Concurrent wafer-level measurement of longitudinal and transverse effective piezoelectric coefficients (d_(33.f) and e_(31,f)) by double beam laser interferometry

摘要

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 (d_(33,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 d_(33.f) is measured when the ratio of the electrode size to the substrate thickness is approximately unity, in the case of a substrate with iso-tropic elastic properties and a Poisson's ratio of ~0.3. In this paper, we further investigate the dependence of the critical ratio (r_c) of the electrode size to the substrate thickness at which the true d_(33,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 (-s_(13)/s_(11)) 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 (e_(31.f)) can also be determined from the same measurement of the electrode size dependence of d_(33.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 e_(31.f) under varying electric field excitation which is important for many applications such as actuators.