Influence of mechanical boundary conditions on ... - Semantic Scholar
JOURNAL OF APPLIED PHYSICS 103, 024104 共2008兲
Influence of mechanical boundary conditions on the electrocaloric properties of ferroelectric thin films G. Akcay, S. P. Alpay,a兲 and G. A. Rossetti, Jr. Materials Science and Engineering Program, University of Connecticut, Storrs, Connecticut 06269, USA and Institute of Materials Science, University of Connecticut, Storrs, Connecticut 06269, USA
J. F. Scott Department of Earth Sciences, University of Cambridge, Cambridge CB2 3EQ, United Kingdom
共Received 24 October 2007; accepted 10 November 2007; published online 25 January 2008兲 A thermodynamic analysis of the electrocaloric 共EC兲 effect in BaTiO3 ferroelectric thin films has been carried out under differing mechanical boundary conditions. It is shown that both the magnitude of the electrocaloric effect and temperature at which it is maximized depend not only on the extent of the applied field change but also on the value of the initial field. For initial fields smaller than a critical value the EC effect is largest at the phase transition temperature but the effect is a strong function of temperature. For external electrical fields larger than this value, conversely, the EC effect is the largest at a higher temperature and is a weak function of temperature. Perfect lateral clamping transforms the first-order phase transition into a second-order transition, lowering the magnitude of the electrocaloric effect and dependence on temperature. Compressive and tensile misfit strains also alter the nature of the phase transition and affect the electrocaloric properties in an analogous way. A compressive misfit strain shifts the maximum in the EC effect to higher temperatures, reduces its magnitude, and reduces its dependence on temperature, while tensile misfit strain results in the opposite effects. Control of the misfit strain by appropriate choice of substrate provides potential means to vary both the magnitude and the temperature sensitivity of the EC effect for use in cooling or thermodielectric power conversion devices. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2831222兴 I. INTRODUCTION
Ferroelectric 共FE兲 perovskites such as BaTiO3 共BT兲, PbTiO3, and solid solutions of PbZrO3 – PbTiO3 关Pb共ZrxTi1−x兲O3, PZT兴 have received great interest because of their potential applications in micro- and nanoelectronics as elements of nonvolatile random access memories 共NVRAMs兲, dynamic random access memories 共DRAMs兲, high dielectric constant capacitors, optical waveguides, tunable dielectric devices, and pyroelectric detectors. To be exploited as components of such devices, FEs must be integrated into thin film structures. A number of theoretical and experimental studies have demonstrated that the nature of the phase transformations between adjacent paraelectric 共PE兲 and FE phases are altered when FEs are fabricated in thin film form and that the dielectric, piezoelectric, and pyroelectric properties are significantly different than those observed in bulk ceramics or crystals 共see, e.g., Refs. 1–7兲. These phenomena can be attributed to the electromechanical coupling between the spontaneous polarization of the FE and the internal stress field. Depending on the specific film-substrate system and the conditions of film deposition, the sources of these internal stresses may include lattice parameter mismatch between film and substrate, the difference between the thermal expansion coefficients of the film and the substrate 共thermal stresses兲, the self-strain of the PE-FE phase transa兲
Author to whom correspondence should be addressed. Electronic mail: [email protected].
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formation if the film is grown at temperatures above the Curie temperature TC, and localized stress fields that emanate from inhomogeneities and defects.8 Although there have been a large number of studies directed at quantifying the ways in which these internal stresses modify the dielectric, pyroelectric, and piezoelectric properties of FE thin films, the influence of mechanical boundary conditions on their electrocaloric 共EC兲 properties has not been fully investigated. Although the EC properties of FEs have long been of interest for use in solid-state cooling devices and power converters,9–12 the EC effect observed in bulk materials has been small and limited by the dielectric breakdown strength of the specimen. However, greatly renewed interest in potential applications for FE-based electrocaloric and thermodielectric devices was simulated after it was reported that large EC responses could be achieved in thin films 共⬃300 nm兲 of PZT 共Ref. 13兲 and Pb共Mg1/3Nb2/3兲O3 – PbTiO3 共PMN-PT兲 共Ref. 14兲 driven at high 共⬃900 kV/ cm兲 electric fields. We have recently computed monodomain contributions to the EC response in mechanically free 共bulk兲 and laterally clamped 共thin film兲 BT using a thermodynamic analysis.15 The results showed that the magnitude of the EC effects observed by experiment in other perovskite thin films are expected to be intrinsic to the stress-free monodomain state under the application of electric fields sufficient to destroy the first-order FE-PE phase transition. Our preliminary calculations indicated that perfect lateral clamping transforms the first-order FE-PE transition in BT into a second-order
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transition, which decreases the magnitude of the EC effect by ⬃20% but reduces its sensitivity to temperature. In this study, building upon these findings and utilizing a similar formalism, we present a more complete theoretical analysis of the influence of mechanical boundary conditions on the EC properties of FE films and take into account lattice mismatch strains and strains that result from thermal expansion mismatch between the film and the substrate. The computations show that as with the dielectric and piezoelectric properties, the magnitude of the EC effect in FE thin films and its dependence on temperature can be controlled by the internal stress state in the films as generated under differing mechanical boundary conditions. II. THERMODYNAMIC THEORY
The reversible change of internal energy 共dU兲 in an elastic dielectric solid follows from the first and second laws of thermodynamics, dU = TdS + du + EdD,
共1兲
where T, , and E are temperature, stress, and electric field, respectively. Here dS, du, and dD are the changes in entropy, strain, and dielectric displacement, respectively. The dielectric displacement D is given by the constitutive relation 共2兲
D = 0E + P S ,
where 0 is the dielectric permittivity of vacuum and PS is the spontaneous polarization. The total free energy of the system is G = U − TS − u − ED,
共3兲
and taking the differential of G and substituting Eq. 共1兲 into Eq. 共3兲 yields dG = − SdT − ud − DdE,
共4兲
the exact differential of which is dG =
冉 冊 G T
dT − ,E
冉 冊 G
d − T,E
冉 冊 G E
dE. T,
共5兲
Equations 共4兲 and 共5兲 give the calorimetric, elastic, and dielectric equations of state,
冉 冊 G T
冉 冊 G
= − S, ,E
冉 冊 G E
= − u, T,E
T,
= − D. 共6兲
Differentiating Eq. 共6兲 results in identities 共Maxwell relations兲 relating the material compliances of the system, e.g.,
冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊
2G − Tu −
−
2G E 2G ET
S = u E =
T
=
u E
D T
u = T E,T =
,T
=
,E
D
S E
,
,T
E,T
,T
ids. Figure 1 shows the change of total entropy as a function of temperature for materials systems with 共a兲 no phase transition, 共b兲 a second-order phase transition, and 共c兲 a firstorder phase transition. While the entropy is continuous for a system with no phase transition, there is a deflection point and a jump for a second-order transition and a first-order transition, respectively. These anomalies disappear at large enough electric fields 共E Ⰷ 0兲 and, theoretically, it is clear that the largest EC response at a given electric field should be observed at TC or T0 for materials undergoing a phase transformation.16 It follows from Eq. 共2兲 that for a linear dielectric having no spontaneous polarization the dielectric displacement is simply D = 0E = P,
共7兲
The last relation provides the most general definition of the electrocaloric effect 共S / E兲,T, and its converse, the pyroelectric effect, 共D / T兲,E, and is valid for all dielectric sol-
共8兲
where P is the polarization induced by an external electric field, E. In this case, the free energy of polarization can be written as G=
,
.
