Corresponding author: Gunnar Suchaneck ( gunnar.suchaneck@tudresden.de ) © 2018 Gunnar Suchaneck, Linda Felsberg, Gerald Gerlach.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Suchaneck G, Felsberg L, Gerlach G (2018) Materials issues in thermal modeling of thin film electrocaloric solidstate refrigerators. Modern Electronic Materials 4(2): 5969. https://doi.org/10.3897/j.moem.4.2.33391

Materials properties affecting EC device operation are discussed based on an analytically tractable model of a layered EC refrigerator. Special attention was paid to thermal and interface thermal resistances. Estimates of the average cooling power of a stacked MEMSbased EC refrigerator were made.
electrocaloric cooling, heat transfer, thermal resistance, heat transfer fluids, cooling power
The electrocaloric (EC) effect is a reversible temperature change of a material which results from an adiabatic application of an electric field. EC refrigeration is an environmentfriendly thermal energy conversion technology. Electric fields required for the EC refrigeration cycle can be supplied much easier and less expensively than the high magnetic fields required for the magnetocaloric (MC) refrigeration. Moreover, electrical energy for EC cooling can be provided by stationary or mobile solar cells, and by electric vehicle batteries. This opens up completely new possibilities for an environmentfriendly industrialization of developing countries [
The extraordinary dielectric response of relaxor ferroelectrics makes them promising for application in electrocaloric refrigerators [
A giant EC temperature change, ΔT_{E}_{C} = 45 K, was obtained for Pb_{0.88}La_{0.08}Zr_{0.65}Ti_{0.35}O_{3}(PLZT) thin films on a Pt/TiO_{2}/SiO_{2}/Si substrate at an electric field of 125 V/μm [
An EC refrigerator, i.e. a heat pump, is able to transport thermal energy against a temperature gradient from the load to the EC layer and from here to the heat sink. It consists of a polarizable EC material contacted by electrodes and thermal boundaries to the load and the heat sink. The thermal connections have to be opened and closed appropriately as the layer is heated or cooled. Heat is transferred from the load or to the heat sink either (i) via controlled heat switches as well as uncontrolled thermal rectifiers, or (ii) by pumping a gaseous or liquid heat transfer agent through the solid refrigerant [
The EC response of a multilayer ceramic capacitor (MLCC) under real environment and operational conditions, but without thermal connections, was considered in [
In [
In [
Recently, an EC device was numerically modeled which comprises multilayers of EC and heat storage material films separated by thermal switches changing the contact thermal conductance by forming a larger gap [
In this work, we derive an analytically tractable model of a layered EC refrigerator, we identify and discuss materials properties affecting EC device operation and we illustrate the impact of these properties on device operation by means of our simplified model.
The variation of temperature within the EC material is described by the heat transfer equation. For the sake of simplicity, we consider in the following an onedimensional model. In the case of linear heat flow, the parabolic heat diffusion equation of a solid comprising a heat source is given by [
where Θ = T − T_{0} is the temperature difference to the environment, α = κ/c the thermal diffusivity, κ the thermal conductivity, and c the volumetric specific heat assumed to be constant (c (E,T) ≈ c), V the volume, A the area and d the thickness of the EC layer, and h the heat transfer coefficient at the boundary describing heat loss to the environment. Introducing a bulk thermal resistance R′_{th,b} = (κA)^{1} and a thermal capacity C′_{th,b} = cA, both per unit length, eq. (1) transforms into
where Q˙′ and h′ are the values of Q˙ and h per unit length, respectively. Since the Biot number Bi = hd/κ (characterizing the ratio of the thermal resistance of the EC materials bulk to that at the boundary) will be less than 0.1 up to values of d in the order of 100 μm, the temperature of the EC element during heat transfer remains nearly constant, i.e. the ∂^{2}Θ/∂x^{2} = 0. This enables a lumped system approximation [
where R″_{th,i} = AR_{th,i} = 1/h is the areaspecific thermal resistance at the interface, C″_{th} = cd the thermal capacitance of the EC layer per unit area, and Φ the heat flux through the interface. Eq. (3) represent the wellknown bolometer equation [
