Ferroelectrics

Overview

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Applications

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Models

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Linear Model

For low input field levels, the hysteretic \(E-P\) and \(E-\varepsilon\) relations can be approximated by the slope at remanence, or about other bias points, to yield linear constitutive relations quantifying the converse piezoelectric effect. Similarly, the polarization and fields generated by the direct piezoelectric effect exhibit a nearly linear dependence on stress for low input levels, thus motivating the use of linear constitutive models for sensor applications.

In the absence of an applied electric field, the polarization produced by a stress \(\sigma\) is \[ P=d\sigma, \] where \(d\) is the piezoelectric charge coefficient. For general materials, \(P\) is vector-valued and related to each component of the stress \(\sigma_{ij}\). Therefore, it follows that \[ P_n = \sum_{i=1}^3 \sum_{j=1}^3 d_{nij}\sigma_{ij}. \] The second contribution to \(P\) is due to the applied electric field \(E\). For linear operating regimes, this component is given by \[ P_n = \chi_{nm}^\sigma E_m \] the polarizibility relation, where \[ \chi_{nm}^\sigma = \overline{\chi}_{nm}^\sigma \epsilon_0 \] and \(\overline{\chi}_{nm}^\sigma\) denotes the dielectric susceptibility measured at constant stress and \(\epsilon_0\) is the permittivity in a vacuum.

Thus within a limited operating range, \[ P_n = d_{nij}\sigma_{ij}+\chi_{nm}^\sigma E_m \] is a linear model for the direct piezoelectric effect.

The converse piezoelectric relation combines Hooke's law (or, in this case, its reciprocal) with the linear \(E-\varepsilon\) relation to express strain as \[ \varepsilon_{ij} = s_{ijkl}^E \sigma_{kl} + d_{nij}E_n, \] where \(s^E\) is the fourth rank compliance tensor measure at constant field.

Domain Wall Model

The construction of \( p_{an}\) from energy analysis follows directly from the minimization of the Gibbs energy \[ G\left(E,P,T\right) = \psi\left(P,T\right) - EP \] with \(\psi\left(P,T\right) \) given by \[ \begin{align} \psi\left(P,T\right) = &\frac{1}{2}Y^P \varepsilon^3 -a_1 \varepsilon P - a_2 \varepsilon P^2 + +\frac{E_h P_S}{2}\lbrack 1-(P/P_S)^2\rbrack + \\ & \frac{E_h T}{2 T_C} \Big \lbrack P \ln \left(\frac{P+P_S}{P_S-P} \right)+P_S \ln \left(1-\left(P/P_S\right)^2\right) \Big \rbrack. \end{align} \] The resulting constitutive relations formulated as a function of strain are given by \[ \sigma = Y^P \varepsilon - a_1 P - a_2 P^2 \] and
\( E = a_1 \varepsilon -2 a\_2 \varepsilon P -\frac{E_h}{P_S} + \frac{E_h T}{T_C} \text{arctanh} \left(P/P_S\right). \) (1)

For transducer models, it is advantageous to reformulate the direct equation in terms of stresses rather than strains. Substitution of the elastic converse relation into (1), neglecting quadratic and cubic couplign terms, and solving for \(P\) yields \[ P_{an}\left( \sigma, E\right) = P_S \text{tanh}\left(\frac{E_e (\sigma,E)}{\alpha(T)}\right), \] where the effective field is given by \[ E_e \left( \sigma,E\right) = E +\alpha_1 \sigma + \left( \alpha + \alpha_2 \sigma\right) P. \] The model parameters must be identified through either a least-squares fit to data or online adaptive estimation techniques for a given material or transducer design.

Homogenized Energy Model

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