FIG. 1. 共Color online兲 Change of entropy as a function of temperature at E = 0 and E Ⰷ 0 for a system with 共a兲 no phase transition, 共b兲 second-order phase transition, and 共c兲 first-order phase transition.
1 2 P , 20
共9兲
where is the dielectric permittivity of the linear dielectric. For FEs, the dielectric displacement includes both the spontaneous 共PS兲 and induced 共0E兲 polarizations. For the case of a monodomain single-crystal FE such as BT in an unclamped, stress-free state, the polarization-dependent part of the free energy can be expressed using a
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Landau–Devonshire17,18 expansion of the total polarization P, Gbulk = G0 + ␣1 P2 + ␣11P4 + ␣111P6 − EP,
共10兲
where P is the component of the polarization vector directed along one of the cube axes of the PE cubic phase; ␣1, ␣11, and ␣111 are the dielectric stiffness and higher-order dielectric stiffness coefficients; and G0 is the energy of the PE phase. The quadratic coefficient 共␣1兲 is given by the Curie– Weiss law, ␣1 = 共T − TC兲 / 共2C0兲, where TC is the Curie temperature, C is the Curie constant, and 0 is the dielectric permittivity of vacuum. The higher-order dielectric stiffness coefficients, ␣11 and ␣111 may, in general, be analytical functions of the temperature as well. The spontaneous polarization when the external field E = 0 follows from the condition of thermodynamic equilibrium dGbulk = ␣1 + 2␣11P2 + 3␣111P4 = 0 dP
共11兲
− ␣11 + 冑␣211 − 3␣1␣111 , 3␣111
共12兲
or P2S =
where PS is the equilibrium value of the spontaneous polarization. When the FE is in thin film form on a cubic substrate with the epitaxial relation 共001兲film 储 共001兲substrate, the free energy has to be modified to take into account the internal stresses that may arise from lattice and thermal expansion mismatches between the film and the substrate, the self-strain of the FE phase transformation, and the clamping effect of the substrate. Considering the mechanical boundary conditions, i.e., equal in-plane biaxial stress components 共in contacted notation兲 1 = 2, no shear stresses 共4 = 5 = 6 = 0兲, and no out-of-plane stress 共3 = 0兲, the free energy density can be expressed as Gfilm = G0 + ␣1 P2 + ␣11P4 + ␣111P6 − EP + Gel .
共13兲
The elastic energy Gel is given by ˜ 共u − Q P2兲2 . Gel = C m 12
共14兲
Here, um is the in-plane polarization-free misfit strain defined as um =
asubstrate − afilm , asubstrate
共15兲
and Q12P2 is the self-strain, Qij are the cubic electrostrictive ˜ is an effective elascoefficients in contracted notation, and C tic modulus, 2 ˜ = C + C − 2C12 , C 11 12 C11
共16兲
where Cij are the elastic coefficients at constant polarization. After some rearrangement we obtain19
˜ , Gfilm = G0 + ˜␣1 P2 + ˜␣11P4 + ␣111P6 − EP + G el
共17兲
with modified dielectric stiffness coefficients given by ˜, ˜␣1 = ␣1 − 2umQ12C
共18兲
˜, ˜␣11 = ␣11 + Q212C
共19兲
˜ = u2 C ˜ G el m ,
共20兲
and
the polarization-free strain energy. The role of the internal stresses and the clamping effect of the substrate can be determined from the modified dielectric stiffness coefficients. There is a change in the phase transformation temperature that varies linearly with the misfit strain through Eq. 共18兲. The two-dimensional clamping is described by Eq. 共19兲 which is not a function of the misfit strain um. Therefore, properties of the FE will be altered regardless of the misfit between the film and substrate de˜ .2 We note here that if the pending on the magnitude of Q212C FE were polycrystalline with a 共001兲 texture, the above relations would still hold if um were replaced by the thermal strain, uT = 共␣film − ␣substrate兲⌬T, where ␣film and ␣substrate are the linear thermal expansion coefficients of the film and the substrate, respectively. For epitaxial films, um contains the contribution of the thermal strain since it is defined as the difference in the lattice parameters of the film and the substrate. The equilibrium polarization P0 of the film, which has contributions both from the spontaneous polarization and the induced polarization, and its temperature dependence as a function of the applied electric field E and the misfit strain um follow from the condition of thermodynamic equilibrium, Gfilm / P = 0. Substitution of P0共T , E , um兲 into Eq. 共17兲 provides the equilibrium energy G0共T , E , um兲 as a function of temperature, applied electric field, and the misfit strain. For a constant applied electric field and stress, the excess entropy SXS and the excess specific heat ⌬C of the FE follow from Eq. 共6兲, XS SE, 共T,E,um兲 = − T
冉 冊 冉 冊 G0 T
⌬CE,共T,E,um兲 = − T
共21兲
,
E,
2G 0 T2
共22兲
. E,
Using the last part of Eq. 共7兲, the electrocaloric coefficient 共p兲 is defined by p共T,E,um兲 =
冉 冊 冉 冊 P0 T
=
E,
S E
, T,
共23兲
which can be rearranged to explicitly determine the temperature change 共⌬T兲 in the FE film due to a change in the applied electric field. By computing values of CE, and P0 as functions of T, E, and um, this temperature change can be determined through
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FIG. 2. 共Color online兲 Free energy curves of a stress-free monodomain FE exhibiting a first-order close to a second-order phase transformation in the vicinity of the FE-PE transformation at critical temperatures. For BT, these temperatures are given by TC = 383 K, T0 = TC + 7.7 K, T1 = TC + 10 K, and T2 = TC + 18 K and follow from the generalized Landau potential such that T1 = TC + 2C0␣211 / 3␣111, and T2 = TC T0 = TC + C0␣211 / 2␣111, + 6C0␣211 / 5␣111 共Ref. 24兲.
