Abstract An analytical study of the frequency-temperature behavior of MEMS AT-cut quartz resonator structures was performed. The piezoelectric, Lagrangian equations for the frequency-temperature behavior of quartz were employed[1]. Two types of resonator structures were stud- ied: A 1 GHz rectangular plate with rectangular electrodes and a 1 GHz rectangular ring electrode mesa (REM) plate. For MEMS resonators, the REM design confers certain advantages in attaining higher resonant frequencies and lower electrode noise. I. Introduction The design of stable resonators is usually done by engineers with years of experience and experimentation. There is however not much experience in designing a sta- ble quartz MEMS thickness shear resonator in the fre- quency range of 3 GHz because the existing conventional resonator designs will not work. A 3 GHz resonator will have an electrode to plate thickness ratio of more than 27%[2]. The frequency-temperature behavior of conventional rectangular plate quartz resonators is well known, however that of a ring electrode mesa (REM) structure is not. A pos- sible application of MEMS AT-cut resonators would be as the reference oscillator in miniature atomic clocks and nan- oresonator development programs. FIG. 1: Position vectors x i , y i and z i of a material point at the respective initial, intermediate and final states. II. Piezoelectric Lagrangean Equations for the Frequency-Temperature Behavior of Quartz Resonators. We state the linear Lagrangean piezoelectric equations of small vibrations superposed on thermal deformations induced by steady, uniform temperature change in quartz which are based upon the three-dimensional mechanical equations of incremental motion superposed on homoge- neous thermal strains[3]. The full details of the develop- ment of the incremental equations is not repeated here, and the interested reader is referred to reference[3]. The formu- lation is similar to those, for example, by Sinha and Tier- sten[4], and Dulmet and Bourquin[5], but we prefer to use the Piola-Kirchhoff stress tensor of the second kind for rea- sons of symmetry. Figure 1 shows the three states of the resonator. The following incremental Lagrangean equa- tions which includes the piezoelectric effect are as follows: IIa. Strain-displacement, and electric field-potential relations. , and (1) , where (2) and (3) . (4) The terms , u k,i , E i and are, respectively, the incremental strains, the first partial derivative of incremental displacement, incremental electric field and incremental electric potential. , , denote the Kronecker delta and nth order temperature coefficients of thermal expansion, respectively. T and T o are the temperatures of the intermediate state and initial (natural) state, respectively, and is the temperature change. The reference temperature T o is set to 25 o C. IIb. Equations of motion and electrostatics. in V, and (5) in V, (6) where is the incremental stress tensor, the mass density, the incremental mechanical displacement and the incremental electric displacement vector. IIc. Initial and boundary conditions. A unique solution is guaranteed by specifying Initial state, T o Intermediate state, T Final state, T x i y i z i U i u i s ij 1 2 -- β kj u ki , β ki u kj , + ( ) = E i φ i , – = β ik δ ik α ik 1 ( ) θ α ik 2 ( ) θ 2 α ik 3 ( ) θ 3 + + + = θ T T o – ( ) = s ij φ δ i k α ik n ( ) θ β ik σ kj j , ρ u ·· i = D ii , 0 = σ k j ρ u i D k FREQUENCY-TEMPERATURE ANALYSIS OF MEMS AT-CUT QUARTZ RESONATORS. Y-K Yong a and J. Vig b , A. Ballato b , R. Kubena c and R. M’Closkey d a Civil & Environmental Engineering Dept., Rutgers University, NJ, U.S.A. yyong@rci.rutgers.edu b U.S. Army CECOM, c HRL Laboratories and d University of California Los Angeles