Patent Publication Number: US-7595708-B2

Title: MEMS resonator array structure

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
   This application is a continuation of and claims priority under 35 U.S.C. § 120 to U.S. application Ser. No. 11/172,143 filed Jun. 30, 2005, now U.S. Pat. No. 7,227,432, the entirety of which is incorporated herein by reference in its entirety. 

   BACKGROUND 
   This invention relates to a microelectromechanical or nanoelectromechanical resonator array structure, and method of designing, operating, controlling and/or using such an architecture; and more particularly, in one aspect, to a plurality of microelectromechanical or nanoelectromechanical resonators (for example, a plurality of resonators at least one of which includes one or more enhanced nodal points that facilitate substrate anchoring in order to minimize influence of packaging stress and/or energy loss via substrate anchoring) that are mechanically coupled to provide one or more output signals having one or more frequencies. 
   Generally, high Q microelectromechanical resonators are regarded as a promising choice for integrated single chip frequency references and filter. In this regard, high Q microelectromechanical resonators tend to provide high frequency outputs that are suitable for many high frequency applications requiring compact and/or demanding space constrained designs. However, while the resonator is being scaled smaller, packaging stress, energy loss into the substrate through substrate anchors, reduced signal strength, and/or instability or movement of the center of gravity during oscillation tend to adversely impact the frequency stability as well as “Q” of the resonator. 
   There are several well-known resonator architectures. For example, one group of conventional resonator architectures employs closed-ended or open-ended tuning fork. For example, with reference to  FIG. 1 , closed-ended or double-clamped tuning fork resonator  10  includes beams or tines  12   a  and  12   b . The beams  12   a  and  12   b  are anchored to substrate  14  via anchors  16   a  and  16   b . The fixed electrodes  18   a  and  18   b  are employed to induce a force to beams  12   a  and  12   b  to cause the beams to oscillate (in-plane). 
   The characteristics and response of tuning fork resonator  10  are well known. However, such resonator architectures are often susceptible to changes in mechanical frequency of resonator  10  by inducing strain into resonator beams  12   a  and  12   b  as a result of packaging stress. In addition, conventional resonator architectures, like that illustrated in  FIG. 1 , experience or exhibit energy loss, though the anchors, into the substrate. 
   Certain architectures and techniques have been described to address Q-limiting loss mechanism of energy loss into the substrate through anchors as well as changes in frequency due to certain stresses. In one embodiment, the beams of the resonator may be “suspended” above the ground plane and sense electrode whereby the vibration mode of the beam is out-of-plane. (See, for example, U.S. Pat. No. 6,249,073). While such architectures may alleviate energy loss through the anchors, resonators that include an out-of-plane vibration mode (i.e., transverse mode) tend to exhibit relatively large parasitic capacitance between drive/sense electrodes and the substrate. Such capacitance may lead to a higher noise floor of the output signal (in certain designs). 
   Other techniques designed to improve the Q-factor of the resonator have been proposed and include designing the spacing between the vibrating beams so that such beams are closely spaced relative to a wavelength associated with their vibrating frequency. (See, for example, the single-ended or single-clamped resonator of U.S. Pat. No. 6,624,726). The vibrating beams are driven to vibrate one-half of a vibration period out of phase with each other (i.e., to mirror each others motion). While these architectures and techniques to improve the Q of the resonator may suppress acoustic energy leakage, such an architecture remain predisposed to packaging stress, energy loss into the substrate through substrate anchors as well as a “moving” of the center of gravity of the resonator during motion by the vibrating beams of the single-ended or single-clamped resonator. 
   Further, other resonator architectures have been described to address energy loss through the anchor, for example, a “disk” shaped resonator design. (See, for example, U.S. Patent Application Publication 2004/0207492). Indeed, an array of identical mechanically-coupled disk-shaped resonators has been proposed to decrease motional resistance while improving linearity. (See, for example, U.S. Pat. No. 6,628,177 and “Mechanically Corner-Coupled Square Microresonator Array for Reduced Series Motional Resistance”, Demirci et al., Transducers 2003, pp. 955-958). 