⌬T共T,E,um兲 = − T
冕
Eb
Ea
冉 冊
1 P0 CE, T
dE, E,
共24兲
where Eb − Ea = ⌬E is the difference in the applied electric field. It is clear that the adiabatic temperature change ⌬T occurring in a FE film in response to an applied electric field is a function of both the temperature and the internal strain. In order to compute the electrocaloric properties of BT film materials, all of the intrinsic material property coefficients entering into the above relations for BT were taken from Pertsev et al.20 All of the coefficients were taken to be independent of temperature, strain, and electric field with the exception of the quadratic and quartic dielectric stiffness coefficients 共␣1 and ␣11, respectively兲 which were given a linear dependence on temperature; in particular, ␣11 varies linearly with temperature as 共T − 448 K兲.21 The absolute value of the heat capacity, CE,共T , E , um兲, was estimated by scaling the computed zero-field values of the excess specific heat ⌬CE,共T , E , um兲 given in Eq. 共22兲 to the lattice or “hard mode” contributions taken from the experimental values22 measured over temperature intervals away from those where the FE phase transitions 共cubic→ tetragonal → orthorhombic→ rhombohedral兲 are located. We note that BT displays a first-order close to a second order transformation.23 There is a hysteresis in the polarization around the phase transformation temperature TC 共i.e., the temperature below which the FE phase is globally stable state兲.24 Schematic free energy curves as a function of polarization at critical temperatures near the PE to tetragonal FE transformation are shown in Fig. 2 where T0 is the first-order transition temperature between absolutely stable PE and FE states where the polarization exhibits a discontinuity, T1 is the highest temperature for which the FE phase can remain as a metastable state in the absence of an applied electric field, and T2 is the highest temperature for which the FE phase can remain as a field-induced metastable phase in the presence of an applied electric field. The values of the critical temperatures can be derived from the generalized Landau potential.24
FIG. 3. 共Color online兲 Change of 共a兲 polarization, 共b兲 specific heat capacity, and 共c兲 entropy of stress-free monodomain uniaxial BT with temperature.
III. RESULTS AND DISCUSSION A. Electrocaloric properties of the stress-free monodomain state
Following from Eq. 共23兲, the way in which the magnitude of the electrocaloric effect varies subject to a change in electric field at constant temperature can be best appreciated by examining the behavior of the excess entropy. The application of the field E conjugate to the order parameter P destroys the FE phase transition and at a sufficiently large field E = E*, the discontinuities at the temperature T0 where the PE and FE phases are in equilibrium will effectively disappear. Figure 3 shows the temperature variation of the polarization, specific heat, and the excess entropy near the first-order cubic to tetragonal FE phase transition for a BT crystal in the unconstrained and stress-free 共bulk兲 monodomain state as a function of applied field. On increasing the field from an initial value Ea = 0 to a nonzero field Eb ⬎ Ea, the maximum change in entropy will always occur at the temperature of the first-order zero-field FE phase transition T = T0. Alternatively, if the field increase is made to take place starting from a much higher value that is sufficient to cause the discontinuities at T0 to effectively disappear as seen in Figs. 3共a兲 and 1共b兲, e.g., at Ea = 50 kV/ cm⬎ E*, the maximum entropy change will always occur at the higher temperature T = T*. On the other hand, when the field is increased from an inter-
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B. Influence of mechanical boundary conditions on the electrocaloric properties
FIG. 4. 共Color online兲 Pseudocolor plots of 共a兲 equilibrium polarization and 共b兲 specific heat capacity of BT as functions of misfit strain and temperature.
mediate nonzero value, e.g., Ea = 10 kV/ cm⬍ E*, the maximum entropy change will occur at some different but intermediate temperature T0 ⬍ Ti ⬍ T*. Consequently, it is clear that when the electric field is changed between two values Ea and Eb both the magnitude of the electrocaloric effect and temperature at which it is maximized depend not only on the extent of the field change ⌬E = Eb − Ea but also on the value of the initial field Ea.15
Using the minimization of the free energy and Eq. 共22兲, the polarization and the specific heat of BT as a function of misfit strain and temperature can be evaluated. Figure 4 shows the pseudocolor plots of the equilibrium polarization and the specific heat of BT at E = 0. The effect of perfect lateral clamping 共e.g., 0% misfit strain兲 due to film-substrate lattice mismatch is to smooth out the discontinuities in polarization and specific heat at the FE phase transition, as seen in Figs. 4共a兲 and 4共b兲. It is also seen in the figure that while the polarization increases strongly with increasing compressive strain, which induces extra charge in BT, the signature discontinuities of the polarization and specific heat near the PE to FE phase transition are diminished. The opposite effect is observed on increasing the tensile misfit strain. With increasing compressive misfit strain the first-order phase transition becomes “second-order-like.” Conversely, increasing of tensile misfit strain reintroduces first-order phase transition behavior. This is shown more clearly in Fig. 5 where the polarization and specific heat of BT at −0.1% 关compressive—Figs. 5共a兲 and 5共d兲兴, 0% 关clamped—Figs. 5共b兲 and 5共e兲兴, and +0.1% 关tensile—Figs. 5共c兲 and 5共f兲兴 misfit strains are plotted as functions of temperature and electric field. Although the phase transition temperature does not change under perfect lateral clamping, it is a function of the misfit strain. When the strain is compressive, the transition temperature is raised due to the enhanced polarization through electrostriction while under tensile strain the polarization is suppressed causing the phase transition to occur at lower temperatures.19
FIG. 5. 共Color online兲 Plots of polarization and specific heat as functions of temperature and applied electric field for BT at 关共a兲 and 共d兲兴 um = −0.1%, 关共b兲 and 共e兲兴 um = 0%, and 关共c兲 and 共f兲兴 um = + 0.1%.
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FIG. 7. 共Color online兲 Maximum change of internal energy 共⌬Qmax兲 at Tmax as a function of misfit strain 共a兲 in three-dimensional 共3D兲 plane, and 共b兲 in two-dimensional 共2D兲 plane together with maximum converse pyroelectric coefficient for comparison.
FIG. 6. 共Color online兲 Three-dimensional plots of temperature change in BT as functions of T and ⌬E 共Ea = 50 kV/ cm兲 at 共a兲 um = −0.1%, 共b兲 um = 0%, and 共c兲 um = + 0.1%; 共d兲 two-dimensional plot of temperature change in BT as a function of temperature at different misfit strains 共⌬E = 100 kV/ cm, Ea = 50 kV/ cm兲.
