Simultaneous determination of thin-film inertia and shear stiffness using thickness-twist and face-shear modes of an AT-cut quartz resonator

We propose the use of thickness-twist (TT) and face-shear (FS) vibration modes of an AT-cut quartz crystal plate resonator for simultaneous determination of the inertia and stiffness of a thin film deposited on a crystal surface. A theoretical analysis using Mindlin's first-order theory for crystal plates is performed to demonstrate the idea. Expressions for the stiffness ratio and mass ratio between the thin film and the resonator are presented in terms of frequency shifts of FS and TT modes, which are experimentally measurable. A numerical example is given.