Study of LGS crystal Micro-resonators using Flexure Mode: Temperature-compensated Cuts G. Douchet, F. Sthal, E. Bigler and R. Bourquin Frequency and Time Department FEMTO-ST, UMR CNRS 6174 Besançon, France Gabrielle.Douchet@ens2m.fr Abstract— In this paper, new experiments have been carried out on LGS crystal to highlight the existence of temperature compensated cuts for flexure vibration. The micro-resonators are square cross section tuning forks whose arms are vibrating in flexure-mode with clamped-free boundary conditions. Frequency versus temperature behaviors have been measured for several cut angles. The results of our experiments show that there is a first order temperature-compensated cut for Langasite crystal resonators vibrating in flexion at room temperature. I. INTRODUCTION Nowadays, quartz resonators used in electronic watches come in the shape of a tuning fork. They have been largely studied in the past, theoretically and in terms of microfabrication [1]. Quartz crystal is the best known piezoelectric crystal but its piezoelectric coefficients are very low. Synthetic piezoelectric materials are now commercially available. Materials such as Langasite crystal (LGS), Langatate crystal (LGT) and Gallium Orthophosphate crystal (GaPO 4 ) are more piezoelectric than quartz crystal and they also have a much better temperature behavior. GaPO 4 crystal has shown its interest for low frequency resonators [2-3]. We already had both theoretical and experimental results concerning LGS crystal length extension resonators and their temperature compensated cuts [4-5]. LGT crystal has also been studied [6-7]. In order to miniaturize the resonators, attempts to get chemical etching of materials like GaPO 4 and LGS have also been investigated [8-11]. Collective micro-fabrication could be very interesting in the future. In this paper, new experiments have been carried out on LGS crystal to highlight the existence of temperature compensated cuts for flexural vibration. Design of the resonators is presented. It concerns tuning fork resonators vibrating in flexure mode with clamped-free boundary conditions. Experimental results are compared with simple analytical model and with finite element analysis. II. THEORETICAL MODEL A tuning fork resonator vibrating in a flexural mode is studied (Fig. 1). In this analytical model, each beam of the tuning fork is considered separately. Figure 1. Tuning fork orientation (X, Y, Z cristallographic axes), θ is the rotation angle. The derivation used in this model is based on the Bernoulli beam model where shear effects are neglected. These assumptions are valid when the beam length is much larger than its width and thickness. The resonant frequencies are: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ ρS s I l π λ = f 2 22 2 2 (1) with S: cross-sectional area of the beam s 22 : Compliance for a beam oriented along the crystallographic Y-axis I: Inertia of the beam with 12 3 t w I ⋅ = ρ: mass density of material λ is solution of an eigenfrequency equation that depends upon the boundary conditions. In our case the boundary conditions are clamped-free. Y X Z X-cut plate (XY) l w t θ 978-1-4244-1795-7/08/$25.00 ©2008 IEEE 332