   There is a need for a resonator array architecture, configuration or structure that overcomes the shortcomings of one, some or all of the conventional architectures, configurations or structures. In this regard, there is a need for improved array of microelectromechanical and/or nanoelectromechanical resonators having improved packaging stress characteristics, reduced and/or minimal energy loss into the substrate though substrate anchors, and/or improved or optimal stability of the center of gravity during oscillation. In this way, the signal to noise of the output signal is increased, the stability and/or linearity of the output frequency of the resonator is enhanced, and/or the “Q” factor of the resonator is relatively high. 
   Further, there is a need for an improved microelectromechanical resonator array architecture, configuration or structure that includes relatively small motional resistance and good linearity, implements full differential signaling and/or possesses a high immunity to on the input signals and/or the output signals. Moreover, there is a need for an improved method of designing, operating, controlling and/or using such a resonator array that overcomes the shortcomings of one, some or all of the conventional resonator array architectures, configurations or structures. 
   SUMMARY OF THE INVENTION 
   There are many inventions described and illustrated herein, as well as many aspects and embodiments of those inventions. This Summary discusses some of the inventions described and claimed herein. By no means is this Summary of the Invention is not exhaustive of the scope of the present inventions. With that in mind, in a first principal aspect, the present invention is a MEMS array structure comprising a plurality of MEMS resonators coupled via one or more resonator coupling sections. In one embodiment, each MEMS resonator includes a plurality of elongated straight beam sections (for example, four elongated straight beam sections), each including first and second ends, and a plurality of curved sections (for example, four curved sections), each including first and second ends, wherein each end of a beam section is connected to an associated end of one of the curved section to thereby form a geometric shape (for example, a rounded square shape). 
   In one embodiment, the MEMS array structure may further include at least one resonator coupling section which is disposed between each of the opposing elongated straight beam sections of adjacent MEMS resonators. 
   In addition, in one embodiment, at least one curved section of at least one MEMS resonator may include a nodal point wherein the MEMS array structure further includes at least one anchor coupling section and a substrate anchor, coupled to the nodal point via the anchor coupling section, to secure the MEMS resonator to a substrate. The MEMS array structure may also include a stress/strain relief mechanism disposed within the anchor coupling section and between the substrate anchor and the nodal point. 
   In another embodiment, at least one curved section of each MEMS resonator includes a nodal point and wherein the MEMS array structure further includes at least one anchor coupling section disposed between an associated nodal point and a substrate anchor and wherein the substrate anchor secures the MEMS resonator to a substrate. A stress/strain relief mechanism maybe disposed within the anchor coupling section and between the substrate anchor and the nodal point. 
   In one embodiment, each resonator coupling section includes voids to reduce the mass of the section. In another embodiment, each resonator coupling section includes a filleted shape at the ends such that the ends of the resonator coupling section have a greater width than the middle of the resonator coupling section. 
   Notably, each curved section of each MEMS resonator may include at least one nodal point. In this embodiment, the at least one nodal point of each MEMS resonator is connected to a substrate anchor via an associated anchor coupling section. The MEMS resonator array structure may include a plurality of stress/strain relief mechanisms disposed within an associated anchor coupling section and between an associated substrate anchor and an associated nodal point. 
   In certain embodiment, the plurality of elongated straight beam sections of each MEMS resonator includes a plurality of slots disposed therein. Moreover, at least one of the plurality of curved sections of each MEMS resonator includes a plurality of slots disposed therein. Indeed, the width of each elongated straight beam section of the MEMS resonator is greater at the ends than in the center thereof. 
