Electro Caloric Effects

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Microscopic theory of the electrocaloric effect in the paraelectric phase ofpotassium dihydrogen phosphateLawrence J. Dunne , Matjaz Valant , George Manos , Anna-Karin Axelsson , and Neil Alford

Citation: Appl. Phys. Lett. 93 , 122906 (2008); doi: 10.1063/1.2991443 View online: http://dx.doi.org/10.1063/1.2991443 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v93/i12 Published by the American Institute of Physics.

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Microscopic theory of the electrocaloric effect in the paraelectric phaseof potassium dihydrogen phosphate

Lawrence J. Dunne, 1 Matjaz Valant, 2,3 ,a George Manos, 4 Anna-Karin Axelsson, 2 andNeil Alford21 Department of Engineering Systems, London South Bank University, London SE1 0AA, United Kingdom2 Department of Materials, Imperial College, Exhibition Road, London SW7 2AZ, United Kingdom3 Laboratory for Electronic and Environmental Materials, University of Nova Gorica, Vipavska 13, 5000

Nova Gorica, Slovenia4 Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom

Received 18 August 2008; accepted 8 September 2008; published online 26 September 2008

Here we present a microscopic theory of the electrocaloric effect in potassium dihydrogenphosphate, KH 2PO 4, based on Slaters lattice model. The model reproduces the essential features of the experimentally observed behavior and also gives a remarkably accurate description of theelectric eld dependence of the electrocaloric effect. The basic principle of the theory also givesguidelines for a theoretical analysis of other dielectrics and for the further development of materialswith an enhanced electrocaloric effect. 2008 American Institute of Physics . DOI: 10.1063/1.2991443

The electrocaloric EC effect occurs when an electriceld applied under reversible and adiabatic conditionschanges the temperature of a polarizable material. 1,2 Becauseof the detrimental impact of conventional vapor compressionrefrigeration system on the environment, effective EC mate-rials are a possible alternative for the development of envi-ronmentally friendly, solid-state refrigeration technology.Considerable effort has been devote d to the discovery of ma-terials displaying a large EC effect; 3 however, the theory of this phenomenon is not well understood.

Application of an electric eld to an electrically polariz-able material under reversible and adiabatic conditions

causes partial alignment of electric dipoles. In turn, to keepthe total entropy constant, an increase in temperature of thematerial occurs. A general theory of the EC effect has not yetbeen developed and a phenomenological approach is some-times used to describe the relationship between the polariza-tion P , strength of external electric eld E , and tempera-ture change T . Such an approximate relationship for thereversible and adiabatic EC effect can be written as

T / E S = T / C E P / T E , 1

where and C E are material den sity and heat capacity atconstant electric eld, respectively. 4

The earliest experimental measurement of the EC ef fectwas performed on Rochelle salt and reported in 1930. 5 Be-cause the temperature change was small 1 K , it was onlylater in 1956 that the EC effect became more widelydiscussed. 6 The investigation of suitable EC materials inten-sied after 1960 when a signicant number of dielectricswere studied. The EC-induced temperature change by an ex-ternal eld of 25 kV cm 1 was generally lower than 1 K.This has been the main reason that the EC effect has not beenconsidered as a candidate for solid-state cooling. Recentwork has focused on ferroelectrics and it is often found thataround Curie point T C the EC effect can be signicantly

greater compared with the effect far away from T C . For ex-ample, an EC effect of 2.3 K was obtained for modiedPbSc 1 /2Ta 1/ 2O3 in a temperature range of T C 5 K.

7

Mischenko et al. 8 reported a giant EC effect in aPbZr 0.95Ti 0.05O3 thin lm. The results were obtained throughan indirect method by measuring the temperature depen-dence of the polarization. The authors inferred an EC effectof 0.48 K / V in a 350 nm lm 12 K at 480 kV / cm . Otherauthors 1,911 have other experimental results, which cannotbe explained withi n an existing theoretical framework. Forexample, Wiseman 1 reported the simultaneous measurementof the polarization and the EC effect in potassium dihydro-gen phosphate KDP both above and below T C as a functionof an external eld. The effect is strongest near ferroelectricphase transitions as shown in Fig. 1 taken from the work of Wiseman . He was able to t the experimental data for theparaelectric state with Eq. 1 and the CurieWeiss law.However, the ts for the ferroelectric state were not goodeven when he applied a high-order Gibbs equation of state.

The basic principle of the reversible and adiabatic ECeffect is that the tendency of the electric eld E to reducethe entropy S E , T by alignment of electric dipoles has to becompensated by a rise in temperature of the sample therebykeeping the total entropy constant. Hence, S E 1 , T 1

= S E 2 , T 2 where the 1,2 subscripts refer to initial and nalstates.To calculate an EC effect under reversible and adiabatic

conditions as a function of electric eld strength, the expres-sion for a eld dependent entropy and lattice specic heat of the particular system should be known. From these two ex-pressions the temperature rise due to the entropy transferfrom pola r to acoustic modes can be calculated.

