# Electrothermal Phenomena in Ferroelectrics

## Abstract

**:**

^{λ}, which is analogous to the electromechanical coupling factor k, relating the elastic compliances under short- and open-circuit conditions, in order to explain the fact that the short-circuit condition exhibited the larger thermal diffusivity than the open-circuit condition. On the other hand, the unpoled specimen exhibits the lowest thermal diffusivity. This tutorial paper was authored for providing comprehensive knowledge on equilibrium and time-dependent thermodynamics in ferroelectrics.

## 1. Background

_{1/3}Nb

_{2/3})O

_{3}-PbTiO

_{3}(PZN-PT) bulk single crystals in 2003 [1], being 0.3 °C under 1 kV/mm electric field applied, ignited the “renaissance” of electrocaloric effect. Successive reports on the high response, 12 °C under an applied electric field of 48 kV/mm in thin film (or ribbon) lead zirconate-titanate (PZT) by Mischenko et al. [2] (that is equivalent to 0.25 °C under 1 kV/mm, lower performance than PZN-PT single crystals), and the demonstration of 12 °C of cooling near room temperature with a ferroelectric polymer by Neese et al. in 2008 [3] also accelerated the research boom on electrocaloric devices.

^{3}[4]. One of the Figures-of-Merit for the high-power density is the “maximum vibration velocity” in the same material species such as PZT’s, which is defined by the vibration velocity generated at the sample edge under its resonance operation, when the maximum temperature rise (i.e., at the nodal point) reaches 20 °C above the room temperature [5,6]. When we operate the piezo-sample under higher vibration levels, even though highly resistive under the off-resonance, the piezoelectric PZT starts generating heat via its dielectric, elastic and piezoelectric losses. Most of the additional input electric energy over the maximum vibration velocity level is converted to heat, like a “ceramic heater”! The current commercially available top data is around ${v}_{max}^{RMS}=0.6\mathrm{m}/\mathrm{s}$ in a rectangular plate PZT’s, which corresponds to the energy density of 40 W/cm

^{3}. We found that the maximum vibration velocity strongly depends on the thermal conductivity. Figure 1 demonstrates the simulation results of the saturated temperature distribution profile difference between two compositions, PZT-5H and PZT-19, under the same vibration velocity operating condition. These two compositions exhibit significant difference in thermal conductivity: 0.14 W/m K versus 2 W/m K [7,8]. The profile curve of PZT-5H (low thermal conductivity) fits a sinusoidal line beautifully, while PZT-19 shows considerable edge temperature rise. You can understand easily that this profile difference is originated primarily from the thermal conductivity or diffusivity difference. Taking the total thermal energy dissipated from the specimen by integrating the temperature rise with respect to the position coordinate, we can expect a similar “mechanical quality factor” Q

_{m}value (i.e., similar intensive elastic loss tan ϕ′) for these two specimens. However, you can notice that the peak temperature at the nodal point is significantly lower for the larger thermal diffusivity material, which means we can excite the vibration more under higher voltage for the PZT-19, since the maximum vibration velocity is defined by the highest temperature rise 20 °C above room temperature at the node (plate center) from the human safety viewpoint in the practical industrial standard (e.g., human fingers will be burned above 50 °C).

^{λ}, which is applied for irreversible thermal flow status and explain the difference in the thermal conductivity between the electric constraint difference, that is, under short- and open-circuit conditions [9].

## 2. Review of Phenomenology in Ferroelectrics

#### 2.1. Static/Equilibrium Ferroelectric Phenomenology

#### 2.1.1. Thermodynamical Functions

≦ TdS + Xdx + EdD

_{1}= U − TS − Xx as examples in this paper; that is,

_{1}is most convenient for discussing the ferroelectric phase transition because of the two merits: (1) by describing higher order Taylor expansion terms of the “order parameter” D (or P), we can derive the spontaneous polarization, and (2) external X can be controlled explicitly, and E is related with easily with ${\left(\frac{\partial {G}_{1}}{\partial P}\right)}_{T,X}$.

#### 2.1.2. Linear Energy Handling

_{R}(room temperature), external X and E (1D case). If the change of parameters is small, we may adopt the three-parameter Taylor expansion approximation up to second derivatives in order to discuss just the linear relationships, based on the description by Mitsui et al. [10]:

_{1}for the phase transition discussion, as in Section 2.1.1.

#### 2.1.3. Nonlinear Energy Handling

_{1}(P) = G

_{1}(0) + a

_{1}P + a

_{2}P

^{2}+ a

_{3}P

^{3}+ a

_{4}P

^{4}+ a

_{5}P

^{5}+ a

_{6}P

^{6}+ ⋯

_{1}(P) = G

_{1}(−P). Thus, the energy expansion series should not contain terms in odd powers of a component, P, or the expansion series include only even powers of P.