   In another principal aspect, the present invention is a MEMS array structure comprising a plurality of MEMS resonators, a plurality of resonator coupling sections and a plurality of anchor coupling sections. Each MEMS resonator includes a plurality of elongated straight beam sections and a plurality of curved sections (for example, four elongated straight beam sections and four curved sections). Each beam section includes a first end and a second end. Further, each curved section includes a first end and a second end, wherein each end of a beam section is connected to an associated end of one of the curved section to thereby form a geometric shape (for example, a rounded square shape). Moreover, at least one curved section includes a nodal point. 
   In this aspect, at least one resonator coupling section is disposed between at least one pair of opposing elongated straight beam sections of adjacent MEMS resonators such that each MEMS resonator is connected to at least one adjacent MEMS resonator. In addition, the at least one nodal point of each MEMS resonator is connected to a substrate anchor via an associated anchor coupling section. 
   In one embodiment, MEMS array structure further includes a plurality of stress/strain relief mechanisms, wherein at least one stress/strain relief mechanism is disposed within an associated anchor coupling section and between the substrate anchor and the nodal point of the MEMS resonator. The resonator coupling sections may include voids to reduce the mass of the section. The resonator coupling sections may, in addition to or in lieu thereof, include a filleted shape at the ends such that the ends of the resonator coupling section have a greater width than the middle of the resonator coupling section. 
   In another embodiment, the plurality of elongated straight beam sections of each MEMS resonator includes a plurality of slots disposed therein. Indeed, the plurality of curved sections of each MEMS resonator may include a plurality of slots disposed therein. 
   The MEMS array structure may also include a plurality of sense electrodes, a plurality of drive electrodes, and sense circuitry. The sense and drive electrodes are juxtaposed the plurality of elongated straight beam sections of the MEMS resonators. The sense circuitry is coupled to the sense electrodes to provide an output signal. 
   The sense electrodes may provide one or more signals to the sense circuitry which, in response, provides a differential output signal. The sense electrodes may provide one or more signals to the sense circuitry which, in response, provides a single ended output signal. 
   In another principal aspect, the present invention is a MEMS array structure comprising a plurality of MEMS resonators wherein each MEMS resonator includes a plurality of elongated straight beam sections, a plurality of curved sections, wherein each end of a beam section is connected to an associated end of one of the curved section to thereby form a geometric shape. The MEMS array structure may further include one or more resonator coupling sections. In this embodiment, each of the opposing elongated straight beam sections of adjacent MEMS resonators includes a resonator coupling section connected therebetween. The MEMS array structure may also include a plurality of sense electrodes, a plurality of drive electrodes, wherein the sense and drive electrodes are juxtaposed one or more of the plurality of elongated straight beam sections of the MEMS resonators. Sense circuitry, coupled to the sense electrodes, provides an output signal (for example, a differential output signal and/or a single ended output signal). 
   In one embodiment, one or more sense electrodes are disposed within the geometric shape of at least one of the MEMS resonators. Indeed, the one or more sense electrode may be juxtaposed a plurality of elongated straight beam sections of the at least one of the MEMS resonator. 
   In one embodiment, at least one curved section of at least one of the plurality of MEMS resonators includes a nodal point. In this embodiment, the MEMS array structure further includes at least one anchor coupling section and a substrate anchor, coupled to the nodal point via the anchor coupling section, to secure the MEMS resonator to a substrate. 
   The MEMS array structure may include a stress/strain relief mechanism disposed within the anchor coupling section and between the substrate anchor and the nodal point. 
