6 121.8 (c) c/J K−1mol−1 T/K FIG. 2. The same as figure 1 but in a narrower temperature interval around the transition point. Bold points stand for the dissipation branch (cd), open points stand for the relaxation branch (cr). (0.44 ±0.3) kV cm−1. III.2. Dielectric measurements In order to relate calorimetric and dielectric behaviour the dielectric permittivity along the ferroelectric axis has been measured simultaneously to the heat flux. In Figure 5 we represent ε(T) for the different electric fields. The dielectric permittivity increases in all cases during the phase transition following the Curie law. In the ferroelectric phase the dielectric permittivity remains in a plateau, indicating also the large domain wall contribution to the dielectric permittivity, but the maximum of permittivity decreases with the applied electric field. It has been previously reported that the maximum value of dielectric permittivity is due to the contribution of domain wall [25]; under electric field the 6 8 4 0 121.5121.4121.3 (a) (i) (ii) φv−1/J K−1 T/K 121.5121.4121.3 (b) T/K 121.6121.5121.4121.3 8 4 0 (c) φv−1/J K−1 T/K FIG. 3. Heat flux divided by rate of change of temperature in the neighborhood of the KDP ferroelectric phase transition. From left to right, top to bottom, (a) 0 V cm−1, (b) 100 V cm−1, (c) 400 V cm−1. Points represent heat flux data given by the fluxmeters —in (a) two similar running labelled (i) and (ii) were carried out—. Stars represent the contribution due to specific heat of Figure 2. Straight lines show the baseline used for determination of the latent heat which happens to be the peak area subtended by experimental points and the baseline. sample becomes close to monodomain state, so number of domain walls decrease and also does the dielectric permittivity. Nevertheless the behaviour at transition temperature is different for 0 V cm−1and 100 V cm−1 —discontinuous transition— than for 400 V cm−1— close-to-critical-point transition— and 1000 V cm−1— continuous transition—. We will relate in Figure 6 the shape of dielectric permittivity for 0 V cm−1and 400 V cm−1. The heat flux and the inverse of the dielectric permittivity versus temperature for E= 0 V cm−1are represented in Figure 6(a). Three regions may be distinguish in this figure: (i) the paraelectric phase where the permittivity follows the Curie law, (ii) the phase transformation interval where the permittivity slightly deviates from the previous behaviour and (iii) the ferroelectric phase where the permittivity shows a plateau. We must point out that the maximum of the permittivity matches with the end of phase transformation. For 400 V cm−1,ε(T) shows a different behaviour — see Figure 6(b)—. At the transition temperature it is present a minimum. This has been observed by Bornarel[19] in crystals of KDP for higher values of electric field (about 1 kV cm−1). In that work Bornarel explains this behaviour in terms of some domains arrangements during the phase transition; ε(T) is described as the sum of the behaviour of ε1(T) contribution that correspond to the behaviour of monodomain sample, and the corresponding ε2(T) due to the contributions of domains. Moreover ε1(T) increases from paraelectric phase and after the maximum at transition temperature decreases to zero in ferroelectric phase; ε2(T) increases from zero at transition temperature and remains in a plateau in ferroelectric phase. The sum of both contributions gives the appearance of ε(T) in figure 6b. It may be seen also from the results presented on Figure 6 that the peak of the heat flux at transition temperature appear at temperature where the production of domains become dominant. IV. CONCLUSIONS The thermal and dielectric behaviour of KDP crystal near the temperature of its ferro-paraelectric phase transition has been simultaneously studied under the influence of electric field. For 0 V cm−1the latent heat has been measured twice. Although both measurements shows a different kinetic, their results are similar and in good agreement with those obtained by Reese[6]. Despite the high increase of the specific heat around the transition temperature, the very small values of the latent heat and the narrow temperature range —ca. 0.1 K— where transition is developed, we have been able to discriminate the contribution of the latent heat to the total change of enthalpy. Finally, simultaneous measurement of heat flux and dielectric susceptibility suggest that the effect of latent heat appears at temperatures range where 7 0 20 40 60 0 200 400 E/V cm−1 ∆h/J mol−1 FIG. 4. Plot of the latent heat ∆hat various electric fields. Critical field is found to be (0.44 ±0.03 kV cm−1 0 5000 10000 15000 120 121 122 123 124 T/K ε 0 V cm−1 100 V cm−1 400 V cm−1 1000 V cm−1 FIG. 5. Plot of the dielectric permittivity at various electric fields. Permittivity data were measured simultaneously to heat flux data of figure 3. domains are growing. Maximum dielectric permittivity was noticed at temperature where phase transition ends. It was established that the latent heat decreases with the field and the critical electric field is estimated to be (0.44 ±0.03) kV cm−1. 4 2 0 121.5121.4121.3 0.0002 0.0001 0 (a) φv−1/J K−1 T/K 1/ε 0 2 4 121.4 121.6 121.8 4000 6000 8000 (b) φv−1/J K−1 T/K ε FIG. 6. (a) Plot of the inverse of the dielectric permittivity (right axis) and φv−1(left axis) for E= 0 V cm−1. (b) Plot of the dielectric permittivity (right axis) and φv−1(left axis) E= 400 V cm−1. ACKNOWLEDGMENTS We wish to thank Prof. Bornarel for fruitful discussions. This work was supported by Spanish Ministerio de Ciencia y Tecnolog´ıa contract number BFM2002-02237. [1] Y. Xu, Ferroelectics materials and their applications (North-Holland, 1991), ISBN 0-444-88354-1. [2] M. Lines and A. Glass, Principles and applications of ferroelectrics and related materials (Clarendon Press, Ox- 8 ford, 1977). [3] C. Stephenson, J. Phys. Chem. 66, 1397 (1944). [4] W. Reese, Physical Review 162, 510 (1967). [5] B. Strukov, M. Amin, and V. 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