#### 2.1.3.1. Second-Order Phase Transition

_{1}(P,T) = (1/2)αP

^{2}+ (1/4)β P

^{4}

^{2}term must be negative for the spontaneously polarized state to be stable, while, in the paraelectric state, it must be positive passing through zero at a certain temperature T

_{0}, called Curie–Weiss temperature. From this concept, we assume α as a linear relation:

_{0}is equal to or lower than the actual transition temperature T

_{C}(Curie temperature), in general. This temperature dependence can also be interpreted as the adoption of the “TP

^{2}” electrothermal coupling term [$\left(\frac{{\partial}^{3}{G}_{1}}{\partial T{\partial}^{2}P}\right)\theta {P}^{2}$] in the Taylor expansion of the G

_{1}energy. The temperature parameter θ here is the deviation from the Curie–Weiss temperature T

_{0}. From $d{G}_{1}=-SdT-xdX+EdP$ [$P\approx D$ in our discussion], the equilibrium polarization under an electric field E should satisfy the condition:

_{1}/∂P) = E = α P + β P

^{3}= P∙(α + β P

^{2})

^{2}= −α/β. The trivial solution in (i) case corresponds to a paraelectric (high temperature) state, while the polarization solutions (P

_{S}= $\pm \sqrt{-\alpha /\beta}$ in (ii) case correspond to a ferroelectric (low temperature, α < 0) state.

_{C}(in this case also T

_{0}). In a lattice vibration model, we can rephrase the above situation as follows: at a high temperature, the nearly “harmonic” atomic (or lattice) vibration (usually an “optical phonon” mode, which generates the polarization deviation) in a sharp convex “parabolic” potential at a high temperature will reduce the vibration frequency with decreasing the temperature approaching to the Curie temperature (i.e., critical slowing down) because of the flat curvature of the potential around T

_{C}. This “phonon softening” phenomenon is interpreted by the nonlinearity of the phonon mode [Refer to Section 4.2]. Because the free energy curve is symmetric with respect to the polarization P when the external field E = 0, the probability of the state +P

_{S}(=+$\sqrt{-\alpha /\beta}$ or −P

_{S}(=$-\sqrt{-\alpha /\beta}$ should be equal. Thus, with decreasing the temperature, passing through T

_{C}, +P

_{S}domains and −P

_{S}domains may arise locally in a specimen with equal volumetric ratio, leading to the “multidomain states” in general. In order to produce the net spontaneous polarization, strain, and piezoelectricity, we need to apply large electric field (higher than “coercive field”, which is given by ${E}_{C}=\sqrt{\frac{-4}{27}{\alpha}^{3}/\beta}$ in the second order phase transition), which is called the “electric poling” process to make a crystal with mono-domain-like status. The double-well potential change with the applied field is illustrated in the inserted figure. Note first that the minimum potential well energy is negative (that is, stabler than the paraelectric phase energy zero), and second that external electric field enforces one spontaneous state more stable. Though the poling process in polycrystalline ceramic samples, composed of many micro single crystals (i.e., “grain”) with random crystal orientations, is not easy, a similar phenomenology is assumed to be adopted qualitatively in this paper.

#### 2.1.3.2. First-Order Phase Transition

_{1}(P,X,T) = (1/2)α P

^{2}+ (1/4)β P

^{4}+ (1/6)γ P

^{6}

− (1/2)s X

^{2}− Q P

^{2}X, (α = (T − T

_{0})/ε

_{0}C)

_{1}/∂P) = α P + β P

^{3}+ γ P

^{5}− 2Q PX minus sign

_{1}/∂X) = sX + QP

^{2}

^{3}+ γ P

^{5}

^{2}

_{0}ε = (∂E/∂P) = α + 3 β P

^{2}+ 5 γ P

^{4}

^{2}= (−β +$\sqrt{{\beta}^{2}-4\alpha \gamma}$)/2γ.

_{S}= 0 or P = ε

_{0}ε E (under small E)

_{0}) (Curie-Weiss law)

_{0}ε)

^{2}E

^{2}

_{S}

^{2}= (−β + $\sqrt{{\beta}^{2}-4\alpha \gamma}$)/2γ or P = P

_{S}+ ε

_{0}ε E (under small E)

_{S}+ ε

_{0}ε E)

^{2}= QP

_{S}

^{2}+ 2 ε

_{0}ε QP

_{S}E (+Q (ε

_{0}ε)

^{2}E

^{2})

_{S}and the piezoelectric constant d as:

_{S}= QP

_{S}

^{2}

_{0}ε QP

_{S}

_{0}

_{S}= 0

_{0}ε = α, then ε = C/(T − T

_{0}) (Curie-Weiss Law)

_{0}

_{0}ε = α + 3 β P

^{2}= −2α, then ε = C/2(T

_{0}− T)

_{S}= QP

_{S}

^{2}= Q (T

_{0}− T)/ε

_{0}Cβ

#### 2.1.4. Isothermal Process—Piezoelectric Coupling

#### 2.1.4.1. Thermodynamical Meaning of Piezoelectric Constant

^{D}x − h D,

_{0}κ

^{x}D,

^{D}is elastic stiffness under constant D, and κ

_{0}κ

^{x}is inverse permittivity (${\kappa}_{0}=1/{\epsilon}_{0}$) under strain free condition, and these coefficients are expressed by:

#### 2.1.4.2. Electromechanical Coupling Factor

^{2}being the ratio of the converted energy over the input energy: when electric to mechanical

^{2}= (Stored mechanical energy/Input electrical energy),

^{2}= (Stored electrical energy/Input mechanical energy)

_{3}is applied to a piezoelectric material in a pseudo-static process. See Figure 4a first, when we apply the electric field on the top and bottom electrodes under stress free conditions (X = 0). Input electric energy must be equal to (1/2)ε

_{0}ε

_{3}

^{X}E

_{3}

^{2}from Equation (4), and the output strain generated by E

_{3}should be d

_{33}E

_{3}from Equation (22a). Since the converted/stored mechanical energy is obtained as (1/2 s

_{33}

^{E}) x

_{3}

^{2}, we obtain:

_{33}

^{2}= [(1/2)−(d

_{33}E

_{3})

^{2}/s

_{33}

^{E}]/[(1/2) ε

_{0}ε

_{3}

^{X}E

_{3}

^{2}]

= d

_{33}

^{2}/ε

_{0}ε

_{3}

^{X}∙s

_{33}

^{E}

_{3}is applied to a piezoelectric material in a pseudo-static process. Refer to Figure 4b. Under short-circuit condition (E

_{3}= 0), the input mechanical energy must be equal to (1/2)s

_{33}

^{E}X

_{3}

^{2}from Equation (4), and the electric displacement D

_{3}(or polarization P

_{3}) generated by X

_{3}should be equal to d

_{33}X

_{3}from Equation (22b). This D

_{3}can be obtained by integrating the short-circuit current in terms of time through the electric lead. Since the converted/stored electric energy is obtained as (1/2 ε

_{0}ε

_{3}

^{X}) D

_{3}

^{2}, we obtain:

_{33}

^{2}= [(1/2 ε

_{0}ε

_{3}

^{X}) (d

_{33}X

_{3})

^{2}]/[(1/2)s

_{3}

^{E}X

_{3}

^{2}]

= d

_{33}

^{2}/ε

_{0}ε

_{3}

^{X}∙s

_{33}

^{E}

^{2}, which has a physical meaning of energy transduction/conversion rate), can be exactly the same for both converse (27a) and direct (27b) piezoelectric effects. The conditions under constant X (free stress) or constant E (short-circuit) are considered to be non-constrained.

#### 2.1.4.3. Constraint Physical Parameters

#### 2.1.5. Adiabatic Process 1—Piezothermal Effect

^{PT}can be defined from Equation (34a,b) by

#### 2.1.6. Adiabatic Process 2—Electrothermal Effect

#### 2.1.6.1. Constraint Specific Heat Capacity

#### 2.1.6.2. Constraint (Adiabatic) Permittivity

#### 2.1.6.3. Electrocaloric Effect

_{C}= T

_{0}) in Section 2.1.3.2:

- (a)
- Specific Heat Capacity$${c}_{p}^{E}=-T{\left(\frac{{\partial}^{2}G}{\partial {T}^{2}}\right)}_{X,E}=\frac{T}{2\beta {\left({\epsilon}_{0}C\right)}^{2}}$$
- (b)
- Pyroelectric CoefficientNoting here ${P}_{S}=\sqrt{\left({T}_{0}-T\right)/{\epsilon}_{0}C\beta}$, the pyroelectric coefficient is obtained as$$p=-{\left(\frac{\partial P}{\partial T}\right)}_{X}=\frac{1}{2\sqrt{{\epsilon}_{0}C\beta}}\frac{1}{\sqrt{\left({T}_{0}-T\right)}}$$
- (c)
- Figure of Merit of Electrocaloric Effect$$\frac{pT}{{c}_{p}^{E}}=\sqrt{\beta}{\left({\epsilon}_{0}C\right)}^{3/2}\frac{1}{\sqrt{\left({T}_{0}-T\right)}}$$

_{C}reduces significantly with approaching the Curie temperature. Thus, in practice, by applying a high positive electric field gradually (pseudo-equilibrium or isothermally), then, a sudden decrease down to zero is conducted to lower the temperature to escape from the depoling problem in a low coercive field material.

#### 2.1.6.4. “Electrothermal Coupling Factor” ${k}^{ET}$

- Permittivity ${\epsilon}^{X}$

#### 2.2. Time-Dependent Ferroelectric Phenomenology

#### 2.2.1. Polarization Relaxation

_{C}. The key assumption is that the change of polarization P with time (i.e., $\frac{\partial P}{\partial t}$) is proportional to internal energy decrease rate with the unit order parameter change (i.e., $\frac{-\partial U}{\partial P}$). By introducing “relaxation time” $\tau $, the inverse of which is taken as a proportional constant:

_{1}(P,T) = (1/2)α P

^{2}+ (1/4)β P

^{4}[α = (T − T

_{0})/ε

_{0}C]

_{0}), the spontaneous polarization is expressed as

- Time constant is proportional to “permittivity” ${\epsilon}_{0}{\epsilon}^{X}$.
- t → $\infty $ gives P → P
_{S}. - With approaching T → T
_{0}, $\tau $ → $\infty $. The recovery time of P is very slow around T_{C}= T_{0}.

#### 2.2.2. Temperature Relaxation

- In a uniform (no space gradient) specimen, temperature change follows an exponential trend with time: $\left(1-\theta /{\theta}_{s}\right)={e}^{-t/\tau}$.
- t → $\infty $ gives $\theta $ → ${\theta}_{s}$. ${\theta}_{s}$ corresponds to the temperature change by the electrocaloric effect.
- Time constant is proportional to temperature, and inversely proportional to the specific heat capacity ${c}_{p}^{E}$. The larger the specific heat capacity ${c}_{p}^{E}$, and the lower the temperature, the lower the time constant $\tau $. The recovery time of θ is quicker. This τ is roughly a suitable rise time period of an applying pseudo-step electric field.

