# 18.3: Variation of Polarisation with Temperature

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With a pyroelectric, the polarisation *P* will typically decrease when its temperature is raised. This is because the increasing disorder results in a reduced segregation of charge, and so the arising dipoles are lessened in magnitude. The drop off in polarisation can be seen on the next slide.

Here the polarisation drops off to the Curie point, where it is zero.

\[ \underline{\Delta P} = \underline{p} \Delta T \]

where *p* = pyroelectric coefficient (C m^{-2} K^{-1}). The pyroelectric coefficient is a vector, with three components:

\[ \begin{matrix} \Delta P_i = p_i \Delta T & i=1,2,3 \end{matrix} \]

Typically, however, the electrodes which measure this are placed along a principal crystallographic direction, and therefore, the coefficient is often measured as a scalar, which is typically negative to represent a polarization falling with increasing temperature.

The pyroelectric coefficient measured under an applied field ** E** is liable to differ from its true value as explained below.

When an electric field, ** E** is applied to a polar material, a moment arises, and the total response

**(as**

*D*measured as a charge per unit area on metallic plates either side of the pyroelectric) is expressed as:

\[ D = \varepsilon E + P_s \]

where ε = electrical permittivity of the pyroelectric and *P*_{s} = spontaneous polarisation.

This means that:

\[ \frac{\partial D}{\partial T} = \frac{\partial P_s}{\partial T} + E \frac{\partial \varepsilon}{\partial T} \]

Since the pyroelectric coefficient *p*_{g} relates changes in ** D** to changes in

*T*, we have a ‘generalised’ pyroelectric coefficient, given by:

\[p_g = \frac{\partial D}{\partial T} = \frac{\partial P_s}{\partial T} + E \frac{\partial \varepsilon}{\partial T} = p + E \frac{\partial \varepsilon}{\partial T}\]

which includes a factor based on the permittivity of the material being temperature dependent. This is what is measured.

The true pyroelectric coefficient is given by:

\[ p =\frac{\partial P_s}{\partial T} \]

as this defines the variation of the spontaneous polarisation with *T*.

The effect due to changes in permittivity can sometimes be comparable in magnitude to true pyroelectricity, and can also be seen above the Curie point.