Concurrent wafer-level measurement of longitudinal and transverse effective piezoelectric coefficients (<i>d</i>33,f and <i>e</i>31,f) by double beam laser interferometry

<jats:p>In a recently published paper [S. Sivaramakrishnan et al., Appl. Phys. Lett. 103, 132904 (2013)], the electrode-size dependence of the longitudinal effective piezoelectric coefficient (d33,f) of piezoelectric thin films measured by double beam laser interferometry was shown to be due to the substrate clamping effects. It was also shown that the true d33,f is measured when the ratio of the electrode size to the substrate thickness is approximately unity, in the case of a substrate with isotropic elastic properties and a Poisson's ratio of ∼0.3. In this paper, we further investigate the dependence of the critical ratio (rc) of the electrode size to the substrate thickness at which the true d33,f is measured on the substrate Poisson's ratio for isotopic substrates and for the important case of the anisotropic Si substrate. It turns out that it is the out-of-plane Poisson's ratio (-s13/s11) that is relevant for this measurement technique and not the in-plane Poisson's ratio which is highly anisotropic for the (001) oriented Si. Furthermore, we show that the transverse effective piezoelectric coefficient (e31,f) can also be determined from the same measurement of the electrode size dependence of d33,f. This provides a convenient non-destructive wafer-level measurement technique for the determination of both the piezoelectric coefficients simultaneously. Moreover, this technique is also capable of measuring e31,f under varying electric field excitation which is important for many applications such as actuators.</jats:p>