## 3. Thermal Diffusivity and Conductivity

#### 3.1. 1D Heat Conduction Model

_{R}) at a point a distance x from one end of rod. We introduce the following three parameters:

- q—Heat flux = quantity of heat passing through cross section of rod per unit area per unit time. The unit of heat quantity is calorie (cal), and 1 cal corresponds to 4.186 Joule in energy.
- λ—Thermal conductivity = thermal or heat conductance per unit length of material. The inverse 1/λ is the resistance which measures the temperature drop per unit length when heat flux is unity.
- C—Heat capacitance of material = specific heat capacity c
_{p}(${c}_{p}^{E}$, ${c}_{p}^{D}$) $\times $ density ρ, number of heat units to raise block of unit area and unit length (i.e., per unit volume) 1 °C in temperature. Note that the specific heat capacity [J/(kg·K)] is determined per unit mass (kg), which must be multiplied by density (kg/m^{3}) to obtain the heat capacitance per unit volume [J/(m^{3}·K)].

- Thermal conductivity $\lambda $—$0.61.3$ [W/(m·K)]
- Specific heat capacity ${c}_{p}$—$340420$ [J/(kg·K)]
- Mass density $\rho $—7600 [kg/m
^{3}] - Thermal diffusivity ${\alpha}_{T}$—$2.75.1\times {10}^{-7}$ [m
^{2}/s]

#### 3.2. Solution of 1D Heat Transfer Equation

^{22)}

_{0}) is attached suddenly (t = 0) on the surface A (x = 0) in Figure 6a; that is, heat flux is given as a step-function with respect to time. Excluding immediately after t = 0 and adjacent to x = 0 point, we consider the region t > δ, x > δ’, and assume the following equation in most of the rod material’s length:

- At x = 0, $\theta \left(x=0,t\right)={\theta}_{0}$ —Temperature is constant at ${\theta}_{0}$, irrelevant to time after attaching to heat source.
- At x = L, $\theta \left(L,t\right)={\theta}_{0}{e}^{-{L}^{2}/4t{\alpha}_{T}}$ —By measuring temperature change with t at x = L, we can obtain the thermal diffusivity ${\alpha}_{T}$.

^{2}/s]. As can been seen, heat flux penetration speed for 2 mm thick sample does not take longer than 0.5 s, and the temperature monitoring finishes in less than 3 s. Even for a thicker specimen with 10 mm, the measuring time is still shorter than 30–40 s.

#### 3.3. Thermal Diffusivity Measurements

^{2}/s). The thermal diffusivity measurements in this experiment have an accuracy of 5% or better for some standard materials such as Fused Quartz and Pyrex 7740, in comparison to the literature values [25].

_{33}m. The thermal diffusivity was measured in the direction of polarization for the poled samples. In parallel, the absolute value of heat capacity of a small specimen was measured using the DSC Q2000 (TA Instruments, New Castle, DE, USA), by comparing it to a sapphire reference sample. The “specific heat capacity (per mass)” and density are respectively c

_{p}= 340 J/kgK and ρ = 7600 kg/ m

^{3}. The specific heat capacity difference between ${c}_{p}^{E}$ and ${c}_{p}^{D}$ could not be detected under electrical boundary condition difference (short- and open-circuit) because the differential scanning calorimetry (DSC) took too long (5 min) to keep the “depolarization field”.

#### 3.4. Thermal Diffusivity under Different Electrical Constraints

_{p}, ρ are thermal conductivity, specific heat capacity, and mass density, respectively. The difference in ${\alpha}_{T}$ must be the combination effect of the $\lambda $ and c

_{p}differences (the density ρ may not be changed by the electric constraint condition).

#### 3.4.1. Specific Heat Capacity—Scalar Parameter

_{t}or k

_{33}piezoelectric resonance modes, where the thickness or length stress distribution generates the depolarization field ${E}_{dep}=-\left(\frac{\Delta P}{{\epsilon}_{0}{\epsilon}^{X}}\right)$ anti-parallel to the induced polarization (∆P = d

_{33}X

_{3}) in order to satisfy the D—constant (constraint) condition. The elastic compliance ${s}_{33}^{D}$ and sound propagation speed (${v}_{33}^{D}=1/\sqrt{\rho {s}_{33}^{D}}$ are modified by the electromechanical coupling factor ${{k}_{t}}^{2}$ or ${{k}_{33}}^{2}$ as ${s}_{33}^{D}={s}_{33}^{E}\left(1-{{k}_{t}}^{2}\right)$. While in the electrothermal material, the polarization modulation is made from the temperature change via the pyroelectric effect. The specific heat capacity is modified by the electrothermal coupling factor ${{k}^{ET}}^{2}$ as ${c}^{D}={c}^{E}\left(1-{{k}^{ET}}^{2}\right)$ depending on the electrical constraint. We may also imagine the heat-flow (i.e., thermal conductivity) modulation according to the electrical constraint condition.

#### 3.4.2. Thermal Conductivity—Tensor Parameter

^{D}

_{33}is 0.57 times smaller than the short-circuit one λ

^{E}

_{33}. We denote the superscripts E and D of the conductivity λ, supposing that the sample is roughly maintained “electric field E constant” and “electric displacement D (or P) constant” during a rather short experimental time period less than 10 s. Subscript “33” stands for the tensor component relating to heat flux vector q

_{3}and x-component of the temperature gradient grad (θ) (i.e., vector) along the spontaneous polarization P

_{S}direction. Second, the unpoled specimen exhibits a crystallographic symmetry of isotropic cubic $m3m$ (different from the poled specimens), and the thermal conductivity λ

^{u}

_{11}shows the smallest value, 25% less than that of the poled open circuit case (see the last column of Table 1). Different from the scalar analysis, the tensor analysis is directly coupled with the material’s crystallographic symmetry. The unpoled specimen cannot be discussed in parallel with the poled specimens because of the crystallographic symmetry difference.