Using Eq. 共24兲, the temperature difference 共⌬T兲 in BT due to the applied electric field change can be determined as a function of temperature and misfit strain for different electrical field changes, ⌬E = Eb − Ea, Ea = 50 kV/ cm⬎ E*, where as shown in Figs. 3 and 5, the discontinuities at the PE to FE transition temperature T0 are fully suppressed. Figure 6 shows three-dimensional diagrams of ⌬T versus temperature and ⌬E at 共a兲 −0.1% 共compressive兲, 共b兲 0% 共clamped兲, and
共c兲 +0.1% 共tensile兲 misfit strains. As expected from Eq. 共24兲, in all cases an increase in the applied electric field difference at constant temperature leads to a linear increase in the magnitude of the temperature change ⌬T due to the electrocaloric effect. As regards the unclamped case described above, the temperature 共Tmax兲 where this change in temperature 共⌬Tmax兲 is largest does not depend on the applied field difference because the initial electric field Ea has been chosen in all cases to be higher than E*. However, both the magnitude of the EC effect as measured by ⌬T and the temperature at which it is maximized do depend on the mechanical boundary conditions. As a consequence of the influence of the mechanical boundary conditions on the temperature and order of the phase transition, a compressive misfit strain shifts the maximum in the EC effect to higher temperatures, reduces its magnitude, and reduces its dependence on temperature. Conversely, a tensile misfit strain shifts the maximum in the EC effect to lower temperatures, increases its magnitude, and increases its dependence on temperature. This is shown more clearly in Fig. 6共d兲 where results are plotted at constant applied field difference ⌬E = 100 kV/ cm. Consequently by appropriately choosing the substrate material to induce a compressive misfit strain in BT film 关e.g., LaAlO3 共LAO兲兴 or a tensile misfit strain 共e.g., MgO or Si兲 the magnitude and temperature dependence of the EC effect can be controlled. Figure 7共a兲 shows the change in internal energy associated with the EC effect in BT film at different misfit strains together with the corresponding temperature Tmax where the change in internal energy is maximum at ⌬E = 100 kV/ cm, Ea = 50 kV/ cm⬎ E*. Noting, as shown in Fig. 5, that the specific heat CE,共T , E , um兲 is a weak function of field at E ⬎ E*, the internal energy as a function of um can be evaluated from
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kV/cm . ⌬Qmax共um兲 = 兩CE,兩T=Tmax 兩⌬Tmax兩⌬E=100 E =50 kV/cm E⬎E*
a
共25兲
As expected, ⌬Qmax exhibits a nearly linear function of the misfit strain and as the misfit strain is varied from compressive to tensile the maximum values of the internal energy change increase. Using the data in Fig. 7共a兲 it is also possible to compute the electrocaloric coefficient p at the temperature Tm where the adiabatic temperature change exhibits its maximum,
冉
冊
⌬Qmax 1 pmax共um兲 ⬵ . Tmax ⌬E
共26兲
Figure 7共b兲 compares the magnitude of the change in internal energy at T = Tmax to the change in magnitude of the maximum pyroelectric effect of BT film as functions of misfit strain. Naturally, the maximum in the electrocaloric coefficient follows the same linear trend as the change in internal energy of the BT film as a function of the misfit strain. IV. CONCLUSIONS
A thermodynamic analysis of the EC effect in BaTiO3 FE thin films under differing mechanical boundary conditions shows the following. 共1兲 Both the magnitude of the electrocaloric effect and temperature at which it is maximized depend not only on the extent of the field change ⌬E = Eb − Ea but also on the value of the initial field Ea. The magnitudes of large EC effects previously observed by experiment10,11 at high electric fields are comparable to the intrinsic magnitude computed for the bulk monodomain state.12 共2兲 For applied electrical fields with initial values Ea ⬍ E*, the EC effect is largest at the phase transition temperature T0; however, the effect is a strong function of temperature. For external electrical fields larger than E*, conversely, the EC effect is the largest at a higher temperature T* ⬎ T0 and it is a weak function of temperature. 共3兲 Perfect lateral clamping transforms the first-order phase transition at T0 into a second-order transition, which lowers the magnitude of the electrocaloric effect but reduces its dependence on temperature. 共4兲 A compressive misfit strain shifts the maximum in the EC effect to higher temperatures, reduces its magnitude, and reduces its dependence on temperature. Conversely,
a tensile misfit strain shifts the maximum in the EC effect to lower temperatures, increases its magnitude, and increases its dependence on temperature. 共5兲 Control of the misfit strain by appropriate choice of substrate provides potential means to vary both the magnitude and the temperature sensitivity of the EC effect. ACKNOWLEDGMENTS
The work at UConn was supported by the U.S. Army Research Office through Grant No. W911NF-05-1-0528. 1
G. R. Bai, S. K. Streiffer, P. K. Baumann, O. Auciello, K. Ghosh, S. Stemmer, A. Munkholm, C. Thompson, R. A. Rao, and C. B. Eom, Appl. Phys. Lett. 76, 3106 共2000兲. 2 K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L. Q. Chen, D. G. Schlom, and C. B. Eom, Science 306, 1005 共2004兲. 3 J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K. M. Rabe, M. Wuttig, and R. Ramesh, Science 299, 1719 共2003兲. 4 W. Chang, S. W. Kirchoefer, J. M. Pond, J. A. Bellotti, S. B. Qadri, J. H. Haeni, and D. G. Schlom, J. Appl. Phys. 96, 6629 共2004兲. 5 N. J. Donnelly, G. Catalan, C. Morros, R. M. Bowman, and J. M. Gregg, J. Appl. Phys. 93, 9924 共2003兲. 6 W. K. Simon, E. K. Akdogan, and A. Safari, J. Appl. Phys. 97, 103530 共2005兲. 7 Z.-G. Ban and S. P. Alpay, Appl. Phys. Lett. 82, 3499 共2003兲. 8 V. Nagarajan, C. L. Jia, H. Kohlstedt, R. Waser, I. B. Misirlioglu, S. P. Alpay, and R. Ramesh, Appl. Phys. Lett. 86, 192910 共2005兲. 9 P. D. Thacher, J. Appl. Phys. 39, 1996 共1968兲. 10 E. Fatuzzo, H. Kiess, and R. Nitsche, J. Appl. Phys. 37, 510 共1966兲. 11 J. D. Childress, J. Appl. Phys. 33, 1793 共1962兲. 12 J. H. He, J. C. Chen, Y. H. Zhou, and J. T. Wang, Energy Convers. Manage. 43, 2319 共2002兲. 13 A. S. Mischenko, Q. Zhang, J. F. Scott, R. W. Whatmore, and N. D. Mathur, Science 311, 1270 共2006兲. 14 A. S. Mischenko, Q. Zhang, R. W. Whatmore, J. F. Scott, and N. D. Mathur, Appl. Phys. Lett. 89, 242912 共2006兲. 15 G. Akcay, S. P. Alpay, J. V. Mantese, and G. A. Rossetti, Jr., Appl. Phys. Lett. 90, 252909 共2007兲. 16 J. F. Scott, Dielectric Properties of Insulators, Encyclopedia of Applied Physics Vol. 5 共VCH, Berlin, 1993兲. 17 A. F. Devonshire, Philos. Mag. 42, 1065 共1951兲. 18 A. F. Devonshire, Philos. Mag. 40, 1040 共1949兲. 19 S. P. Alpay, I. B. Misirlioglu, A. Sharma, and Z.-G. Ban, J. Appl. Phys. 95, 8118 共2004兲. 20 N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Phys. Rev. Lett. 80, 1988 共1998兲. 21 Coefficients for BT in SI Units: C = 1.7⫻ 105, TC = 383 K, ␣11 = 3.6共T − 448兲 ⫻ 106, ␣111 = 6.6⫻ 109, Q11 = 0.11, Q12 = −0.043, C11 = 1.75⫻ 1011, and C12 = 8.46⫻ 1010. 22 F. Jona and G. Shirane, Ferroelectric Crystals 共Dover, New York, 1962兲. 23 J. F. Scott, J. Phys. Soc. Jpn. 58, 4487 共1989兲. 24 E. Fatuzzo and W. J. Merz, Ferroelectricity 共Wiley, New York, 1967兲.