   In another embodiment, each curved section of each MEMS resonator includes at least one nodal point. In this embodiment, at least one nodal point of each MEMS resonator is connected to a substrate anchor via an associated anchor coupling section. A plurality of stress/strain relief mechanisms may be disposed within an associated anchor coupling section and between an associated substrate anchor and an associated nodal point. The resonator coupling sections may include voids to reduce the mass of the section to a filleted shape at the ends such that the ends of the resonator coupling section have a greater width than the middle of the resonator coupling section. 
   Again, there are many inventions, and aspects of the inventions, described and illustrated herein. This Summary discusses some of the inventions described and claimed herein. By no means is this Summary of the Invention is not exhaustive of the scope of the present inventions. Moreover, this Summary of the Invention is not intended to be limiting of the invention and should not be interpreted in that manner. While certain embodiments have been described and/or outlined in this Summary of the Invention, it should be understood that the present invention is not limited to such embodiments, description and/or outline. Indeed, many others embodiments, which may be different from and/or similar to, the embodiments presented in this Summary, will be apparent from the description, illustrations and claims, which follow. In addition, although various features, attributes and advantages have been described in this Summary of the Invention and/or are apparent in light thereof, it should be understood that such features, attributes and advantages are not required whether in one, some or all of the embodiments of the present inventions and, indeed, need not be present in any of the embodiments thereof. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the course of the detailed description to follow, reference will be made to the attached drawings. These drawings show different aspects of the present invention and, where appropriate, reference numerals illustrating like structures, components, materials and/or elements in different figures are labeled similarly. It is understood that various combinations of the structures, components, materials and/or elements, other than those specifically shown, are contemplated and are within the scope of the present invention. 
       FIG. 1  is a block diagram (top view) representation of a conventional microelectromechanical tuning fork resonator device; 
       FIG. 2A  is a schematic representation of MEMS resonator array having an N×M MEMS resonator configuration, according to one aspect of the present inventions, wherein each MEMS resonator of the array is coupled to the adjacent resonator; 
       FIG. 2B  is a schematic representation of MEMS resonator array having an N×M MEMS resonator configuration, according to one aspect of the present inventions, wherein the MEMS resonators of the array are coupled to at least one adjacent resonator; 
       FIG. 3A  is a top view of one embodiment of a rounded triangle shaped MEMS resonator, having three elongated beam sections that are connected via rounded or curved sections, according to an embodiment of one aspect of the MEMS resonator array of the present inventions; 
       FIG. 3B  is a top view of one embodiment of a rounded square shaped MEMS resonator, having four elongated beam sections that are connected via rounded or curved sections, according to an embodiment of one aspect of the MEMS resonator array of the present inventions; 
       FIG. 3C  is a top view of one embodiment of a rounded hexagon shaped MEMS resonator, having six elongated beam sections that are connected via rounded or curved sections, according to an embodiment of one aspect of the MEMS resonator array of the present inventions; 
       FIGS. 4A-4I  illustrate top views of exemplary MEMS resonator arrays having a plurality of rounded square shaped MEMS resonators according to certain embodiments of the present inventions wherein the plurality of rounded square shaped MEMS resonators are mechanically coupled to one or more adjacent MEMS resonators of the MEMS resonator array employing various resonator coupling sections; 
       FIGS. 5A and 5B  illustrate top views of exemplary MEMS resonator arrays having a plurality of rounded square shaped MEMS resonators according to certain embodiments of the present inventions wherein the plurality of rounded square shaped MEMS resonators are mechanically coupled to one or more adjacent MEMS resonators of the MEMS resonator array employing various resonator coupling sections that include one or more loading relief mechanisms which are mechanically disposed within the resonator coupling section; 