^{λ}

_{33}below, in the relationship between the open-circuit λ

^{D}

_{33}and short-circuit thermal conductivity λ

^{E}

_{33}such as:

^{λ}

_{33}” with using a “space-gradient model” illustrated in Figure 9. Analogous to Equation (39a,b),

^{2}] and $\nabla \left(\theta \right)=\frac{\partial \theta}{\partial x}$ [K/m], and gradient of vector $\nabla \left(E\right)$ [V/m

^{2}]. Nabra $\nabla $ is the gradient, $\nabla =i\frac{d}{dx}+j\frac{d}{dy}+k\frac{d}{dz}$. In our simple model, we focus only on the x-direction component. Knowing the facts in 1D model: (1) ${q}_{x}=-\lambda \nabla \left(\theta \right)$, and (2) heat flux seems to be controlled by the electric field E (because of the difference between ${\lambda}_{33}^{D}$ and ${\lambda}_{33}^{E}$), we assume the following constitutive equations, resembling Equation (39a,b) above, but taking the gradient with respect to the heat flux direction x:

^{2}]. Because T and $\nabla \left(\theta \right)$ have [K] and [K/m], respectively, $\lambda $ should possess the unit of [W/K·m] (normalized by time). While $\left(\frac{1}{v}\dot{{E}_{x}}\right)$ has [V/m

^{2}], ${p}^{\prime}$ should have [Cm/K·s]. On the contrary, Equation (78b) requires a tricky explanation: we assume $\left(\frac{1}{v}\dot{{D}_{x}}\right)$ is total charge flow (multiplied by cross-section area) per second with the unit [C/s]. In this case, taking $\nabla \left(\theta \right)$ as [K/m], ${p}^{\prime}$ should have [Cm/K·s]. The remaining problem is the last term: ${\epsilon}_{0}{\epsilon}^{X}$ has [C/Vm] and $\left(\frac{1}{v}\dot{{E}_{x}}\right)$ has [V/m

^{2}], thus the product should have the unit of [C/m

^{3}]. We need to translate this term by multiplying the volume, such as ${\epsilon}_{0}{\epsilon}^{X}\dot{{E}_{x}}\to C\dot{V}$ (here C: capacitance, V: voltage). The new coupling parameter p′ seems to be analogous to pyroelectric coefficient p [C/K·m

^{2}] but is the total charge flow (multiplied by volume) per second because we are discussing basically space-gradient temperature profile. From Equation (78a,b), we may define a new “electrothermal coupling coefficient” k

^{λ}

_{33}as

_{3}= constant (i.e., open-circuit), we get $\left(-\nabla \left({E}_{x}\right)\right)=\frac{{p}^{\prime}}{{\epsilon}_{0}{\epsilon}^{X}}\left(-\nabla \left(\theta \right)\right)$ from Equation (77b), then Equation (77a) is transformed into

^{D}

_{33}= 0.57 λ

^{E}

_{33}, gives ${{k}_{33}^{\lambda}}^{2}=\frac{Tp{\prime}^{2}}{{\lambda}_{33}^{E}{\epsilon}_{0}{\epsilon}^{X}}=0.43$. Knowing ${\lambda}_{33}^{E}=2.13W/K\xb7m$ and ${\epsilon}^{X}=1375$, T = 300 K, we can obtain ${p}^{\prime}=6.1\times {10}^{-6}\left[\mathrm{Cm}/\mathrm{s}\xb7\mathrm{K}\right]$. The remaining problem is how to integrate the volume and unit time calibration into the ${\epsilon}_{0}{\epsilon}^{X}$ term to satisfy all units.

#### 3.5. Thermal Conductivity in Pb-Free Piezoelectrics

_{31}type plate specimens. Table 2 summarized the preliminary data on the thermal conductivity in the NKN-based material with the PZT’s value (both unpoled samples). Pb-free piezoceramics such as (Na,K)NbO

_{3}− and (Bi,Na)TiO

_{3}− based materials show much higher maximum vibration velocity than the PZT’s [26,27]. Much larger thermal conductivity in the NKN-based material than the PZT’s may also contribute to this good high-power performance in NKN-ceramics because of the definition of the maximum vibration velocity, as we have discussed in Figure 1 in Section 1.

## 4. Nonlinear Elastic Performances and “Phonons”

#### 4.1. Electrostriction and Thermal Expansion

_{0}):

#### 4.2. Lattice Vibration and Phonon

#### 4.2.1. Lattice Vibration—One-Atom Chain Model

**u**is an N-Dimensional vector with the components of ${u}_{n}$, and $\left[F\right]$ is a symmetric matrix with the components:

^{23}per mol) of atoms of N, the energy gap among phonons is quantum-mechanically small. According to the Gibbs statistical dynamics for the equilibrium status, thermo-dynamical properties in a crystal lattice is determined by these eigenvalues of the phonons. The internal energy U is expressed by using quantum-mechanically small unit $\hslash $ as

#### 4.2.2. Lattice Vibration—Two-Atom Chain Model

**e**satisfies

^{+}and Cl

^{-}ions move in opposite directions, leading to the electric polarization deviation. This phonon mode can easily couple with the external electric field such as “optical light” electric wave; thus, this is called “optical branch.” The reader can imagine that this interaction may be a part of the origin of “electrothermal coupling” in ferroelectric materials.

