Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators.

Elastic constants c11, c12, and c44 of degenerately doped silicon are studied experimentally as a function of the doping level and temperature. First-and second-order temperature coefficients of the elastic constants are extracted from measured resonance frequencies of a set of MEMS resonators fabricated on seven different wafers doped with phosphorus (carrier concentrations 4.1, 4.7, and 7.5 x 10(19) cm(-3)), arsenic (1.7 and 2.5 x 10(19) cm(-3)), or boron (0.6 and 3 × 10(19) cm(-3)). Measurements cover a temperature range from -40°C to +85°C. It is found that the linear temperature coefficient of the shear elastic parameter c11 - c12 is zero at n-type doping level of n ~ 2 x 10(19) cm(-3), and that it increases to more than 40 ppm/K with increasing doping. This observation implies that the frequency of many types of resonance modes, including extensional bulk modes and flexural modes, can be temperature compensated to first order. The second-order temperature coefficient of c11 - c12 is found to decrease by 40% in magnitude when n-type doping is increased from 4.1 to 7.5 × 10(19) cm(-3). Results of this study enable calculation of the frequency drift of an arbitrary silicon resonator design with an accuracy of ±25 ppm between the calculated and real(ized) values over T = -40°C to +85°C at the doping levels covered in this work. Absolute frequency can be estimated with an accuracy of ±1000 ppm.

Parametric study of laterally acoustically coupled bulk acoustic wave filters.

Acoustically coupled thin-film bulk acoustic wave resonator filters, in which the coupling takes place mechanically in the lateral direction between closely-spaced narrow resonators, are a promising approach to passband filtering at gigahertz frequencies. In this paper, filters with interdigital electrode structures are studied. Electrode number, electrode width, and coupling gap width are varied. The resonators are solidly mounted, having an acoustic mirror isolating the resonator from a Si substrate and providing the means to engineer the acoustic dispersion properties of the resonators. The center frequency of the filters is around 2 GHz. Electrical frequency responses of the filters are measured and the strength of the lateral acoustic coupling is calculated from the measurements. The effects of device parameters on the acoustic coupling and the obtainable filter bandwidth are analyzed in detail. A bandpass filter with 4.9% bandwidth, minimum insertion loss of 2 dB and sharp transition from passband to suppression band, is presented.

Measurement of evanescent wave properties of a bulk acoustic wave resonator.

Acoustic wave fields in a thin-film bulk acoustic wave resonator are studied using a heterodyne laser interferometer. The measurement area is extended outside the active electrode region of the resonator, so that wave fields in both the active and surrounding regions can be characterized. At frequencies at which the region surrounding the resonator does not support laterally propagating acoustic waves, the analysis of the measurement data shows exponentially decaying amplitude fields outside the active resonator area, as suggested by theory. The magnitude of the imaginary wave vectors is determined by fitting an exponential function to the measured amplitude data, and thereby the experimentally determined dispersion diagram is extended into the region of imaginary wave numbers.

Dispersion and mirror transmission characteristics of bulk acoustic wave resonators.

A heterodyne laser interferometer is used for a detailed study of the acoustic wave fields excited in a 932-MHz solidly mounted ZnO thin-film BAW resonator. The sample is manufactured on a glass substrate, which also allows direct measurement of the vibration fields from the bottom of the acoustic mirror. Vibration fields are measured both on top of the resonator and at the mirror-substrate interface in a frequency range of 350 to 1200 MHz. Plate wave dispersion diagrams are calculated from the experimental data in both cases and the transmission characteristics of the acoustic mirror are determined as a function of both frequency and lateral wave number. The experimental data are compared with 1-D and 2-D simulations to evaluate the validity of the modeling tools commonly used in mirror design. All the major features observed in the 1-D model are identified in the measured dispersion diagrams, and the mirror transmission characteristics predicted for the longitudinal waves, by both the 1-D and the 2-D models, match the measured values well.

2-D modeling of laterally acoustically coupled thin film bulk acoustic wave resonator filters.