The driving force of heat flux is the temperature difference Θ caused by charging or discharging the EC element which itself represents a dielectric capacitor. When charging a capacitor possessing a dielectric permittivity ε under isothermal conditions, the heat dQ delivered by a small change of the electric field dE is given by [
Since the EC temperature change is usually small, ΔΘ_{EC} << Θ, the corresponding thermal flux Φ(t) = dQ (t)/(Adt) might be written as
where <Τ> is the average temperature of the cooling cycle and where the field dependence of the temperature coefficient of dielectric permittivity 1/ε·∂ε/∂T is neglected. Note, that above the temperature T_{m} of maximum dielectric permittivity, 1/ε·∂ε/∂T is negative. The rise time of an output signal of a voltage pulse generator is mainly dependent on the switching speed of the power amplifier, while the fall time is determined by the resistorcapacitor (RC) circuitry at the generator output and the load. Here, we will not consider the physical origin of power amplifier switching. Assuming for the rising pulse
with τ_{e} = RC the time constant of the electronic circuit, an approximate solution of eq. (3) is given by
where τ_{th} = R″_{th,i}C″_{th} and where the temperature coefficient of the dielectric permittivity was assumed to be constant. For very short rise times of the electric field, τ_{e}→0, eq. (7) reduces to
A pulse decay given by
yields for t´ = t – t_{0}
transforming in terms of t′ which for τ_{e} → 0 is equivalent to eq. (8) with opposite sign. A flexible EC cooling device with electrostatic actuation reported in [
For sake of simplicity, we describe for a moment the thermodynamic cycles of the EC element by a periodic temperature change with a fundamental angular frequency ω = 2π/τ_{c}, where τ_{c} is the cycle time. Since the parabolic heat diffusion equation (1) has the same structure as the equation of an electrical RCtransmission line, multilayer problems are best treated by the matrix methods commonly used in electrical engineering [
Θ′ = LΘ + MΦ,
Φ′ = NΘ + OΦ, (11)
with
L = cosh(kd), M = –(κk)^{1}sinh(kd),
N = –κksinh(kd), O = cosh(kd), (12)
and
where d_{D} = (2D/ω)^{1/2} = (Dτ_{c}/π) is the penetration depth of the thermal oscillation. Note, that for dielectric thin and thick films with d < 100 µm we get L = O ≈ 1, M = –d/κ, and N = –κk^{2}d for ω = 1 Hz since D is in the order of 10^{6} m^{2}/s. The value of M represents the areaspecific thermal resistance of the film bulk.
Eq. (11) may be considered as matrix equation
In eq. (14), Θ and Φ must be multiplied by their time dependence exp(iωt) which is omitted here entirely. It should be again included when real or imaginary parts have to be taken at the end of the calculation. Assuming perfect thermal contact between the faces of the slabs a comprising n sublayers, the r_{th} being of thickness d_{r}, thermal conductivity κ_{r} and thermal diffusivity D_{r} with Θ_{r} and Φ_{r} and Θ′_{r} and Φ′_{r} at its lefthand and righthand faces, respectively, recursive application of eq. (14) gives for a onedimensional heat flow across several sublayers
Given any two of Θ_{r}, Φ_{r}, Θ′_{r} and Φ′_{r} the other two can be found using this method. The multiplication of matrices in eq. (15) can be carried out successively. However, deriving explicit formulas for slabs of n layers is very cumbersome [
In the presence of thermal contact resistances between the slabs or at the surfaces, they may be expressed also in matrix notation and included into eq. (15) [
and, correspondingly,
The final result of this calculation is are linear relations between the temperature and fluxes Θ_{1}, Θ′_{n}, Φ_{1}, Φ′_{n} at the two surfaces of the composite slab. The surface conditions will provide two more equations so that all four quantities Θ_{1}, Θ′_{n}, Φ_{1}, Φ′_{n} will be determined. Recursive application of this matrix scheme yields for a onedimensional heat flow across j slabs
where M is the heat transfer matrix. Thus, the matrix method allows numerically a very easy analysis of multilayer structure by multiplying numerical matrices.
The relative thickness of the EC layer in prototype EC cooling devices amounts to 0.30 to 0.35 d_{D} [
According to eq. (8), the EC temperature change given by ΔT_{EC} = Θ(t = 0) is defined by the volumetric specific heat c and the temperature coefficient dε/dT of dielectric permittivity. Since the values of c at room temperature approach the high temperature limit, it is expected that they do not vary strongly among various pyroelectric materials [
Temperature coefficient of dielectric permittivity above T_{m} for BZT20 and selected relaxor ferroelectrics.
Refrigerant  ε(T_{m})  dε/dT, K^{1}  Ref. 