       FIGS. 6A ,  6 B,  6 D- 6 H,  7 A- 7 H,  8 A,  8 B,  9 A- 9 C,  10 A and  10 B illustrate top views of exemplary MEMS resonator arrays having a plurality of rounded square shaped MEMS resonators according to certain embodiments of present the inventions wherein one or more of the plurality of rounded square shaped MEMS resonators are mechanically coupled to one or more substrate anchors using various anchoring techniques and/or configurations; 
       FIG. 6C  illustrates an oblique view of the MEMS resonator array of  FIG. 6D ; 
       FIGS. 11A and 11B  illustrate top views of a portion of exemplary MEMS resonator arrays including a rounded square shaped MEMS resonator according to certain embodiments of the present inventions wherein the MEMS resonator array includes stress/strain relief mechanisms which are mechanically coupled between a rounded square shaped MEMS resonator and a substrate anchor; 
       FIGS. 12A-12C  and  13 A- 13 C illustrate top views of exemplary MEMS resonator arrays including a plurality of rounded square shaped MEMS resonators according to certain embodiments of the present inventions wherein each MEMS resonator array includes stress/strain relief mechanisms which are mechanically coupled between one or more of the rounded square shaped MEMS resonators and one or more substrate anchor(s); 
       FIGS. 14 and 15  are top views of a portion of exemplary embodiments of rounded square shaped MEMS resonator, according to certain embodiments of MEMS resonator array of the present inventions, wherein the rounded or curved sections have different radii, and a plurality of anchor coupling sections that connect the rounded or curved sections to one or more anchors; 
       FIGS. 16-18  are top views of various embodiments of anchor coupling sections in conjunction with a section of a MEMS resonator, according to certain embodiments of the present inventions; 
       FIGS. 19-21  are top views of various embodiments of anchor coupling sections and stress/strain mechanisms, in conjunction with a section of a MEMS resonator, according to certain embodiments of the present inventions; 
       FIGS. 22A and 22B  are top views of a ring oscillator that is oscillating in plane in a breathing-like mode or motion, wherein the ring oscillator expands ( FIG. 22A ) and contracts ( FIG. 22B ) in relation to a non-induced state; 
       FIGS. 23A and 23B  are top views of one embodiment of a rounded square shaped MEMS resonator, including in-plane vibration of elongated beam sections, according to one aspect of present invention, wherein the MEMS resonator oscillates between a first deflected state ( FIG. 23A ) and a second deflected state ( FIG. 23B ) and wherein each deflected state is superimposed over (or illustrated relative to) the stationary state of MEMS resonator; 
       FIGS. 24A and 24B  are top views of an exemplary embodiment of a MEMS resonator array including four rounded square shaped MEMS resonators, having in-plane vibration of elongated beam sections, according to one aspect of present invention, wherein the MEMS resonators oscillate between deflection states and wherein each deflected state is superimposed over (or illustrated relative to) the stationary state of MEMS resonator; 
       FIG. 25  illustrates an exemplary embodiment of a MEMS resonator array including four rounded square shaped MEMS resonators, in conjunction with drive and sense electrodes and drive and sense circuitry, according to an aspect of present invention; 
       FIGS. 26A and 26B  illustrate exemplary embodiment of a MEMS resonator array including rounded square shaped MEMS resonators, in conjunction with a differential output signaling technique and embodiment, having drive and sense electrodes and differential drive and sense circuitry, according to exemplary embodiments of the present invention; 
       FIGS. 27A and 27B  illustrate exemplary embodiments of a MEMS resonator array, including four rounded square shaped MEMS resonators, in conjunction with a differential output signaling technique and embodiment, having drive and sense electrodes and differential drive and sense circuitry, according to another embodiment of the present invention; 
       FIGS. 28A ,  28 B and  29 A- 29 F illustrate exemplary embodiments of a MEMS resonator array, including four rounded square shaped MEMS resonators, in conjunction with various embodiments of drive and sense electrodes, according to exemplary embodiments of the present invention; 
       FIGS. 30A ,  30 B and  31 - 42  are top views of embodiments of a MEMS resonator array (or portions thereof) according to an aspect of the invention, wherein the MEMS resonator device includes openings, voids or slots for improved manufacturability (for example, faster release of the mechanical structures in those instances where the opening, void or slot extends the entire height/thickness of the beam section) and/or to improve temperature management techniques (for example, decrease thermo elastic energy dissipation) implemented in one or more elongated beam sections, one or more curved sections, and/or one or more anchor coupling sections; 