A 2-D model is developed for calculating lateral acoustical coupling between adjacent thin film BAW resonators forming an electrical N-port. The model is based on solution and superposition of lateral eigenmodes and eigenfrequencies in a structure consisting of adjacent regions with known plate wave dispersion properties. Mechanical and electrical response of the device are calculated as a superposition of eigenmodes according to voltage drive at one electrical port at a time while extracting current induced in the other ports, leading to a full Y-parameter description of the device. Exemplary cases are simulated to show the usefulness of the model in the study of the basic design rules of laterally coupled thin film BAW resonator filters. Model predictions are compared to an experimental 1.9-GHz band-pass filter based on aluminum nitride thin film technology and lateral acoustical coupling. Good agreement is obtained in prediction of passband behavior. The eigenmode-based model forms a useful tool for fast simulation of laterally coupled acoustic devices. It allows one to gain insight into basic device physics in a very intuitive fashion compared with more detailed but heavier finite element method. Shortcomings of this model and possible improvements are discussed.

Experimental investigation of acoustic substrate losses in 1850-MHz thin film BAW resonators.

After optimizing for electromechanical coupling coefficient K(2), the main performance improvement in the thin film bulk acoustic wave resonators and filters can be achieved by improving the Q value, i.e., minimizing the losses. In Bragg-reflector-based solidly mounted resonator technology, a significant improvement of Q has been achieved by optimizing the reflector not only for longitudinal wave, the intended operation mode, but also for shear waves. We have investigated the remaining acoustic radiation losses to the substrate in so-optimized 1850-MHz AlN resonators by removing the substrate underneath the resonators and comparing the devices with and without substrate by electrical characterization before and after the substrate removal. Several methods to extract Q-values of the resonators are compared. Changes caused by substrate removal are observed in resonator behavior, but no significant improvement in Q-values can be confirmed. Loss mechanisms other than substrate leakage are concluded to dominate the resonator Q-value. Difficulties of detecting small changes in the Q-values of the resonators are also discussed.

Spurious resonance suppression in gigahertz-range ZnO thin-film bulk acoustic wave resonators by the boundary frame method: modeling and experiment.

Zinc-oxide-based thin-film bulk acoustic wave (BAW) resonators operating at 932 MHz are investigated with respect to variation of dimensions of a boundary frame spurious mode suppression structure. A plate wave dispersion-based semi-2-D model and a 2-D finite element method are used to predict the eigenmode spectrum of the resonators to explain the detailed behavior. The models show how the boundary frame method changes the eigenmodes and their coupling to the driving electrical field via the modification of the mechanical boundary condition and leads to emergence of a flat-amplitude piston mode and suppression of spurious modes. Narrow band suppression of a single mode with a nonoptimal boundary frame is observed. Reduction of the effective electromechanical coupling coefficient k2eff as a function of the boundary width is observed and predicted by both models. The simple semi-2-D plate model is shown to predict the device behavior very well, and the 2-D finite element method results are shown to coincide with them with some additional effects. Breaking the resonator behavior down to eigenmodes, which are not directly observable in measurements, by the models, yields insight into the physics of the device operation.

Estimating materials parameters in thin-film BAW resonators using measured dispersion curves.

The dispersion curves of Lamb-wave modes propagating along a multilayer structure are important for the operation of thin-film bulk acoustic wave (BAW) devices. For instance, the behavior of the side resonances that may contaminate the electrical response of a thin-film BAW resonator depends on the dispersion relation of the layer stack. Because the dispersion behavior depends on the materials parameters (and thicknesses) of the layers in the structure, measurement of the dispersion curves provides a tool for determining the materials parameters of thin films. We have determined the dispersion curves for a multilayer structure through measuring the mechanical displacement profiles over the top electrode of a thin-film BAW resonator at several frequencies using a homodyne Michelson laser interferometer. The layer thicknesses are obtained using scanning electron microscope (SEM) measurements. In the numerical computation of the dispersion curves, the piezoelectricity and full anisotropy of the materials are taken into account. The materials parameters of the piezoelectric layer are determined through fitting the measured and computed dispersion curves.