Influence of mechanical boundary conditions on the electrocaloric properties of ferroelectric thin films G. Akcay, S. P. Alpay,a兲 and G. A. Rossetti, Jr. Materials Science and Engineering Program, University of Connecticut, Storrs, Connecticut 06269, USA and Institute of Materials Science, University of Connecticut, Storrs, Connecticut 06269, USA
J. F. Scott Department of Earth Sciences, University of Cambridge, Cambridge CB2 3EQ, United Kingdom
共Received 24 October 2007; accepted 10 November 2007; published online 25 January 2008兲 A thermodynamic analysis of the electrocaloric 共EC兲 effect in BaTiO3 ferroelectric thin films has been carried out under differing mechanical boundary conditions. It is shown that both the magnitude of the electrocaloric effect and temperature at which it is maximized depend not only on the extent of the applied field change but also on the value of the initial field. For initial fields smaller than a critical value the EC effect is largest at the phase transition temperature but the effect is a strong function of temperature. For external electrical fields larger than this value, conversely, the EC effect is the largest at a higher temperature and is a weak function of temperature. Perfect lateral clamping transforms the first-order phase transition into a second-order transition, lowering the magnitude of the electrocaloric effect and dependence on temperature. Compressive and tensile misfit strains also alter the nature of the phase transition and affect the electrocaloric properties in an analogous way. A compressive misfit strain shifts the maximum in the EC effect to higher temperatures, reduces its magnitude, and reduces its dependence on temperature, while tensile misfit strain results in the opposite effects. Control of the misfit strain by appropriate choice of substrate provides potential means to vary both the magnitude and the temperature sensitivity of the EC effect for use in cooling or thermodielectric power conversion devices. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2831222兴 I. INTRODUCTION
Ferroelectric 共FE兲 perovskites such as BaTiO3 共BT兲, PbTiO3, and solid solutions of PbZrO3 – PbTiO3 关Pb共ZrxTi1−x兲O3, PZT兴 have received great interest because of their potential applications in micro- and nanoelectronics as elements of nonvolatile random access memories 共NVRAMs兲, dynamic random access memories 共DRAMs兲, high dielectric constant capacitors, optical waveguides, tunable dielectric devices, and pyroelectric detectors. To be exploited as components of such devices, FEs must be integrated into thin film structures. A number of theoretical and experimental studies have demonstrated that the nature of the phase transformations between adjacent paraelectric 共PE兲 and FE phases are altered when FEs are fabricated in thin film form and that the dielectric, piezoelectric, and pyroelectric properties are significantly different than those observed in bulk ceramics or crystals 共see, e.g., Refs. 1–7兲. These phenomena can be attributed to the electromechanical coupling between the spontaneous polarization of the FE and the internal stress field. Depending on the specific film-substrate system and the conditions of film deposition, the sources of these internal stresses may include lattice parameter mismatch between film and substrate, the difference between the thermal expansion coefficients of the film and the substrate 共thermal stresses兲, the self-strain of the PE-FE phase transa兲
Author to whom correspondence should be addressed. Electronic mail: [email protected].
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formation if the film is grown at temperatures above the Curie temperature TC, and localized stress fields that emanate from inhomogeneities and defects.8 Although there have been a large number of studies directed at quantifying the ways in which these internal stresses modify the dielectric, pyroelectric, and piezoelectric properties of FE thin films, the influence of mechanical boundary conditions on their electrocaloric 共EC兲 properties has not been fully investigated. Although the EC properties of FEs have long been of interest for use in solid-state cooling devices and power converters,9–12 the EC effect observed in bulk materials has been small and limited by the dielectric breakdown strength of the specimen. However, greatly renewed interest in potential applications for FE-based electrocaloric and thermodielectric devices was simulated after it was reported that large EC responses could be achieved in thin films 共⬃300 nm兲 of PZT 共Ref. 13兲 and Pb共Mg1/3Nb2/3兲O3 – PbTiO3 共PMN-PT兲 共Ref. 14兲 driven at high 共⬃900 kV/ cm兲 electric fields. We have recently computed monodomain contributions to the EC response in mechanically free 共bulk兲 and laterally clamped 共thin film兲 BT using a thermodynamic analysis.15 The results showed that the magnitude of the EC effects observed by experiment in other perovskite thin films are expected to be intrinsic to the stress-free monodomain state under the application of electric fields sufficient to destroy the first-order FE-PE phase transition. Our preliminary calculations indicated that perfect lateral clamping transforms the first-order FE-PE transition in BT into a second-order
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transition, which decreases the magnitude of the EC effect by ⬃20% but reduces its sensitivity to temperature. In this study, building upon these findings and utilizing a similar formalism, we present a more complete theoretical analysis of the influence of mechanical boundary conditions on the EC properties of FE films and take into account lattice mismatch strains and strains that result from thermal expansion mismatch between the film and the substrate. The computations show that as with the dielectric and piezoelectric properties, the magnitude of the EC effect in FE thin films and its dependence on temperature can be controlled by the internal stress state in the films as generated under differing mechanical boundary conditions. II. THERMODYNAMIC THEORY
The reversible change of internal energy 共dU兲 in an elastic dielectric solid follows from the first and second laws of thermodynamics, dU = TdS + du + EdD,
共1兲
where T, , and E are temperature, stress, and electric field, respectively. Here dS, du, and dD are the changes in entropy, strain, and dielectric displacement, respectively. The dielectric displacement D is given by the constitutive relation 共2兲
D = 0E + P S ,
where 0 is the dielectric permittivity of vacuum and PS is the spontaneous polarization. The total free energy of the system is G = U − TS − u − ED,
共3兲
and taking the differential of G and substituting Eq. 共1兲 into Eq. 共3兲 yields dG = − SdT − ud − DdE,
共4兲
the exact differential of which is dG =
冉 冊 G T
dT − ,E
冉 冊 G
d − T,E
冉 冊 G E
dE. T,
共5兲
Equations 共4兲 and 共5兲 give the calorimetric, elastic, and dielectric equations of state,
冉 冊 G T
冉 冊 G
= − S, ,E
冉 冊 G E
= − u, T,E
T,
= − D. 共6兲
Differentiating Eq. 共6兲 results in identities 共Maxwell relations兲 relating the material compliances of the system, e.g.,
冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊
2G − Tu −
−
2G E 2G ET
S = u E =
T
=
u E
D T
u = T E,T =
,T
=
,E
D
S E
,
,T
E,T
,T
ids. Figure 1 shows the change of total entropy as a function of temperature for materials systems with 共a兲 no phase transition, 共b兲 a second-order phase transition, and 共c兲 a firstorder phase transition. While the entropy is continuous for a system with no phase transition, there is a deflection point and a jump for a second-order transition and a first-order transition, respectively. These anomalies disappear at large enough electric fields 共E Ⰷ 0兲 and, theoretically, it is clear that the largest EC response at a given electric field should be observed at TC or T0 for materials undergoing a phase transformation.16 It follows from Eq. 共2兲 that for a linear dielectric having no spontaneous polarization the dielectric displacement is simply D = 0E = P,
共7兲
The last relation provides the most general definition of the electrocaloric effect 共S / E兲,T, and its converse, the pyroelectric effect, 共D / T兲,E, and is valid for all dielectric sol-
共8兲
where P is the polarization induced by an external electric field, E. In this case, the free energy of polarization can be written as G=
,
.
