Vibration modes of mass-loaded planoconvex quartz crystal resonators

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Abstract
Layered piezoelectric resonators exhibit a complex spectrum of eigenfrequencies, which can be understood better if the associated vibration patterns are examined. For a closer insight into such a system, planoconvex AT-cut quartz crystals with a thick layer of copper on one side are investigated experimentally. Since presently no comprehensive three-dimensional theory of layered resonators is available, the experimental results can be compared only with one-dimensional descriptions of layered resonators or with theoretical models for unloaded contoured quartz crystals. The anharmonic spectrum of the crystals is used to find a formula by which the eigenfrequency of the corresponding infinite plate can be determined from measured frequencies of planoconvex crystals. If this correction is applied to experimental frequency spectra of composite quartz-film resonators, excellent agreement with the one-dimensional model is found. As theoretically predicted, also even harmonic overtones of heavily loaded crystals can be excited electrically. The vibration patterns are examined with an ultrasensitive laser-speckle technique. The measurements of the odd harmonic overtones agree well with the theory of energy trapping in planoconvex quartz crystals. An increase of the crystal’s active area with increasing mass load is observed. The even overtones frequently exhibit no such energy trapping; in these cases, unusual vibration patterns occur.

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