**e**satisfies the following condition at k = 0:

#### 4.2.3. Phonon Spectral Density

#### 4.3. Anharmonic Phonon Modes and Electrothermal Coupling

#### 4.3.1. Anharmonic Phonon Modes and Wave Packet

#### 4.3.2. Heat Flow and Electrothermal Coupling

^{28)}Assuming the spirit of the Drude model, the phonon collision occurs in a distance $l={v}_{x}\tau $ (${v}_{x}$: energy transport velocity) in terms of the relaxation time τ. Though the microscopic origin is not clear, this relaxation time concept has also been integrated in Section 2.2:

^{E}

_{33}and λ

^{D}

_{33}), according to the crystallographic symmetry change from $m3m$ to $\infty mm$. Third, because the elastic compliance under the short-circuit condition (s

^{E}

_{33}) is softer than the elastic compliance under open-circuit conditions (s

^{D}

_{33}) via the relation s

^{E}

_{33}(1 − k

_{33}

^{2}) = s

^{D}

_{33}, and the sound velocity ${v}_{33}^{E}$ is smaller than ${v}_{33}^{D}$, the lattice vibration and domain wall dynamic motion are expected to be larger in the polarization direction under the short-circuit condition. In the thermal conductivity measurement, the “depolarization field” in parallel to the spontaneous polarization direction under open-circuit condition (i.e., D-constant), originated from the pyroelectric charge during the temperature rise, stabilizes the crystal vibration. The internal electric field in piezo-materials in general stabilizes the lattice vibration (in particular optical branch) and domain wall motion. The larger lattice vibration and domain wall dynamics in the short-circuit condition may also introduce a larger phonon transportation, i.e., larger thermal conductivity (λ

^{E}

_{33}). This speculation can also suggest the analogous relations among two coupling factors: electromechanical k

_{33}and secondary electrothermal k

^{λ}

_{33}.

## 5. Thermal Analysis on Piezoelectric Transducers

#### 5.1. Pseudo-DC Piezoelectric Actuators

#### 5.1.1. Heat Generation from Multilayer Actuators

_{e}/A, where V

_{e}is the effective volume (electrode overlapped part, abL in the figure) and A is the all surface area. Because the temperature was uniformly generated in a bulk sample (no significant stress distribution, except for the small inactive portion of the external electrode sides), this linear relation is reasonable because the volume V

_{e}generates the heat, which is dissipated through the area A. Thus, if we need to suppress the temperature rise, a small V

_{e}/A design is preferred. Instead of one ML, four (1/4) small-ML’s connected in parallel are preferred.

#### 5.1.2. Thermal Analysis on ML Actuators

_{g}, and that at which the heat is dissipated, q

_{d}, can be expressed as

_{g}− q

_{d}= V ρ c

_{p}(dT/dt),

_{p}is the specific heat capacity (per mass) of the specimen. The heat generation in the piezoelectric is attributed to losses. Thus, the rate of heat generation, q

_{g}, is expressed as:

_{g}= w f V

_{e},

_{e}is the effective volume of active ceramic (no-electrode parts are omitted). According to the measurement conditions (no significant stress in the sample at off-resonance), this w may correspond primarily to the dielectric hysteresis loss (i.e., P-E hysteresis), w

_{e}, which is expressed in terms of the intensive dielectric loss tan δ’ as:

_{e}= π ε

^{X}ε

_{0}E

_{0}

^{2}tan δ’.

_{d}) from the sample is the sum of the rates of heat flow by radiation (q

_{r}) and by convection (q

_{c}):

_{d}= q

_{r}+ q

_{c}= eAσ (T

^{4}− T

_{o}

^{4}) + h

_{c}A(T − T

_{o}),

_{o}is the initial sample temperature, h

_{c}is the average “convective heat transfer coefficient”. Thus, Equation (98) can be written in the form:

_{e}− A k(T) (T − T

_{o}) = V ρ C

_{p}(dT/dt),

^{2}+ T

_{o}

^{2})(T + T

_{o}) + h

_{c}

_{o}= [w f V

_{e}/k(T) A] (1 − e

^{−t/τ}),

_{p}V/k(T) A.

_{e}/k(T) A,

_{T}f V

_{e}/ρC

_{p}V) = ΔT/τ,

_{T}can be regarded under these conditions as a measure of the total loss of the piezoelectric. The dependence of k(T) on applied electric field and frequency are shown in Figure 15a,b, respectively. Note that k(T) is almost constant, as long as the driving voltage or frequency is not very high (E < 1 kV/mm, f < 2 kHz). The total loss, w

_{T}, as calculated from Equation (137b) is given for three multilayer specimens in Table 3, three values of which are almost the same in less than 2% deviation. In parallel, we measured the P-E hysteresis losses under stress-free conditions. The w

_{e}values obtained (Equation (130)) are also listed in Table 3 for comparison. It is intriguing that the extrinsic P-E hysteresis loss contributes more than 90% of the calculated total loss associated with the heat generated in the operating piezoelectric specimen [34,35]. We can conclude that the heat generation of the piezoelectric specimen under high-electric field off-resonance operation is primarily originated from the intensive dielectric loss factor, tan δ’.

