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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.