Piezoelectric resonators have been in use as mass-sensing devices for almost half a century. More recently it was recognized that shifts in frequency and bandwidth can come about by a diverse variety of interactions with the sample. The classical "load" consists of a thin film, which shifts the resonance frequency due to its inertia. Other types of loading include semi-infinite viscoelastic media, rough objects contacting the crystal via isolated asperities, mechanically nonlinear contacts, and dielectric films. All these interactions can be analyzed within the small-load approximation. The small-load approximation provides a unified frame for interpretation and thereby opens the way to new applications of the QCM.

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Viscoelastic, mechanical, and dielectric measurements on complex samples with the quartz crystal microbalance

A resonator structure (FBAR) made of electrodes sandwich a piezoelectric material. The intersection of the two conducting electrodes defines the active area of the acoustic resonator. The active area is divided into two concentric areas; a perimeter or frame, and a central region. An annulus is added to one of the two conducting electrodes to improve the electrical performance (in terms of Q).

Thin film bulk acoustic resonator with a mass loaded perimeter

Disclosed is an acoustic resonator that includes a substrate, a first electrode, a layer of piezoelectric material, a second electrode, and an alternating frame region. The first electrode is adjacent the substrate, and the first electrode has an outer perimeter. The piezoelectric layer is adjacent the first electrode. The second electrode is adjacent the piezoelectric layer and the second electrode has an outer perimeter. The alternating frame region is on one of the first and second electrodes.

Acoustic resonator performance enhancement using alternating frame structure

This paper studies the application of lateral bulk acoustic thin-film piezoelectric-on-substrate (TPoS) resonators in high-frequency reference oscillators Low-motional impedance TPoS resonators are designed and fabricated in 2 classes--high-order and coupled-array Devices of each class are used to assemble reference oscillators and the performance characteristics of the oscillators are measured and discussed Since the motional impedance of these devices is small, the transimpedance amplifier (TIA) in the oscillator loop can be reduced to a single transistor and 3 resistors, a format that is very power-efficient The lowest reported power consumption is ~350 muW for an oscillator operating at ~106 MHz A passive temperature compensation method is also utilized bThis paper studies the application of lateral bulk acoustic thin-film piezoelectric-on-substrate (TPoS) resonators in high-frequency reference oscillators Low-motionalimpedance TPoS resonators are designed and fabricated in 2 classes--high-order and coupled-array Devices of each class are used to assemble reference oscillators and the performance characteristics of the oscillators are measured and discussed Since the motional impedance of these devices is small, the transimpedance amplifier (TIA) in the oscillator loop can be reduced to a single transistor and 3 resistors, a format that is very power-efficient The lowest reported power consumption is ~350 muW for an oscillator operating at ~106 MHz A passive temperature compensation method is also utilized by including the buried oxide layer of the silicon-on-insulator (SOI) substrate in the structural resonant body of the device, and a very small (-24 ppm/degC) temperature coefficient of frequency is obtained for an 82-MHz oscillatory including the buried oxide layer of the silicon-on-insulator (SOI) substrate in the structural resonant body of the device, and a very small (-24 ppm/degC) temperature coefficient of frequency is obtained for an 82-MHz oscillator

Thin-film piezoelectric-on-silicon resonators for high-frequency reference oscillator applications

This review article summarizes the development of higher order theories for the dynamic analysis of piezoelectric plates, and describes their applications, especially for crystal resonators. The theories are categorized according to how the displacement components and electric potential are assumed to vary through the plate thickness. The greatest attention has been given to ordinary polynomial variations, especially the efforts of RD Mindlin and HF Tiersten and their coworkers. Considerable progress has also been made using trigonometric series representations, especially by PCY Lee and his coworkers. Legendre polynomial and normal mode expansions have also been developed. Both analytical and finite element methods of dealing with problems are described. The article contains 174 references.

Higher-Order Theories of Piezoelectric Plates and Applications

Closed‐form asymptotic expressions for the frequency–wavenumber dispersion relations in doubly rotated quartz plates vibrating in the vicinity of the odd pure thickness frequencies are derived from the equations of linear piezoelectricity and the associated boundary conditions on the major surfaces. The usual assumptions of small piezoelectric coupling and small wavenumbers along the plate are made and it is supposed that the pure thickness frequencies are sufficiently different that one pure thickness wave is dominant at a time. In the treatment the mechanical displacement is decomposed along the eigenvector triad of the pure thickness solution to facilitate the asymptotic analysis. The fact that the wavenumbers along the plate are restricted to be small significantly reduces the complexity of the equations without neglecting any transformed elastic constants. The resulting asymptotic dispersion equation enables the construction of a scalar differential equation describing the transverse behavior of essentially thickness modes of vibration in doubly rotated quartz plates. The scalar equation is applied in the analysis of both trapped energy resonators with rectangular electrodes and contoured crystal resonators using established procedures. In particular, calculations performed for the contoured SC cut and a number of other doubly rotated orientations are shown to be in excellent agreement with experiment. Since the differential equation for each harmonic family depends on the order of the harmonic and in the general doubly rotated case contains mixed derivatives in the plane of the plate, a different transformation is required for each harmonic family to obtain the coordinate system in which the mixed derivatives do not appear and, hence, the equation is separable. An interesting consequence of this transformation is that since the nodal planes of the anharmonics of each harmonic family of the contoured SC‐cut quartz resonator are oriented along the transformed coordinate system for that harmonic family, they are oriented differently for each harmonic family.