       FIGS. 43A and 43B  illustrate top views of exemplary MEMS resonator arrays having a plurality of rounded triangle shaped MEMS resonators according to certain exemplary embodiments of the present inventions wherein the plurality of triangle shaped MEMS resonators are mechanically coupled to one or more adjacent triangle shaped MEMS resonators of the MEMS resonator array; 
       FIGS. 43C and 43D  illustrate top views of exemplary MEMS resonator arrays having different shaped MEMS resonators including, a rounded triangle shaped MEMS resonator mechanically coupled to a rounded square shaped MEMS resonator ( FIG. 43C ) and rounded hexagon shaped MEMS resonators mechanically coupled to a rounded square shaped MEMS resonator ( FIG. 43D ); 
       FIGS. 44-46  are top views of various embodiments of exemplary MEMS resonator arrays including various exemplary anchor coupling sections and stress/strain mechanisms, in conjunction with a curved section of a MEMS resonator, according to certain embodiments of the present inventions; 
       FIGS. 47 and 48  are top views of a portion of an exemplary MEMS resonator arrays including various exemplary anchoring techniques to anchor the MEMS resonator array (and/or the MEMS resonators thereof) to the substrate; 
       FIGS. 49-52  are top views of exemplary MEMS resonator arrays including various exemplary anchoring techniques and stress/strain mechanisms in conjunction with various exemplary embodiments of resonator mechanical coupling techniques, according to certain embodiments of the present inventions; 
       FIGS. 53-55  are top views of exemplary MEMS resonator arrays including various exemplary anchoring techniques and stress/strain mechanisms in conjunction with various exemplary embodiments of resonator mechanical coupling techniques and loading relief mechanisms, according to certain embodiments of the present inventions; 
       FIG. 56A  is a top view of a MEMS frame array structure having a plurality of square shaped MEMS resonators, wherein each square shaped MEMS resonator of the array is coupled to the adjacent square shaped MEMS resonator and shares a beam section therewith, according to another aspect of the present inventions; 
       FIG. 56B  illustrates an oblique view of the MEMS frame array structure of  FIG. 56A ; 
       FIGS. 57A ,  58  and  59  illustrate top views of exemplary MEMS frame array structures having a plurality of square shaped MEMS resonators wherein one or more of the plurality of rounded square shaped MEMS resonators are mechanically coupled to an associated one of the substrate anchors using various anchoring techniques and/or configurations; 
       FIG. 57B  illustrates an oblique view of the MEMS frame array structure of  FIG. 57A ; 
       FIG. 60A and 60B  illustrate top views of a portion of exemplary MEMS frame array structures including a plurality of square shaped MEMS resonators according to one embodiment of present inventions wherein the MEMS frame array structure includes stress/strain relief mechanisms which are mechanically coupled between (i) one or more of the square shaped MEMS resonators and (ii) to a substrate anchor; 
       FIG. 61  is a top view of a MEMS frame array structure having a four by four array of square shaped MEMS resonators, wherein each square shaped MEMS resonator of the array is coupled to the adjacent square shaped MEMS resonator, according to one aspect of present invention; 
       FIG. 62  is a top view of the MEMS frame array structure of  FIG. 61  wherein the square shaped MEMS resonators oscillate between deflection states (only one illustrated herein) and wherein each deflected state is superimposed over (or illustrated relative to) the stationary state of MEMS resonator; 
       FIGS. 63 and 64  illustrate top views of an exemplary MEMS frame array structure (in oscillation) having a plurality of square shaped MEMS resonators wherein two rounded square shaped MEMS resonators are mechanically coupled to an associated substrate anchor using various anchoring techniques and/or configurations; and 
       FIG. 65  illustrates an exemplary embodiment of a MEMS frame array structure including four square shaped MEMS resonators, in conjunction with a differential output signaling technique and embodiment, having drive and sense electrodes and differential drive and sense circuitry, according to one embodiment of the present invention. 
   