#### 5.2. Resonance Drive Piezoelectric Transducers

#### 5.2.1. Heat Generation from a Resonating Piezoelectric Specimen

_{31}plates (Figure 16) when driven at the resonance [37]. The temperature distribution profile in a PZT-based plate sample was observed with a pyroelectric infrared camera (FLIR Systems ThermaCAM S40 outfitted with a 200 mm lens), as shown in Figure 17, where the temperature variations are shown in the sample driven at the first (28.9 kHz) (a) and second resonance (89.7 kHz) (b) modes, respectively. The highest temperature (bright spot) is evident at the nodal line areas of the length resonance for the specimen in Figure 17a,b. This observation supports that the heat generated in a resonating sample is primarily originated from the intensive elastic loss, tanφ’. Remember that the maximum vibration velocity is defined as the velocity under which a 20 °C temperature increase occurs at the maximum temperature point (i.e., nodal line!) in the sample.

#### 5.2.2. Heat Generation at the Antiresonance Mode

_{A}at the resonance under constant voltage drive, while by Q

_{B}at the antiresonance under constant current drive. In the k

_{31}mode, we derived the mechanical quality factors Q

_{A}and Q

_{B}as follows [37]:

_{33′}, tanϕ

_{11′}, and tanθ

_{31′}are intensive loss factors for ε

_{33}

^{X}, s

_{11}

^{E}, d

_{31}, respectively, and Ω

_{B,}

_{31}is the normalized antiresonance frequency:

_{B,}

_{31}is k

_{31}dependent, and can be obtained from the equation $1-{{\mathit{k}}_{31}}^{2}+{{\mathit{k}}_{31}}^{2}\frac{\mathrm{tan}{\Omega}_{\mathit{B}}}{{\Omega}_{\mathit{B}}}=0$. Because the intensive dielectric loss tanθ

_{31′}is larger than $(\mathrm{tan}{{\mathbf{\delta}}_{33}}^{\prime}+\mathrm{tan}{{\mathit{\varphi}}_{11}}^{\prime})/2$ in PZT piezoceramics, Q

_{B}at antiresonance is higher than Q

_{A}at resonance; that is, the antiresonance operation seems to be more efficient than the resonance drive.

_{31}= 30% [37]. In comparison with the fundamental resonance mode, though the antiresonance mode exhibits the strain-zero lines (i.e., anti-node lines) a little inside from the plate edges for this small k

_{31}case, the overall vibration configurations for resonance and antiresonance modes are rather close to each other. Namely, as long as the vibration velocity at the plate edge is the same, the total mechanical energy is assumed to be the same for both resonance and antiresonance operations. Numerical profiles of the temperature distribution for the A- and B-type resonance modes are shown in Figure 18c for various vibration velocity, which seems to be pseudo-sinusoidal curves in terms of the length position coordinate. Under the same vibration velocity, 550 mm/s RMS, the resonance nodal line area shows the temperature up to 100 °C, while the antiresonance mode shows the maximum around 60 °C; dramatic heat generation reduction under the antiresonance drive

#### 5.2.3. Thermal Analysis on the Resonance Mode

#### 5.2.3.1. Heat Transfer Modeling

_{31}mode piezoelectric rectangular plate around the resonance/antiresonance frequency range. In comparison with the off-resonance model, where the heat is generated primarily from the dielectric loss and the uniform temperature distribution profile due to no particular stress in the specimen, the resonance case generates the heat originated from the intensive elastic loss on a sinusoidal stress distribution of a specimen. Because of the initial heat source being sinusoically distributed in the specimen, we need to integrate the thermal diffusivity of the piezo-ceramic in order to analyze the temperature distribution of the sample.

- (1)
- 1D heat conduction in the specimen.
- (2)
- Heat generation is proportional to strain squared (i.e., elastic energy), distributed on the specimen.
- (3)
- Heat dissipation via convection (to air) and radiation. Conduction is neglected.

_{p}(c

^{E}in the k

_{31}case) is specific heat (unit: J/kg K), ρ density, and $\frac{\lambda}{{c}_{p}\rho}={\alpha}_{T}$ is called thermal diffusivity [unit: m

^{2}/s].

^{4,7)}The first term of the right-hand-side of Equation (140) describes the temperature distribution in respect of the position x, which changes the shape with time. The second term corresponds to the temperature increment caused by heat generation per unit volume (devided by ${c}_{p}\rho $), which may exhibit a sinusoidal distribution. The third term indicates the heat dissipation proportional to the temperature difference ΔT from the ambient temperature ${T}_{air}$.

_{d}is a proportional constant to [T − T

_{air}], similar to “heat transfer coefficient” introduced in Equation (131).

#### 5.2.3.2. Temperature Distribution Profile Change with Time

_{31}rectangular plate specimen with $80\times 14\times 2{\mathrm{mm}}^{3}$ in size with a hard PZT composition (APC 841, American Piezo Company, Mill Hall, PA, USA) for both admittance and thermal imaging observation purposes. We used a constant vibration velocity method for the measurements under 300 mm/s RMS. Figure 20a shows the temperature distribution profile change with time after driving. You can notice that the plate edge temperature increases significantly with time lapse, primarily due to the thermal diffusion in the PZT from the nodal highest point to the edge lowest temperature point. The saturated temperature distribution profile for the k

_{31}specimen is shown in Figure 20b, which can be used for calculating the total thermal dissipation energy g

_{h}. A small temperature dent at x = 0 originated from the heat dissipation by the center sample-holding rod conduction, which is neglected in the simulation.