An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation

It has recently been shown that highly uniform thin layers can be etched in a small well‐defined region of a silicon wafer and that good quality thin piezoelectric films such as zinc‐oxide can be deposited along with the electrodes to form high frequency trapped energy resonant device structures. The accompanying analytical work has been for pure thickness vibrations only. In this work an analysis of essentially thickness‐extensional trapped energy modes in the thin piezoelectric film on silicon composite structure is performed. It is shown that for small wavenumbers along the plate the dispersion equation is isotropic in the plane of the plate even though the silicon is severely anisotropic in that plane. From the resulting dispersion relation an asymptotic differential equation describing the mode shape along the surface of the composite plate vibrating in the vicinity of a thickness‐extensional resonance is obtained along with the associated edge conditions. Since the mode is essentially thickness‐extensional, trapping does not usually occur in the electroded regions for the fundamental mode of the flat composite plate. However, by appropriately thickening the silicon outside the electroded regions trapping in the electroded regions can be made to occur for the fundamental mode. Furthermore, in the case of zinc‐oxide on silicon, trapping in the electroded regions can be made to occur for the fundamental mode of a flat plate simply by making the zinc‐oxide sufficiently thicker than the etched silicon. The aforementioned asymptotic differential equation and edge conditions are applied in the analyses of the steady‐state vibrations of trapped energy resonators and acoustically coupled two‐pole resonator filters, both with rectangular electrodes. Lumped parameter representations of the admittance in the case of the single‐pole device and admittance matrix in the case of the two‐pole device, which are valid in the vicinity of resonances, are obtained. The analyses hold for the first few thickness‐extensional modes and their accompanying lateral overtones.

An analysis of thickness‐extensional trapped energy resonant device structures with rectangular electrodes in the piezoelectric thin film on silicon configuration

A system of approximate equations recently obtained for the determination of thermal stresses in electroded piezoelectric plates is applied to contoured AT‐cut quartz crystal resonators. The changes in resonant frequency resulting from the thermally induced biasing stresses and strains, the temperature derivatives of the fundamental elastic constants of quartz, and the very important temperature dependence of the thickness piezoelectric constant for the AT cut are determined from an equation for the perturbation of the eigenfrequency of the piezoelectric solution due to a bias. The changes in resonant frequency with temperature are calculated for the fundamental and some of the harmonic overtone thickness modes, which were obtained in a recent analysis of overtone modes in contoured crystal resonators. The influences of both the contouring and the electrode size are clearly exhibited.

Temperature dependence of the resonant frequency of electroded contoured AT-cut quartz crystal resonators

Acoustic Surface Wave Accelerometer and Rotation Rate Sensor

Electroelastic equations containing terms up to cubic in the small mechanical displacement field, but no higher than linear in the electric variables, are applied in the analysis of nonlinear resonance in doubly rotated contoured‐quartz resonators and all terms present in the general anisotropic case are included. Since the modes of motion in contoured‐quartz resonators are essentially thickness modes varying slowly along the plate, only the thickness dependence of the elastic nonlinearities are retained in the equations, as in earlier work. The linear portions of the equations are the same as those that have recently been derived and employed in the analysis of doubly rotated contoured‐quartz resonators. The steady‐state solutions are obtained by means of an asymptotic iterative procedure and an expansion in the linear eigensolutions while retaining the nonlinear correction to the eigensolution that has a resonant frequency in the vicinity of the driving frequency. The slow variations in the mode along the plate are included in the nonlinear correction by averaging over the plate. Lumped parameter representations of the solutions, which are valid in the vicinity of a resonance and relate the amplitude of the mode nonlinearly to the voltage across the electrodes, are obtained. In each instance the expression for the current through the crystal is determined, the external circuitry is incorporated in the description and an equation relating the mode amplitude nonlinearly to the driving voltage and other circuit parameters is obtained. The analysis holds for the fundamental and odd harmonic overtones. Nonlinear resonance curves are calculated for AT‐cut quartz using the known nonlinear coefficients. In particular, it is shown that the order of the harmonic has a more significant influence on the shift in resonant frequency from the linear value than the current through the crystal. An equation relating the change in resonant frequency resulting from the nonlinearity to the current through the crystal, independent of the external circuitry, is derived. This latter equation is employed in the evaluation of the coefficient of nonlinear resonance for SC‐cut quartz from measurements of the shift in resonant frequency with current level in contoured resonators.

D.S. Stevens

Papers

An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation

An analysis of thickness‐extensional trapped energy resonant device structures with rectangular electrodes in the piezoelectric thin film on silicon configuration

Temperature dependence of the resonant frequency of electroded contoured AT-cut quartz crystal resonators

Acoustic Surface Wave Accelerometer and Rotation Rate Sensor

An analysis of nonlinear resonance in contoured‐quartz crystal resonators