FIG. 1. 共Color online兲 Change of entropy as a function of temperature at E = 0 and E Ⰷ 0 for a system with 共a兲 no phase transition, 共b兲 second-order phase transition, and 共c兲 first-order phase transition.
1 2 P , 20
共9兲
where is the dielectric permittivity of the linear dielectric. For FEs, the dielectric displacement includes both the spontaneous 共PS兲 and induced 共0E兲 polarizations. For the case of a monodomain single-crystal FE such as BT in an unclamped, stress-free state, the polarization-dependent part of the free energy can be expressed using a
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Landau–Devonshire17,18 expansion of the total polarization P, Gbulk = G0 + ␣1 P2 + ␣11P4 + ␣111P6 − EP,
共10兲
where P is the component of the polarization vector directed along one of the cube axes of the PE cubic phase; ␣1, ␣11, and ␣111 are the dielectric stiffness and higher-order dielectric stiffness coefficients; and G0 is the energy of the PE phase. The quadratic coefficient 共␣1兲 is given by the Curie– Weiss law, ␣1 = 共T − TC兲 / 共2C0兲, where TC is the Curie temperature, C is the Curie constant, and 0 is the dielectric permittivity of vacuum. The higher-order dielectric stiffness coefficients, ␣11 and ␣111 may, in general, be analytical functions of the temperature as well. The spontaneous polarization when the external field E = 0 follows from the condition of thermodynamic equilibrium dGbulk = ␣1 + 2␣11P2 + 3␣111P4 = 0 dP
共11兲
− ␣11 + 冑␣211 − 3␣1␣111 , 3␣111
共12兲
or P2S =
where PS is the equilibrium value of the spontaneous polarization. When the FE is in thin film form on a cubic substrate with the epitaxial relation 共001兲film 储 共001兲substrate, the free energy has to be modified to take into account the internal stresses that may arise from lattice and thermal expansion mismatches between the film and the substrate, the self-strain of the FE phase transformation, and the clamping effect of the substrate. Considering the mechanical boundary conditions, i.e., equal in-plane biaxial stress components 共in contacted notation兲 1 = 2, no shear stresses 共4 = 5 = 6 = 0兲, and no out-of-plane stress 共3 = 0兲, the free energy density can be expressed as Gfilm = G0 + ␣1 P2 + ␣11P4 + ␣111P6 − EP + Gel .
共13兲
The elastic energy Gel is given by ˜ 共u − Q P2兲2 . Gel = C m 12
共14兲
Here, um is the in-plane polarization-free misfit strain defined as um =
asubstrate − afilm , asubstrate
共15兲
and Q12P2 is the self-strain, Qij are the cubic electrostrictive ˜ is an effective elascoefficients in contracted notation, and C tic modulus, 2 ˜ = C + C − 2C12 , C 11 12 C11
共16兲
where Cij are the elastic coefficients at constant polarization. After some rearrangement we obtain19
˜ , Gfilm = G0 + ˜␣1 P2 + ˜␣11P4 + ␣111P6 − EP + G el
共17兲
with modified dielectric stiffness coefficients given by ˜, ˜␣1 = ␣1 − 2umQ12C
共18兲
˜, ˜␣11 = ␣11 + Q212C
共19兲
˜ = u2 C ˜ G el m ,
共20兲
and
the polarization-free strain energy. The role of the internal stresses and the clamping effect of the substrate can be determined from the modified dielectric stiffness coefficients. There is a change in the phase transformation temperature that varies linearly with the misfit strain through Eq. 共18兲. The two-dimensional clamping is described by Eq. 共19兲 which is not a function of the misfit strain um. Therefore, properties of the FE will be altered regardless of the misfit between the film and substrate de˜ .2 We note here that if the pending on the magnitude of Q212C FE were polycrystalline with a 共001兲 texture, the above relations would still hold if um were replaced by the thermal strain, uT = 共␣film − ␣substrate兲⌬T, where ␣film and ␣substrate are the linear thermal expansion coefficients of the film and the substrate, respectively. For epitaxial films, um contains the contribution of the thermal strain since it is defined as the difference in the lattice parameters of the film and the substrate. The equilibrium polarization P0 of the film, which has contributions both from the spontaneous polarization and the induced polarization, and its temperature dependence as a function of the applied electric field E and the misfit strain um follow from the condition of thermodynamic equilibrium, Gfilm / P = 0. Substitution of P0共T , E , um兲 into Eq. 共17兲 provides the equilibrium energy G0共T , E , um兲 as a function of temperature, applied electric field, and the misfit strain. For a constant applied electric field and stress, the excess entropy SXS and the excess specific heat ⌬C of the FE follow from Eq. 共6兲, XS SE, 共T,E,um兲 = − T
冉 冊 冉 冊 G0 T
⌬CE,共T,E,um兲 = − T
共21兲
,
E,
2G 0 T2
共22兲
. E,
Using the last part of Eq. 共7兲, the electrocaloric coefficient 共p兲 is defined by p共T,E,um兲 =
冉 冊 冉 冊 P0 T
=
E,
S E
, T,
共23兲
which can be rearranged to explicitly determine the temperature change 共⌬T兲 in the FE film due to a change in the applied electric field. By computing values of CE, and P0 as functions of T, E, and um, this temperature change can be determined through
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FIG. 2. 共Color online兲 Free energy curves of a stress-free monodomain FE exhibiting a first-order close to a second-order phase transformation in the vicinity of the FE-PE transformation at critical temperatures. For BT, these temperatures are given by TC = 383 K, T0 = TC + 7.7 K, T1 = TC + 10 K, and T2 = TC + 18 K and follow from the generalized Landau potential such that T1 = TC + 2C0␣211 / 3␣111, and T2 = TC T0 = TC + C0␣211 / 2␣111, + 6C0␣211 / 5␣111 共Ref. 24兲.