Slater 12 introduced a lattice model of hydrogen bondedKDP treating the model by m ean eld theory. The model hassince been widely discussed 1315 and the two-dimens ionalvariant of the model has been solved exactly by Lieb. 16 Afragment of the crystal structure of KDP is shown in Fig. 2.

Each phosphorous atom is tetrahedrally coordinated by foura Electronic mail: [email protected].

APPLIED PHYSICS LETTERS 93 , 122906 2008

0003-6951/2008/93 12 /122906/3/$23.00 2008 American Institute of Physics93 , 122906-1

http://dx.doi.org/10.1063/1.2991443http://dx.doi.org/10.1063/1.2991443http://dx.doi.org/10.1063/1.2991443http://dx.doi.org/10.1063/1.2991443http://dx.doi.org/10.1063/1.2991443http://dx.doi.org/10.1063/1.2991443http://dx.doi.org/10.1063/1.2991443


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oxygen atoms with a hydroge n atom join ing each tetrahedronwith its neighbors. The Slater 12 and Lieb 16 models have beenwidely discussed as classes of so-called ice models where

ice rules apply to the conguration of hydrogen bondsaround each phosphate group. Following Slater we supposethat there are N phosphate groups in the crystal and N hy-drogen atoms. Of the six arrangements of the hydrogen at-oms about a phosphate group, two of these m and p have adipole with energy E of 4.3 1030 C m calculatedfrom the density and saturation polarization, which points inthe c direction, while the other four nonpolar arrangements o lie at right angles to these polar congurations. is thedipole moment and E is the electric eld. The transition tem-perature T C in the model is

T C = k ln2

. 2

Experimentally, T C for KDP is 122.68 K. In a somewhat dif-ferent notation from Slater, we dene the variables

m = N p N

, p = N p N

, o = N o N

. 3

These represent the fraction of dipoles in the m and p con-gurations and the fraction of nonpolar o congurations.

Slater then dened the polarization parameter x by

x = p m , 4

which is a measure of the net dipole moment of the crystaland ranges from 1 to +1. Above the Curie point, Slaterderived an approximation for the electric eld dependence of x given by

x = E

2kT exp

kT kT

, 5

and the following relations describing the temperature de-pendencies of the polar and nonpolar fractions given by

o =4

4 exp2

kT

2 x 2 4 exp

2

kT + exp

2

kT

1 / 2

4 exp2

kT

, 6

p = 1 o + x

2, 7

FIG. 1. Color online Experimental measurements of polarization and ECeffect of KDP as a function of electric eld strength for a the system aboveT C and b below T C as reported by Wiseman Ref. 1 . T C of KDP is122.68 K. The shaded area shows the eld strength range, in which theferroelectric domain growth represents the major contribution to polariza-tion. No signicant EC effect was measured in this range.

FIG. 2. A fragment of the crystal structure of KH 2PO 4 showing phosphatetetrahedra linked by the hydrogen ions. The OHO groups are polar withmainly covalent bond on one oxygen and hydrogen bond on another. Thetwo adjacent blue spheres in the structure represent the two possible posi-tions of the hydrogen atom in the double well potential as shown in thelower gure. The polarity distribution of the OHO groups over the crystalcorresponds to Slaters model.

122906-2 Dunne et al. Appl. Phys. Lett. 93 , 122906 2008



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m =1 o x

2. 8

The electric eld dependent contribution to the entropy fromthe hydrogen bonded system is given by

S = p ln p +

1

2 o2

p + m ln m +

1

2 o2

m + o ln 4 p +

1

2 o

m +12

o

o. 9

In addition to the entropy due to the hydrogen bonded sys-tem, the acoustic phonons in the lattice have also an entropycontribution, which can be derived from the lattice specicheat. The lattice specic heat C G for KDP has been estimatedby calorimetric measurements above T C to be

14

C G = 21.43 + 0.3466 T J mol 1 K1 . 10

Integration of C G / T produces an entropy change per phos-phate group in units of k due to population of the acousticphonon modes given by

S lattice =21.43

R

T

T + 0.3466

T

R. 11

T is the temperature rise assumed small from an initialtemperature T . The two entropy contributions given in Eqs. 9 and 11 enable a temperature rise with electric eld to becalculated using a numerical search procedure.

In Fig. 3 a we show a decrease in entropy due to a

partial alignment of dipoles under an inuence of an externalelectric eld. Under reversible and adiabatic conditions thisdecrease must be compensated by an increase in temperatureto keep the total entorpy constant. The temperature rises areshown in Fig. 3 b , which may be compared to the experi-mental results of Wiseman, 1 as shown in Fig. 1. Consideringthe limitations of the model the agreement is very satisfac-tory.