BaZr_{0.2}Ti_{0.8}O_{3}  10600  100  [5] 
87  [24]  
BaZr_{0.25}Ti_{0.75}O_{3}  6950  72.5  [25] 
18940  377  [26]  
BaZr_{0.3}Ti_{0.7}O_{3}  33400  560  [27] 
BaZr_{0.35}Ti_{0.65}TiO_{3}  11550  81.5  [28] 
BaSn_{0.24}Ti_{0.76}TiO_{3}  16000  30  [29] 
Ba_{0.20}Pb_{0.80}ZrO_{3}  11080  340  [30] 
0.9PbMg_{1/3}Nb_{2/3}O_{3}0.1PbTiO_{3}  11400  130  [31] 
Recently, BaZr_{0.2}Ti_{0.8}O_{3} (BZT20) was demonstrated to be a promising material for EC application [
EC refrigerants are evaluated by a material criterion which characterizes the efficiency of the physical cooling process and, therefore, is independent on the performance of different thermodynamic cycles [
Here, εε_{0}·E^{2}·tanδ is the nonrecoverable electrical loss with tanδ the loss tangent, and cΔT_{EC} the heat transferred from the load to the heat sink within one refrigeration cycle with c the specific heat of the cooling element and ΔT_{EC} the EC temperature change. At this point, the tanδ comes into play as an additional material parameter since – following eq. (8) – the dielectric permittivity cancels out. It has to been noted that tanδ of eq. (20) needs to be investigated in the frequency range of EC device operation (0.1 to 10 Hz). In the presence of a ferroelectric hysteresis, it mainly consists of losses due to minor hysteresis loops described by an apparent loss tangent [
where P–(E) and P^{+}(E) are the field dependence of dielectric polarization at the upper and lower branch of the minor hysteresis loop, respectively. Obviously, tanδ depends significantly on materials synthesis. Values of tanδ < 0.07 provide refrigerant efficiencies of Φ > 0.95 exceeding the ones of any other solid state cooling technology.
At thin film interfaces, the highest value of R^{"}_{th,i} measured at room temperature to date is 1.2·10^{5} m^{2}K/W for Bi/Hydrogenterminated diamond. The thermal interface resistance at room temperature of other metal/dielectric combinations falls within a relatively narrow range, 3.3·10^{8} m^{2}K/W < R^{"}_{th,i} < 12·10^{8} m^{2}K/W [
The minimum possible R^{"}_{th,i}^{phph} produced by a purely harmonic process involving two phonons (one phonon on each side of the interface) is given by the phonon radiation limit which in the limit of high enough temperatures, where the DulongPetit law holds, is given by
Here, k is the Boltzmann constant, f_{max} = kΘ/h the vibrational cutoff frequency of the metal, h the Planck constant, Θ the Debye temperature, and v_{D},
the Debye velocity of a two dimensional dielectric with v_{l} and v_{t} the average longitudinal and transverse sound velocities, respectively. A compilation of all data used for calculations is given in Table
Material parameters used for the calculation of the interface thermal resistance.
Material  Parameter  Value  Ref. 

Ni  Θ  495 K  [35] 
v_{l}  6040 m/s  [36]  
ρ  8.902 g/cm^{3}  [36]  
M  58.69 g/mol  [36]  
l _{ph}  1.2 nm  [37]  
c _{e}  0.3174 MJ/m^{3}K  [38]  
Cu  τ_{eph}  1…4 ps  [39] 
W  ~0.5 ps  [40]  
BaTiO_{3}  v_{l}  6860 m/s  [41] 
v_{t}  3870 m/s  
κ  2.61 W/mK  [42]  
BaSr_{0.3}Ti_{0.7}O_{3}  ε_{b}  420  [43] 
BaSr_{0.48}Ti_{0.52}O_{3}  380  [44]  
BaSr_{0.55}Ti_{0.45}O_{3}  566  [45]  
BaSr_{0.65}Ti_{0.35}O_{3}  488  [46]  
Pt/BaSr_{0.3}Ti_{0.7}O_{3}/Pt  C_{i}  0.032 F/m^{2}  [43] 
Pt/BaSr_{0.48}Ti_{0.52}O_{3}/Pt  d_{if}/ε_{if}  0.076  [44] 
Pt/BaSr_{0.55}Ti_{0.45}O_{3}/Pt  0.048  [45]  
Pt/BaSr_{0.65}Ti_{0.35}O_{3}/Pt  0.174  [46] 
While phonons dominate the heat conduction in dielectrics, electrons dominate it in metals. Hence, an energy transfer occurs between electrons and phonons during heat transport across the metaldielectric interface. In result, two thermal resistances  volumetric electronphonon (R^{"}_{th}^{eph}) in the metal and interfacial phonon (R^{"}_{th,i}^{phph}) – are in series. The former one is given by [