   DETAILED DESCRIPTION 
   There are many inventions described and illustrated herein, as well as many aspects and embodiments of those inventions. In one aspect, the present invention is directed to a plurality of mechanically coupled resonators that are arranged in an N×M MEMS array structure (where N and M are integers). Each of the resonators includes a plurality of straight (or substantially straight) elongated beam sections that are connected by curved or rounded sections. Each elongated beam section of a given resonator is connected to another elongated beam section at a distal end via the curved or rounded sections thereby forming a geometric shape having at least two elongated beam sections that are interconnected via curved or rounded sections. 
   Each resonator is mechanically coupled to at least one other resonator of the MEMS array via a resonator coupling section. The resonator coupling sections are disposed or connected between elongated beam sections of mechanically coupled resonators. In this way, all of the resonators, when induced or during operation, vibrate at the same or substantially the same frequency. That is, in one embodiment, each beam section of each resonator of the array oscillates or vibrates at the same or substantially the same frequency oscillates or vibrates at the same or substantially the same frequency. 
   In one embodiment, each MEMS resonator of a MEMS array of the present invention includes three elongated beam sections that are interconnected via curved sections to form a rounded triangle shape. In another embodiment, the MEMS array of the present invention includes a plurality of resonators having four straight (or substantially straight) elongated beams that are connected, at distal ends, to rounded sections thereby forming a rounded square or rectangle shape. 
   In operation, when induced or during operation; each MEMS resonator of the array oscillates in a combined elongating (or breathing) mode and bending mode. In this regard, the beam sections of each MEMS resonator of the array exhibit an elongating-like (or breathing-like) motion and a bending-like motion. Further, when induced or during operation, each beam section of the MEMS resonators oscillates or vibrates at the same or substantially the same frequency. The beam sections of the MEMS resonators of the array all exhibit the same or substantially the same elongating-like (or breathing-like) motion and bending-like motion to thereby produce the same or substantially the same frequency. 
   The design and motion of each MEMS resonator of the array structure is such that the resonator includes one or more nodal points or areas (i.e., portions of the resonator structure that are stationary, experience little movement, and/or are substantially stationary in one or more degrees of freedom (whether from a rotational and/or translational perspective) during oscillation of the resonator structure). The nodal points are located in one or more portions or areas of the curved sections of the resonator structure. The nodal points are suitable and/or preferable locations to anchor the resonator structure and/or the array structure to the substrate. In this way, energy loss into the substrate may be minimized, limited and/or reduced, thereby enhancing the Q-factor of the resonator structure and/or the array structure. Notably, such a configuration may minimize and/or reduce communication of stress and/or strain between the resonating beams of one or more resonators of the array and the substrate. 
   In addition, although the beam sections of each MEMS resonator of the array, when induced or during operation, move in an elongating-like (or breathing-like) manner (for example, like that of a ring oscillator) and a bending-like manner (for example, like that of a beam of a double-claimed tuning fork), each MEMS resonator tends to maintain a relatively stable or fixed center of gravity. In this way, the resonators may avoid energy loss and thereby provide an array structure having a higher Q-factor. 
   Notably, the present inventions are described in the context of microelectromechanical systems. The present inventions, however, are not limited in this regard. Rather, the inventions described herein are applicable to, for example, nanoelectromechanical systems. Thus, the present inventions are pertinent to microelectromechanical and nanoelectromechanical (herein collectively “MEMS” unless specifically noted to the contrary) systems, for example, gyroscopes, resonators, and/or accelerometers, implementing one or more of the MEMS resonator array structures of the present inventions. 
   As mentioned above, in one aspect, the present invention is an array of N×M MEMS resonators (where N and M are integers) coupled to one or more of the adjacent MEMS resonators. 
   Each MEMS resonator is mechanically coupled to at least one other resonator of the array via a resonator coupling section. With reference to  FIG. 2A , in one embodiment, MEMS resonator array  100  includes a plurality of MEMS resonators  102   a - d  which are mechanically coupled, via resonator coupling sections  104 , to each adjacent MEMS resonator. In this way, each MEMS resonator  102  is coupled to all adjacent MEMS resonator(s)  102 . 
   With reference to  FIG. 2B , in another embodiment, MEMS resonator array  100  includes a plurality of MEMS resonators  102   a - d  which are mechanically coupled, via resonator coupling sections  104 , to at least one adjacent MEMS resonator. For example, MEMS resonator  102   e  is mechanically coupled to adjacent MEMS resonators  102   b ,  102   d ,  102   f  and  102   h . In contrast, MEMS resonator  102   h  is mechanically coupled to adjacent MEMS resonators  102   e  and  102   k . In this embodiment, MEMS resonator  102   h  is not coupled to adjacent MEMS resonators  102   g  and  102   i.    
   As mentioned above, each MEMS resonator of the MEMS resonator array, according to one aspect of the present invention, includes a plurality of elongated beam sections that are connected by curved or rounded sections. Each elongated beam section is connected to another beam section of the MEMS resonator at each distal end via the curved or rounded sections thereby forming a geometric shape having at least two elongated beams that are interconnected via curved or rounded sections. In one embodiment, with reference to  FIG. 3A , MEMS resonator  102  includes three elongated beam sections  106   a - c  that are connected via curved sections  108   a - c  to form a rounded triangle shape. With reference to  FIG. 3B , in another embodiment, MEMS resonator  102  includes four elongated beam sections  106   a - d  that are connected via curved sections  108   a - d  to form a rounded square shape. 
   Notably, MEMS resonator  102  of the present inventions may include more than four elongated beam sections, for example, MEMS resonator  102  may include six elongated beam sections  106   a - f  that are connected together via curved sections  108   a - f  to form a rounded hexagon shape (see,  FIG. 3C ). Indeed, the resonator structure of the present inventions may take any geometric shape whether now know or later developed that includes two or more straight elongated beam sections which are interconnected by two or more curved or rounded sections. 
   The length and width of each beam section  106  and inner radii of the curved sections  108  (and/or, more generally the shape of the radii of the curved sections) may determine one or more resonant frequencies of MEMS resonator  102 . The beam sections  106  oscillate or vibrate at the same frequency. TABLE 1 provides a resonant frequency in conjunction with exemplary dimensions of the length and width of each beam section  106  and inner radii of the curved sections  108  of rounded square MEMS resonator  102  which is fabricated from a polycrystalline silicon material. Notably, in these exemplary embodiments, the width of elongated beam sections  106  and curved sections  108  are the same or substantially the same. 
   
