To derive the piezoelectric effect in crystals

[etotheipi]

For a nice cubic non-centrosymmetric crystal like quartz/$\mathrm{SiO_2}$ we can imagine that on application of stress the tetrahedral coordination polyhedra become distorted, and the central ion is displaced by a fraction $\lambda$ of the cell parameter $a$. The total polarisation $\mathbf{P} = \mathbf{P}_1 + \mathbf{P}_2$ is the sum of two contributions: $\mathbf{P}_1 \propto q (\lambda a)/a^3 \boldsymbol{e}_1$ due to the 'frozen-in' displacement of the central ions in each unit cell, and $\mathbf{P}_2 = \varepsilon_0 (\kappa - 1) \mathbf{E}$ due to polarisation of the rest of the crystal (approximated to be linear and isotropic) due to the net field $\mathbf{E}$. With the Gauss relation $\nabla \cdot \mathbf{E} = \rho / \varepsilon_0$ applied in integral form to a pillbox at the surface then we can show for a linear piezoelectric we get something like$$V = \frac{\sigma Ld}{\kappa \varepsilon_0}$$where $\sigma$ is the magnitude of stress, $L$ the width of the crystal and $d$ a piezoelectric coefficient. But for the general piezoelectric we have the constitutive equations$$\begin{align*}
\sigma_{ij} &= c_{ijkl} S_{kl} - e_{kij} E_k \\
D_k &= e_{kij} S_{ij} + \epsilon_{ki} E_i\end{align*}$$with $\sigma_{ij}$ the stress tensor, $S_{ij}$ the strain tensor, $\epsilon_{ij}$ the dielectric tensor, $c_{ijkl}$ elastic constants, $e_{jik}$ piezoelectric constants and $E_i$ & $D_i$ as usual. Does anyone have reference to derive these equations? I don't know anything about elasticity theory. Thanks

 

[Charles Link]

See https://www.amazon.com/dp/0198511655/?tag=pfamazon01-20
The book by Nye is very good reading for a number of topics including tensors.

 

[etotheipi]

See https://www.amazon.com/dp/0198511655/?tag=pfamazon01-20
The book by Nye is very good reading for a number of topics.

Thanks, that looks great. I'll see if I can borrow it

 

[etotheipi]

Okay, I think I understand the argument now. I'll write up the main points in case anyone else is interested. First some notational housekeeping; in the more modern notation, we use $\epsilon_{ij}$ to refer to the strain tensor, $\sigma_{ij}$ the stress tensor, $d_{ijk}$ the piezoelectric moduli, and $s_{ijkl}$ the compliance tensor (the twin of the stiffness tensor $c_{ijkl}$).

In the most general case, thermal, elastic and electrical quantities all depend on each other. We can choose a set of independent variables in which to describe the quantities $\epsilon_{ij}$ , $D_i$ and $S$ (entropy); we will here choose to express these as functions of the variables $(\sigma_{ij}, E_i, T)$, i.e.$$d\epsilon_{ij} = \left(\frac{\partial \epsilon_{ij}}{\partial \sigma_{kl}}\right) d\sigma_{kl} + \left(\frac{\partial \epsilon_{ij}}{\partial E_k}\right) dE_k + \left(\frac{\partial \epsilon_{ij}}{\partial T}\right) dT$$ $$dD_i = \left(\frac{\partial D_i}{\partial \sigma_{jk}}\right) d\sigma_{jk} + \left(\frac{\partial D_i}{\partial E_j}\right) dE_j + \left(\frac{\partial D_i}{\partial T}\right) dT$$ $$dS = \left(\frac{\partial S}{\partial \sigma_{ij}}\right) d\sigma_{ij} + \left(\frac{\partial S}{\partial E_i}\right) dE_i + \left(\frac{\partial S}{\partial T}\right) dT$$Since all processes are assumed reversible,$$dU = \sigma_{ij} d\epsilon_{ij} + E_i dD_i + T dS$$Consider the function $\Phi = \Phi(\sigma_{ij}, E_i, T)$ such that$$\Phi = U - \sigma_{ij} d\epsilon_{ij} - E_i D_i -TS \implies d\Phi = - \epsilon_{ij} d\sigma_{ij} -D_i dE_i - SdT$$but we also have$$d\Phi = \left(\frac{\partial \Phi}{\partial \sigma_{ij}}\right) d\sigma_{ij} + \left(\frac{\partial \Phi}{\partial E_i}\right) dE_i + \left(\frac{\partial \Phi}{\partial T}\right) dT$$and hence we have the relationships $\left(\frac{\partial \Phi}{\partial \sigma_{ij}}\right) = -\epsilon_{ij}$, then $\left(\frac{\partial \Phi}{\partial E_i}\right) = -D_i$, and finally $\left(\frac{\partial \Phi}{\partial T}\right) = -S$. But then, because partial derivatives commute, note that for instance$$\left(\frac{\partial \epsilon_{ij}}{\partial E_k}\right) = \left(\frac{\partial D_k}{\partial \sigma_{ij}}\right) \overset{\text{def}}{=} d_{kij}$$and a similar reasoning for the other combinations. This is interesting, because it means that the same components $d_{ijk}$ are the moduli for both the direct and converse piezoelectric effects! That is,$$\begin{align*} &\text{Direct effect:} \, P_i = d_{ijk} \sigma_{jk} \\ &\text{Converse effect:} \, \epsilon_{jk} = d_{ijk} E_i \end{align*}$$The general constitutive equations can be obtained by simply integrating the equations for $\epsilon_{ij}$ and $D_i$ written at the start of this post, and substituting in relational tensors like $d_{ijk}$, etc. in place of the partial derivatives.