⌬T共T,E,um兲 = − T
冕
Eb
Ea
冉 冊
1 P0 CE, T
dE, E,
共24兲
where Eb − Ea = ⌬E is the difference in the applied electric field. It is clear that the adiabatic temperature change ⌬T occurring in a FE film in response to an applied electric field is a function of both the temperature and the internal strain. In order to compute the electrocaloric properties of BT film materials, all of the intrinsic material property coefficients entering into the above relations for BT were taken from Pertsev et al.20 All of the coefficients were taken to be independent of temperature, strain, and electric field with the exception of the quadratic and quartic dielectric stiffness coefficients 共␣1 and ␣11, respectively兲 which were given a linear dependence on temperature; in particular, ␣11 varies linearly with temperature as 共T − 448 K兲.21 The absolute value of the heat capacity, CE,共T , E , um兲, was estimated by scaling the computed zero-field values of the excess specific heat ⌬CE,共T , E , um兲 given in Eq. 共22兲 to the lattice or “hard mode” contributions taken from the experimental values22 measured over temperature intervals away from those where the FE phase transitions 共cubic→ tetragonal → orthorhombic→ rhombohedral兲 are located. We note that BT displays a first-order close to a second order transformation.23 There is a hysteresis in the polarization around the phase transformation temperature TC 共i.e., the temperature below which the FE phase is globally stable state兲.24 Schematic free energy curves as a function of polarization at critical temperatures near the PE to tetragonal FE transformation are shown in Fig. 2 where T0 is the first-order transition temperature between absolutely stable PE and FE states where the polarization exhibits a discontinuity, T1 is the highest temperature for which the FE phase can remain as a metastable state in the absence of an applied electric field, and T2 is the highest temperature for which the FE phase can remain as a field-induced metastable phase in the presence of an applied electric field. The values of the critical temperatures can be derived from the generalized Landau potential.24
FIG. 3. 共Color online兲 Change of 共a兲 polarization, 共b兲 specific heat capacity, and 共c兲 entropy of stress-free monodomain uniaxial BT with temperature.
III. RESULTS AND DISCUSSION A. Electrocaloric properties of the stress-free monodomain state
Following from Eq. 共23兲, the way in which the magnitude of the electrocaloric effect varies subject to a change in electric field at constant temperature can be best appreciated by examining the behavior of the excess entropy. The application of the field E conjugate to the order parameter P destroys the FE phase transition and at a sufficiently large field E = E*, the discontinuities at the temperature T0 where the PE and FE phases are in equilibrium will effectively disappear. Figure 3 shows the temperature variation of the polarization, specific heat, and the excess entropy near the first-order cubic to tetragonal FE phase transition for a BT crystal in the unconstrained and stress-free 共bulk兲 monodomain state as a function of applied field. On increasing the field from an initial value Ea = 0 to a nonzero field Eb ⬎ Ea, the maximum change in entropy will always occur at the temperature of the first-order zero-field FE phase transition T = T0. Alternatively, if the field increase is made to take place starting from a much higher value that is sufficient to cause the discontinuities at T0 to effectively disappear as seen in Figs. 3共a兲 and 1共b兲, e.g., at Ea = 50 kV/ cm⬎ E*, the maximum entropy change will always occur at the higher temperature T = T*. On the other hand, when the field is increased from an inter-
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B. Influence of mechanical boundary conditions on the electrocaloric properties
FIG. 4. 共Color online兲 Pseudocolor plots of 共a兲 equilibrium polarization and 共b兲 specific heat capacity of BT as functions of misfit strain and temperature.
mediate nonzero value, e.g., Ea = 10 kV/ cm⬍ E*, the maximum entropy change will occur at some different but intermediate temperature T0 ⬍ Ti ⬍ T*. Consequently, it is clear that when the electric field is changed between two values Ea and Eb both the magnitude of the electrocaloric effect and temperature at which it is maximized depend not only on the extent of the field change ⌬E = Eb − Ea but also on the value of the initial field Ea.15
Using the minimization of the free energy and Eq. 共22兲, the polarization and the specific heat of BT as a function of misfit strain and temperature can be evaluated. Figure 4 shows the pseudocolor plots of the equilibrium polarization and the specific heat of BT at E = 0. The effect of perfect lateral clamping 共e.g., 0% misfit strain兲 due to film-substrate lattice mismatch is to smooth out the discontinuities in polarization and specific heat at the FE phase transition, as seen in Figs. 4共a兲 and 4共b兲. It is also seen in the figure that while the polarization increases strongly with increasing compressive strain, which induces extra charge in BT, the signature discontinuities of the polarization and specific heat near the PE to FE phase transition are diminished. The opposite effect is observed on increasing the tensile misfit strain. With increasing compressive misfit strain the first-order phase transition becomes “second-order-like.” Conversely, increasing of tensile misfit strain reintroduces first-order phase transition behavior. This is shown more clearly in Fig. 5 where the polarization and specific heat of BT at −0.1% 关compressive—Figs. 5共a兲 and 5共d兲兴, 0% 关clamped—Figs. 5共b兲 and 5共e兲兴, and +0.1% 关tensile—Figs. 5共c兲 and 5共f兲兴 misfit strains are plotted as functions of temperature and electric field. Although the phase transition temperature does not change under perfect lateral clamping, it is a function of the misfit strain. When the strain is compressive, the transition temperature is raised due to the enhanced polarization through electrostriction while under tensile strain the polarization is suppressed causing the phase transition to occur at lower temperatures.19
FIG. 5. 共Color online兲 Plots of polarization and specific heat as functions of temperature and applied electric field for BT at 关共a兲 and 共d兲兴 um = −0.1%, 关共b兲 and 共e兲兴 um = 0%, and 关共c兲 and 共f兲兴 um = + 0.1%.
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FIG. 7. 共Color online兲 Maximum change of internal energy 共⌬Qmax兲 at Tmax as a function of misfit strain 共a兲 in three-dimensional 共3D兲 plane, and 共b兲 in two-dimensional 共2D兲 plane together with maximum converse pyroelectric coefficient for comparison.
FIG. 6. 共Color online兲 Three-dimensional plots of temperature change in BT as functions of T and ⌬E 共Ea = 50 kV/ cm兲 at 共a兲 um = −0.1%, 共b兲 um = 0%, and 共c兲 um = + 0.1%; 共d兲 two-dimensional plot of temperature change in BT as a function of temperature at different misfit strains 共⌬E = 100 kV/ cm, Ea = 50 kV/ cm兲.