where R_{e} is the electron cooling rate or electrontophonon energy transfer per unit volume, R_{e} = c_{e}/τ with c_{e} the electron heat capacity per unit volume and τ_{eph} is the relaxation time characterizing electronphonon energy loss or electron cooling. The value κ_{ph} is the phonon thermal conductivity,
with ρ the density, N_{A} the Avogadro constant, M the molar mass, and l_{ph} the phonon meanfreepath of the metal. Assuming τ_{eph} = 2 ps (cf. values of W and Cu in Table
From a microscopic point of view, all surfaces in contact have deviations from their idealized geometry. Because of these imperfections, two bodies in contact touch actually only at a few discrete points. The heat flux close to the interface is then constricted in these microcontact regions [
The thermal contact resistance is defined as the ratio between the temperature drop at the interface and the heat flux crossing the interface. Similarly, electrical capacitance C is the ratio of the total charge on a conductor to its electric potential (potential difference to a reference electrode). Both capacitance and thermal conductance are affected by edge effects. Therefore, there is a close relation between R_{th,i} and the interface capacitance C_{i} given by [
where ε_{b} is the bulk dielectric permittivity. The arearelated value R″_{th,i} = AR_{th,i} is then given by
with d_{i} the thickness and ε_{i} the dielectric permittivity of the interface layer. With regard to the values of C_{i} and ε_{b} in Pt/BaSr_{x}Ti_{1}_{x}O_{3}/Pt structures (cf. Table
In macroscopic cooling systems, the thermal interface resistance is generally negligible since it is in series with other thermal resistances exhibiting much larger dimensions and, thus, much larger resistance. Here, heat is transferred from the load or to the heat sink (i) via thermal switches or (ii) by a gaseous or liquid heat transfer agents [
Thermal contact resistances of solidsolid and liquidsolid configurations.
Contact  R″_{th,c}, m^{2}K/W  Parameter  Ref. 

Si/Si  ~2·10^{4}  0.169 MPa  [50] 
Cu/Si  ~1·10^{4}  
CuIn/Si  2.72·10^{5}  
CuPCM^{1)}/Si  5.22·10^{5}  
CuCNT/Si  ~2.5·10^{5}  
CuPCM^{1)}CNT/Si  ~3.4·10^{5}  
polySi/SiN_{x}  1.7·10^{5}  [51]  
Cu/Cu (sheets)  1.59·10^{5}  3.6 MPa  [52] 
1.05·10^{5}  14.4 MPa  
Al/Al (sheets)  4.76·10^{5}  3.6 MPa  
4.00·10^{5}  14.4 MPa  
Si/Hg 
1.09·10^{6}  K = 110 (0.1N)  [53] 
Si/Hg  2.6·10^{6}  224 (1N)  
CNT columns/Si  5.7·10^{5}   (0.1 N)  
CNT columns/Si  2.04·10^{5}  8.4 (1N)  
CNT columnsAu/Si  4.16·10^{5}  4.4  
CNT/Si  4.6 (0.1N)  
CNT/Si  8.87·10^{5}  27.8 (1N)  
CNT/CNT  4.23·10^{4}  6.1  
CNTPolyimide  ~2·10^{3 2)}  [15]  
Hybrid solidliquid (CuH_{2}O)PTFE/Si  1.3·10^{5}  d _{H2O} = 75 µm  [54] 
SiH_{2}OPTFE/Glass  1.0·10^{5}  d _{H2O} = 6.1 µm  [55] 
2.0·10^{5}  11.6 µm  
3.2·10^{5}  19.3 µm  
Cu/PDMS  3.25·10^{5}  0.46 MPa  [56] 
ZnO, BN/PDMS  ~9·10^{6}  
Carbon fiber/epoxy resin  0.31.8·10^{5}  [57]  
Pt/SiO_{2}  3.2·10^{9}  d _{SiO2} = 26, 440 nm  [58] 
Si/SiO_{2}  < 1.7·10^{9}  d _{SiO2} = 26 nm  
Al/SiO_{2}  0.71·10^{8}  d _{Al} = 50120 nm  [59] 
Si/SiO_{2}  68·10^{8}  d _{SiO2} = 110518 nm  
Al/epoxy resin  4.6·10^{4}  [60]  
Glass/epoxy resin  8.6·10^{4}  
Grease  0.62·10^{5}  [61]  
Galistan  7.7·10^{6}  0.172 MPa  [62] 
Galistan  2.8·10^{5}  0.284 MPa  [63] 
In an active EC regenerator configuration, liquid or gaseous heat transfer agents are pumped through the EC refrigerant. They absorb heat from the bed and transfer it to the cold sink. Thereby, an electric field is applied solely to the refrigerant. The flow channels are usually rectangular ducts requiring a twodimensional flow analysis. On the other hand, in a onedimensional model, the determination of a heat transfer coefficient h (the inverse of h represents the thermal interface resistance) describing the heat transfer between the fluid and the solid is sufficient. The heat transfer across a boundary is defined by the Nusselt number Nu – the ratio of convective to conductive heat transfer normal to the considered boundary:
with d_{H} the hydraulic diameter. For steady state, laminar flow of an incompressible liquid with constant physical properties in a rectangular duct of constant crosssectional area characterized by its aspect ratio AR (the ratio of thesmallertothe larger duct size), the value of Nu is approximately given by [
Nu = 8.235(1 – 2.0421AR + 3.0853AR^{2} –
2.4765AR^{3} + 1.0578AR^{4} – 0.1861AR^{5}). (29)
At Bi < 0.1, the temperature of the EC element during heat transfer remains nearly constant (cf. section 2.1). Here, a duct high of z = 0.6 mm and AR = 0.15 yield thermal resistances of
The thermal resistance of the fluid, R″_{th,f}, is determined by the inverse of the product of its mass flow rate dm/dt and its specific heat at constant pressure c_{p} [
For calculation, an upper limit of the fluid velocity of v = 0.1 m/s was chosen providing a back and forth fluid motion of 1 cm at a cycle time of τ_{c} = 0.2 s. This results in vc >> h leading to R″_{th,f} ≈ R″_{th,sf}. Values of R″_{th,sf} are compiled in Table
Physical properties of heat transfer fluids, thermal resistance at the solidfluid interface R″_{th,sf} for laminar flow, and values of d_{D}_{,1Hz}.
Heat transfer fluid  κ, W/mK  c, MJ/m^{3}K  R″_{th,sf} , 10^{3}m^{2}K/W  d_{D} _{,1Hz}, mm 

Silicone oil, 20 cSt  0.142^{a}  1.52^{a}  1.36  0.30 
Water  0.606^{b}  4.19^{b}  0.32  0.38 
HT 70  0.07^{c}  1.62  2.76  0.21 
Summing up, the thermal resistance at the interface to the heat transfer fluid limits the heat transfer of active EC regenerators.
A MEMSbased EC element with a stack of three cooling cells is illustrated in Figure
(a) Electrocaloric element: 1, 2 – outer electrodes, 3 – EC MLC, 4 – SiN_{x} membrane, 5 – Si wafer, 6 – cavity sidewall coating and (b) stack of EC three cells.
Bulk thermal resistances of the constituent layers of the EC device in Figure
Layer  κ, W/mK  d, µm  R″_{th}, 10^{3}m^{2}K/W  Ref. 

Thermal insulation  10  [69]  
Ni electrodes  90.7  200×2  4.41·10^{2}  [7] 
BZT20 layer  3.4  200×6.5  0.5  [5] 
Pt electrode  71.6  0.1  1.4·10^{6}  [36] 
SiN_{x} membran  10  0.2  2.0·10^{5}  [70] 
PDMS coating  0.2  1000  5  [71] 
The thermodynamic cycle of an EC refrigeration process in such a cooler design includes four steps [
For a cycle time τ_{c} = 2mC″_{th}R″_{th} proportional to the thermal time constant of the EC device, the cycleaveraged cooling power of a single cell thermally isolated at the top is given by [
whith ΔΘ^{*} ≈ ΔT_{EC} and
Here, TI denotes the thermal insulation layer. Neglecting the small bulk and interface thermal resistances, assuming R″_{th,b}^{TI} → ∞ – equivalent to zero temperature gradient at the boundary between the cells  and taking into account, that the heat flux is divided between the heat transfer fluid and its PDMS environment, we arrive at
Taking water as heat transfer fluid, assuming ΔT_{EC} = 5 K and m = 2, the cycleaveraged cooling power of a single cell amount to 180 mW/cm^{2}. Following Table
For a cell stack, only the bottom and top cells should be thermally insulated to the environment by a layer of low thermal conductivity. Here, MEMSstructures consisting of vacuum cavities [
Figure
The estimations made in this work illustrate that a solid state EC refrigerator with a cooling power of up to 2 W/cm^{2} can be realized using a stack of 10 single MEMSbased cells.
A simplified analytical model of solid state EC refrigerators allows analysis of the materials impact on device operation, enables the indentification of the key material parameters and makes it possible to predict the device performance. For a stack of MEMSbased EC cells in an active regenerator mode, cycleaveraged cooling powers of up to 2 W/cm^{2} were estimated.
This work was supported by the German Research Foundation (DFG) within the Priority Program «Ferroic Cooling» (SPP1599, project B6). The authors thank R. Liebschner for providing Figure