Below T C , the external electric eld causes ferroelectricdomain growth and the theory presented above does notseem to be applicable A theory of the EC effect in this re-gime has a number of challenges and will need to considerthe coupling between the elastic and electric and polarization

elds. However, some observations seem pertinent. It is re-markable that within the domain growth regime o nly a smallEC effect can be measured as shown in Fig. 1 b ,1 and onlywhen saturation polarization is reached does the EC effectstart to appear. For an estimation of the EC effect from mea-sured polarization values, one should be aware of differentcontributions as there are two types of polarization mecha-nisms occuring: domain growth and dipole alignment. Equa-tion 1 may give a reliable result only when the polarizationchange occurs within a linear regime.

In summary, in this paper we have presented a micro-scopic theory of the EC effect in the paraelectric phase of KDP based on Slaters lattice model. The model reproducesthe essential features of the behavior observed experimen-tally by Wiseman and also provides a satisfactory account of the electric eld dependence of the EC effect. The underly-ing principle of the phenomenon above T C is the propensityof the electric eld to reduce the entropy by aligning the

electric dipoles, and this is compensated by a rise in tempera-ture of the sample thereby keeping the total entropy constantunder adiabatic conditions. Our model also suggests furtherapproaches for a theoretical analysis of the EC effect in otherdielectrics and for the development of materials with an en-hanced EC effect.

1G. G. Wiseman, IEEE Trans. Electron Devices ED-16 , 588 1969 .2J. F. Nye, Physical Properties of Crystals Oxford University Press, Lon-don, 1957 .

3G. Akcay, S. P. Alpay, J. V. Mantese, and G. A. Rosetti, Jr., Appl. Phys.Lett. 90 , 25909 2007 .

4B. A. Tuttle, Ph.D. thesis, University of Illinois at Urbana-Champaign,Urbana, Illinois, 1981.

5

P. Kobeco and Y. Kurtshatov, Z. Phys. 66 , 192 1930 .6H. Granicher, Helv. Phys. Acta 29 , 210 1956 .7L. Shebanovs, K. Borman, W. N. Lawless, and A. Kalvane, Ferroelectrics

273 , 2515 2002 .8S. Mischenko, Q. Zhang, J. F. Scott, R. W. Whatmore, and N. D. Mathur,Science 311 , 1270 2006 .

9G. G. Wiseman and J. K. Kuebler, Phys. Rev. 131 , 2023 1963 .10G. Akcay, S. P. Alpay, J. V. Mantese, and G. A. Rosetti, Jr., Appl. Phys.

Lett. 90 , 252909 2007 .11J. H. Qiu and Q. Jiang, J. Appl. Phys., 103 , 84105 2008 .12J. C. Slater, J. Chem. Phys. 9 , 16 1941 .13H. B. Silsbee and E. A. Uehling, Phys. Rev. 133 , A165 1964 .14W. Reese and L. F. May, Phys. Rev. 162 , 510 1967 .15F. Y. Wu and Z. R. Yang, J. Phys. C 16 , L125 1983 .16E. H. Lieb, Phys. Rev. Lett. 19 , 108 1967 .

FIG. 3. Calculated a decrease in entropy of KH 2PO 4crystal structure per phosphate group caused by dipolealignment at a xed temperature. To compensate forthis, the crystal temperature must rise to keep the totalentropy constant, under adiabatic conditions. b Crystaltemperature increase due to the EC effect as a function

of the electric eld strength. The circles in the plotrepresent the experimental data obtained by Wiseman Ref. 1 .

122906-3 Dunne et al. Appl. Phys. Lett. 93 , 122906 2008

l d d 06 b 2013 14 139 6216 di ib i bj A li i h h // l i / b / i h d i i
http://dx.doi.org/10.1126/science.1123811http://dx.doi.org/10.1103/PhysRev.131.2023http://dx.doi.org/10.1063/1.2750546http://dx.doi.org/10.1063/1.2750546http://dx.doi.org/10.1063/1.1750821http://dx.doi.org/10.1103/PhysRev.162.510http://dx.doi.org/10.1088/0022-3719/16/4/008http://dx.doi.org/10.1103/PhysRevLett.19.108http://dx.doi.org/10.1103/PhysRevLett.19.108http://dx.doi.org/10.1088/0022-3719/16/4/008http://dx.doi.org/10.1103/PhysRev.162.510http://dx.doi.org/10.1063/1.1750821http://dx.doi.org/10.1063/1.2750546http://dx.doi.org/10.1063/1.2750546http://dx.doi.org/10.1103/PhysRev.131.2023http://dx.doi.org/10.1126/science.1123811

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Electro Caloric Effects