Using Eq. 共24兲, the temperature difference 共⌬T兲 in BT due to the applied electric field change can be determined as a function of temperature and misfit strain for different electrical field changes, ⌬E = Eb − Ea, Ea = 50 kV/ cm⬎ E*, where as shown in Figs. 3 and 5, the discontinuities at the PE to FE transition temperature T0 are fully suppressed. Figure 6 shows three-dimensional diagrams of ⌬T versus temperature and ⌬E at 共a兲 −0.1% 共compressive兲, 共b兲 0% 共clamped兲, and
共c兲 +0.1% 共tensile兲 misfit strains. As expected from Eq. 共24兲, in all cases an increase in the applied electric field difference at constant temperature leads to a linear increase in the magnitude of the temperature change ⌬T due to the electrocaloric effect. As regards the unclamped case described above, the temperature 共Tmax兲 where this change in temperature 共⌬Tmax兲 is largest does not depend on the applied field difference because the initial electric field Ea has been chosen in all cases to be higher than E*. However, both the magnitude of the EC effect as measured by ⌬T and the temperature at which it is maximized do depend on the mechanical boundary conditions. As a consequence of the influence of the mechanical boundary conditions on the temperature and order of the phase transition, a compressive misfit strain shifts the maximum in the EC effect to higher temperatures, reduces its magnitude, and reduces its dependence on temperature. Conversely, a tensile misfit strain shifts the maximum in the EC effect to lower temperatures, increases its magnitude, and increases its dependence on temperature. This is shown more clearly in Fig. 6共d兲 where results are plotted at constant applied field difference ⌬E = 100 kV/ cm. Consequently by appropriately choosing the substrate material to induce a compressive misfit strain in BT film 关e.g., LaAlO3 共LAO兲兴 or a tensile misfit strain 共e.g., MgO or Si兲 the magnitude and temperature dependence of the EC effect can be controlled. Figure 7共a兲 shows the change in internal energy associated with the EC effect in BT film at different misfit strains together with the corresponding temperature Tmax where the change in internal energy is maximum at ⌬E = 100 kV/ cm, Ea = 50 kV/ cm⬎ E*. Noting, as shown in Fig. 5, that the specific heat CE,共T , E , um兲 is a weak function of field at E ⬎ E*, the internal energy as a function of um can be evaluated from
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kV/cm . ⌬Qmax共um兲 = 兩CE,兩T=Tmax 兩⌬Tmax兩⌬E=100 E =50 kV/cm E⬎E*
a
共25兲
As expected, ⌬Qmax exhibits a nearly linear function of the misfit strain and as the misfit strain is varied from compressive to tensile the maximum values of the internal energy change increase. Using the data in Fig. 7共a兲 it is also possible to compute the electrocaloric coefficient p at the temperature Tm where the adiabatic temperature change exhibits its maximum,
冉
冊
⌬Qmax 1 pmax共um兲 ⬵ . Tmax ⌬E
共26兲
Figure 7共b兲 compares the magnitude of the change in internal energy at T = Tmax to the change in magnitude of the maximum pyroelectric effect of BT film as functions of misfit strain. Naturally, the maximum in the electrocaloric coefficient follows the same linear trend as the change in internal energy of the BT film as a function of the misfit strain. IV. CONCLUSIONS
A thermodynamic analysis of the EC effect in BaTiO3 FE thin films under differing mechanical boundary conditions shows the following. 共1兲 Both the magnitude of the electrocaloric effect and temperature at which it is maximized depend not only on the extent of the field change ⌬E = Eb − Ea but also on the value of the initial field Ea. The magnitudes of large EC effects previously observed by experiment10,11 at high electric fields are comparable to the intrinsic magnitude computed for the bulk monodomain state.12 共2兲 For applied electrical fields with initial values Ea ⬍ E*, the EC effect is largest at the phase transition temperature T0; however, the effect is a strong function of temperature. For external electrical fields larger than E*, conversely, the EC effect is the largest at a higher temperature T* ⬎ T0 and it is a weak function of temperature. 共3兲 Perfect lateral clamping transforms the first-order phase transition at T0 into a second-order transition, which lowers the magnitude of the electrocaloric effect but reduces its dependence on temperature. 共4兲 A compressive misfit strain shifts the maximum in the EC effect to higher temperatures, reduces its magnitude, and reduces its dependence on temperature. Conversely,
a tensile misfit strain shifts the maximum in the EC effect to lower temperatures, increases its magnitude, and increases its dependence on temperature. 共5兲 Control of the misfit strain by appropriate choice of substrate provides potential means to vary both the magnitude and the temperature sensitivity of the EC effect. ACKNOWLEDGMENTS
The work at UConn was supported by the U.S. Army Research Office through Grant No. W911NF-05-1-0528. 1
G. R. Bai, S. K. Streiffer, P. K. Baumann, O. Auciello, K. Ghosh, S. Stemmer, A. Munkholm, C. Thompson, R. A. Rao, and C. B. Eom, Appl. Phys. Lett. 76, 3106 共2000兲. 2 K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L. Q. Chen, D. G. Schlom, and C. B. Eom, Science 306, 1005 共2004兲. 3 J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. Vaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K. M. Rabe, M. Wuttig, and R. Ramesh, Science 299, 1719 共2003兲. 4 W. Chang, S. W. Kirchoefer, J. M. Pond, J. A. Bellotti, S. B. Qadri, J. H. Haeni, and D. G. Schlom, J. Appl. Phys. 96, 6629 共2004兲. 5 N. J. Donnelly, G. Catalan, C. Morros, R. M. Bowman, and J. M. Gregg, J. Appl. Phys. 93, 9924 共2003兲. 6 W. K. Simon, E. K. Akdogan, and A. Safari, J. Appl. Phys. 97, 103530 共2005兲. 7 Z.-G. Ban and S. P. Alpay, Appl. Phys. Lett. 82, 3499 共2003兲. 8 V. Nagarajan, C. L. Jia, H. Kohlstedt, R. Waser, I. B. Misirlioglu, S. P. Alpay, and R. Ramesh, Appl. Phys. Lett. 86, 192910 共2005兲. 9 P. D. Thacher, J. Appl. Phys. 39, 1996 共1968兲. 10 E. Fatuzzo, H. Kiess, and R. Nitsche, J. Appl. Phys. 37, 510 共1966兲. 11 J. D. Childress, J. Appl. Phys. 33, 1793 共1962兲. 12 J. H. He, J. C. Chen, Y. H. Zhou, and J. T. Wang, Energy Convers. Manage. 43, 2319 共2002兲. 13 A. S. Mischenko, Q. Zhang, J. F. Scott, R. W. Whatmore, and N. D. Mathur, Science 311, 1270 共2006兲. 14 A. S. Mischenko, Q. Zhang, R. W. Whatmore, J. F. Scott, and N. D. Mathur, Appl. Phys. Lett. 89, 242912 共2006兲. 15 G. Akcay, S. P. Alpay, J. V. Mantese, and G. A. Rossetti, Jr., Appl. Phys. Lett. 90, 252909 共2007兲. 16 J. F. Scott, Dielectric Properties of Insulators, Encyclopedia of Applied Physics Vol. 5 共VCH, Berlin, 1993兲. 17 A. F. Devonshire, Philos. Mag. 42, 1065 共1951兲. 18 A. F. Devonshire, Philos. Mag. 40, 1040 共1949兲. 19 S. P. Alpay, I. B. Misirlioglu, A. Sharma, and Z.-G. Ban, J. Appl. Phys. 95, 8118 共2004兲. 20 N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Phys. Rev. Lett. 80, 1988 共1998兲. 21 Coefficients for BT in SI Units: C = 1.7⫻ 105, TC = 383 K, ␣11 = 3.6共T − 448兲 ⫻ 106, ␣111 = 6.6⫻ 109, Q11 = 0.11, Q12 = −0.043, C11 = 1.75⫻ 1011, and C12 = 8.46⫻ 1010. 22 F. Jona and G. Shirane, Ferroelectric Crystals 共Dover, New York, 1962兲. 23 J. F. Scott, J. Phys. Soc. Jpn. 58, 4487 共1989兲. 24 E. Fatuzzo and W. J. Merz, Ferroelectricity 共Wiley, New York, 1967兲.
