Improved analytical modelling and finite element verification of stressed GaN microbeam resonators by piezoelectric actuation

The characteristics of piezoelectric micro-resonators based on vibrating beams essentially depend on two basic physical coefficients, an effective Young's modulus (Y-e) and a piezoelectric coupling factor (e(31e)). An improved analytic model is proposed with newly derived expressions of Y-e and e(31e) that account for the anisotropic properties of the III-nitride materials and beam width, W.The analytic model applicable to the only axial stress is completed by finite element (FE) simulations that allow any spatial patterns of pre-stress in wafers to be studied. The value of e(31e) for wider beams is analytically demonstrated to be much higher than the usual e(31), and a strong dependence of e(31e) on W is also confirmed by FE simulations. Resonance frequency (f(r)) and actuation efficiency (eta) are numerically studied for several pre-stress patterns and beam dimensions. The integral(r) is found to be sensitive to the beam width only for resonators under 2D pre-stress while the eta to the stress magnitude regardless of stress pattern. Compared with measurements published for some fabricated resonators, both analytic and FE approaches agree well quantitatively for the resonance frequency and qualitatively for the dynamic amplitude. The results of this study can help design optimization, such as appropriate electrode length and suitable beam width, to gain better performance for this type of resonators.