1116 IEEE TRANSACTIONS ON MAGNETICS, VOL. 45, NO. 3, MARCH 2009 Prototype Design of a Thin-Film Bulk Acoustic-Wave Resonator by the Finite Element Method V. De Santis, M. Feliziani, C. Buccella, and P. Tognolatti Department of Electrical and Computer Engineering, University of L’Aquila, 67040 L’Aquila AQ, Italy A thin film bulk acoustic wave resonator (FBAR) used in the RF frequency region of a few gigahertz is considered and its impedance is evaluated by using a harmonic analysis with the three-dimensional finite element method. In particular, the spurious characteristics caused by variations in the electrode area, as well as all the resonant modes and the mode shapes are analyzed. A design procedure is presented by using a prediction software tool. An experimental prototype is built and the measured results are compared to the numerical ones. Index Terms—Film bulk acoustic wave resonator (FBAR), finite element analysis, multiphysics, piezoelectric material. I. INTRODUCTION T HE mobile communication systems require smaller sizes and higher performances for radio frequency (RF) passive devices such as resonators or filters. Thus, a new technology based on the film bulk acoustic wave resonators (FBARs) has been developed in recent years [1], [2]. This technology presents a lot of advantages because FBAR resonators are at least an order of magnitude smaller than dielectric or lumped chip counter- parts, and they have higher quality factors and power endurance than surface acoustic wave (SAW) devices [3]–[5]. Also, FBAR resonators can be integrated with other active RF devices, such as monolithic microwave amplifiers and mixers, which are fabricated using the Silicon or GaAs substrate. In this paper, the design of a FBAR by using a simulation soft- ware tool based on a three-dimensional finite element method (FEM) is presented. The resonant characteristics, mode shapes and spurious responses of the resonator are predicted by har- monic and eigenvalue analyses. The design procedure is pre- sented and a FBAR prototype is fabricated. The electrical per- formances of the realized resonator are evaluated by experi- mental measurements. Then, the results obtained by simulation and experiments, in terms of impedance, are compared to vali- date the design procedure and to verify the fabrication process. II. PROBLEM FORMULATION Conventional FBAR resonators can have different topologies and different materials. However, the basic FBAR structure con- sists of a resonant acoustic cavity formed by a piezoelectric film sandwiched between two electrode films, as shown in Fig. 1. Due to the piezoelectric capability of the cavity some mechan- ical displacements occur when subjects to an electric field and vice versa [6]. Therefore, the multiphysics problem coupling the mechanical quantities with the electrical one must be analyzed. As concern the mechanical behaviour of the resonator, the equations for linear elasticity must be considered: (1a) Manuscript received October 07, 2008. Current version published February 19, 2009. Corresponding author: V. De Santis (e-mail: desantis.valerio@ing. univaq.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2009.2012646 Fig. 1. Basic configuration of a piezoelectric resonator. (1b) where is the mass density, the applied body force, the mechanical displacement vectors, the acceleration, the stress tensor, the strain tensors, and the derivation operator defined as (2) In the piezoelectric material the quasi-static approximation for the Maxwell equations can be assumed and therefore the electrical problem is governed by (3a) (3b) where is the electric field, the electrical displacement vector, the volume charge density and the electric potential. The piezoelectric constitutive equations that couple the me- chanical and electrical quantities can be expressed as: (4a) (4b) where is the piezoelectric tensor, the elastic stiffness tensor (evaluated at constant electric field) and is the dielec- tric permittivity matrix (evaluated at constant strain). The piezo- electric materials usually employed in the FBAR resonators (i.e. AlN, ZnO and PZT) belong to the hexagonal crystal systems. The matrices and vectors in (4) are then given by [6] (5a) 0018-9464/$25.00 © 2009 IEEE