Effects of electrodes on frequency-temperature characteristics of singly and doubly rotated thickness-mode resonators

The use of metal electrodes deposited on major surfaces of a quartz crystal resonator vibrating in a thickness mode has been common in many applications. Conventional analyses assumed that the thickness of electrodes was thin enough, so that only the mass loading of electrodes were taken into account. The thickness of a crystal plate has been reduced to meet recent demands for devices operable in a higher frequency range. The thickness of electrodes, however, cannot be proportionally reduced, because adequate conductivity has to be maintained. Then electrodes behave not as simple masses but as continuous bodies, in which elastic waves propagate. This causes marked changes in plate back, Q and temperature characteristics, which previous analyses have failed to explain. This paper presents an analysis based on equivalent circuits, consisting of three transmission lines for doubly rotated resonators and a single line for singly rotated resonators, taking full elastic properties of electrodes into account. A good agreement is obtained between calculations and experiments using AT and SC cuts. The analysis is also useful to determine properties of thin film, which have been difficult to measure.