_{m}from

_{m}= 507 by a 3-dB-down method on an admittance spectrum. Then, the sample was excited under the vibration velocity of 400 mm/s for 30 s, which corresponds to the heat dissipation of 11.6 W/m

^{2}. Figure 21 shows the Q

_{m}obtained at three frequencies slightly above the resonance frequency (20.04 kHz) [7]. The Q

_{m}values around 550 agree with the extraporated values from the above 507. Thus, we can conclude that this thermal method can determine the Q

_{m}reasonably at any frequency around the resonance and anti-resonance region. An increase of Q

_{m}with increasing the frequency suggests that a maximum-Q

_{m}frequency (i.e., the highest efficiency) exists between the resonance and antiresonance frequencies [38].

## 6. Conclusions

^{λ}for the thermal conductivity analysis, relating the parameter change under short- and open-circuit conditions, in order to explain the fact that the short-circuit condition exhibited larger thermal diffusivity than the open-circuit condition. On the other hand, the unpoled specimen exhibits the lowest thermal conductivity. This tutorial paper was authored for providing comprehensive knowledge on equilibrium and time-dependent thermodynamics in ferroelectrics. Electrothermal coupling may be categorized as primary and secondary effects in general. The primary effect includes “pyroelectric” and “electro-caloric” phenomena, and the secondary effect includes the space-gradient (i.e., first-derivative) phenomena, such as “thermal conductivity” and local “electric displacement current”. The author is expecting further theoretical expansion in this electrothermal coupling area, and industrial application products.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Calculation models of electro-mechanical coupling factor k for (

**a**) electric input under stress free, and (

**b**) stress input under short-circuit condition.

**Figure 5.**Schematic representation of the response of a piezoelectric material under: Top constant applied electric field and Bottom constant applied stress conditions.

**Figure 6.**(

**a**) 1D thermal diffusion model, (

**b**) experimental setup for determining directional thermal diffusivity.

**Figure 7.**(

**a**) temperature distribution profiles as a function of position x for various time lapses, and (

**b**) temperature change with time at the position x = L.

**Figure 8.**Depolarization field generation during temperature rise by pyroelectric effect: (

**a**) poled specimen under short-circuit condition, (

**b**) poled specimen under open-circuit condition, and (

**c**) unpoled specimen.

**Figure 11.**ω–k Dispersion curve for (

**a**) one-atom 1D lattice chain model, and (

**b**) two-atom 1D lattice change model.

**Figure 14.**(

**a**) device temperature change with driving time for ML actuators of various sizes; (

**b**) temperature rise at off-resonance versus V

_{e}/A in various size soft PZT ML actuators, where V

_{e}and A are the effective volume and the surface area.

**Figure 15.**Overall heat transfer coefficient, k(T), plotted as a function of applied electric field (

**a**), and of frequency (

**b**) for a PZT ML actuator with 7 × 7 × 2 mm

^{3}driven at 400 Hz.

**Figure 16.**A rectangular piezo-ceramic plate (L ≫ w ≫ b) for a longitudinal mode through the transverse piezoelectric effect (d

_{31}).

**Figure 17.**Temperature variations in a PZT-based plate sample observed with an infrared camera, driven at: (

**a**) first resonance and (

**b**) second resonance mode.

**Figure 18.**Temperature variations in a PZT-based plate specimen observed with a pyroelectric infrared camera, when driven at the antiresoance (

**a**) and resonance frequency (

**b**); (

**c**) numerical temperature profile for the A- and B-type resonance modes.

**Figure 20.**(

**a**) Temperature distribution profile change with time after driving; (

**b**) the saturated temperature distribution profile for k

_{31}specimen with $80\times 14\times 2$ mm

^{3}.

**Table 1.**Thermal diffusivity, specific heat capacity, and thermal conductivity of a Hard PZT, for unpoled and poled under different electrical boundary conditions.

Hard PZT | Thermal Diffusivity ${\mathit{\alpha}}_{\mathit{T}}$ (10 ^{−7} m^{2}/s) | +/− | Specific Heat Capacity c _{p} (J/kg K) | Thermal Conductivity $\mathit{\lambda}$ (W/m K) | +/− |
---|---|---|---|---|---|

Unpoled | 4.32 | 0.34 | ${c}_{p}^{D}$ = 279 | 0.91 | 0.10 |

Open Circuit | 5.02 | 0.23 | $\frac{1}{3}\left({c}_{p}^{D}+2{c}_{p}^{E}\right)$ = 320 | 1.22 | 0.06 |

Short Circuit | 8.25 | 0.78 | ${c}_{p}^{E}$ = 340 | 2.13 | 0.23 |

Thermal Properties | c_{p} (J/kg K) | λ (W/m/K) |
---|---|---|

Hard-PZT | 420 | 1.25 |

NKN-Cu | 580 | 3.10 |

**Table 3.**Loss and overall heat transfer coefficient for PZT multilayer samples (under E = 3 kV/mm, f = 300 Hz) [34].

Actuator | 4.5 × 3.5 × 2.0 mm^{3} | 7.0 × 7.0 × 2.0 mm^{3} | 17 × 3.5 × 1.0 mm^{3} |
---|---|---|---|

wT (kJ/m^{3}) [$\frac{\rho {C}_{p}V}{f{V}_{e}}{\left(\frac{dT}{dt}\right)}_{t\to 0}$] | 19.2 | 19.9 | 19.7 |

P-E hysteresis loss (kJ/m^{3}) | 18.5 | 17.8 | 17.4 |

k(T) (W/m^{2} K) | 38.4 | 39.2 | 34.1 |

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Uchino, K. Electrothermal Phenomena in Ferroelectrics. *Actuators* **2020**, *9*, 93.
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