Mechanically vibrating devices are widely used in communication and other systems that require frequency selection or a frequency reference for various operations.  Although quartz-based resonant devices and other acoustic devices, such as film bulk acoustic resonator (FBAR) technologies, have been used in the many different products, these devices require a significant amount of area.  Moreover, these devices are difficult to integrate with the associated control circuitry.  On the other hand, silicon-based microelectromechanical systems (MEMS) are attractive for their potential to be compact, single-chip integrated frequency references and/or filters.  MEMS generally refer to an integration of mechanical elements, sensors, actuators, and sometimes electronics on a common silicon substrate.  The electronics portions of the MEMS may be fabricated using known integrated circuit (IC) processes, while the micromechanical components are typically fabricated by selectively etching portions of or adding new structural layers to a silicon substrate. 
This technology is becoming more sophisticated and continues to expand into new areas, fostering a wide range of new applications, in the wireless industry for example.  Vibrating micromechanical resonators are emerging as attractive candidates for on-chip versions of the high-Q mechanical passive components used in transceivers for wireless communication applications.  Micromechanical resonators are particularly attractive due to the small size, close to zero direct current (DC) power consumption, and the potential use of IC fabrication technologies to enable on-chip integration of frequency selective components having high quality factors (Qs).  Recently, MEMS resonators have demonstrated mechanical Qs over 10,000 for frequencies in the gigahertz (GHz) range.  Also, MEMS resonators having parts-per-million (ppm) level long term stability and temperature stability have been demonstrated.  These and other features make the MEMS resonator feasible for communication grade applications. 
Unfortunately, however, the resonant frequency of a MEMS resonator is generally dependent on its geometry.  It follows that the geometrical dependence may result in resonant frequencies that are sensitive to unavoidable process variations.  Moreover, the resonant frequencies may be unrepeatable due to the process variations.  For example, an undesirable resonant frequency shift may result from certain geometric process variations, such as an undercut.  Geometric process variations may inadvertently increase or decrease the operational frequency of a device as the dimensions fluctuate.  Thus, unavoidable process variations during device manufacture may lead to unrepeatable resonant and operational frequencies, resulting in a device or system that has unpredictable operational characteristics.