     
       
         
             
             
             
             
           
             
                 
               TABLE 1 
             
           
          
             
                 
                 
             
             
                 
                 
                 
               Resonant 
             
             
                 
               Elongated Beam Section 
               Curved Section 
               Frequency 
             
          
         
         
             
             
             
             
             
          
             
                 
               Width (μm) 
               Length (μm) 
               Inner Radius (μm) 
               (MHz) 
             
             
                 
                 
             
          
         
         
             
             
             
             
             
          
             
               Example 1 
               24 
               122.43 
               34.787 
               5.3034 
             
             
                 
             
          
         
       
     
   
   TABLE 2 provides a resonant frequency in conjunction with exemplary dimensions of the length and width of each beam section  106  and inner radii of the curved sections  108  of a rounded square MEMS resonator  102  which is fabricated from a monocrystalline silicon material. Again, in these exemplary embodiments, the width of elongated beam sections  106  and curved sections  108  are the same or substantially the same. 
   
     
       
         
             
             
             
             
           
             
                 
               TABLE 2 
             
           
          
             
                 
                 
             
             
                 
                 
                 
               Resonant 
             
             
                 
               Elongated Beam Section 
               Curved Section 
               Frequency 
             
          
         
         
             
             
             
             
             
          
             
                 
               Width (μm) 
               Length (μm) 
               Inner Radius (μm) 
               (MHz) 
             
             
                 
                 
             
          
         
         
             
             
             
             
             
          
             
               Example 1 
               8 
               209.61 
               7.1944 
               1.1903 
             
             
               Example 2 
               24 
               129.89 
               31.055 
               4.8286 
             
             
                 
             
          
         
       
     
   
   Notably, the dimensions of the MEMS resonators set forth in Tables 1 and 2 are merely exemplary. The dimensions, characteristics and/or parameters of a MEMS resonator according to the present invention may be determined using a variety of techniques including modeling and simulation techniques (for example, a finite element modeling and/or simulation process implemented via a computer driven analysis engine, such as FEMLab (from Consol), ANSYS (ANSYS INC.), IDEAS and/or ABAKUS) and/or empirical data/measurements. For example, a finite element analysis engine, using or based on a set of boundary conditions (for example, the size of the resonator structure), may be employed to design, determine and assess the dimensions, characteristics and/or parameters of (i) elongated beam sections  106 , (ii) curved sections  108 , and (iii) other elements or properties of the resonator structure that are discussed below. Notably, an empirical approach may also be employed (in addition to or in lieu of a finite element analysis (or the like) approach) to design, determine and assess the dimensions, characteristics and/or parameters of(i) elongated beam sections  106 , (ii) curved sections  108 , and (iii) other elements or properties of the resonator structure. 
   The MEMS resonators  102  of MEMS resonator array  100  are mechanically coupled via one or more resonator coupling sections  104 . With reference to  FIGS. 4A-4C , in one embodiment, resonator coupling sections  104  may be substantially straight beams having relatively uniform width. 
   Further, each of resonator coupling section  104  may have the same or substantially the same length and the same or substantially the same shape. For example, with reference to  FIGS. 4B and 4C , resonator coupling section  104  that mechanically couples MEMS resonators  102   a  and  102   b  is substantially identical in shape and dimensions as resonator coupling sections  104  that mechanically couples MEMS resonators  102   b  and  102   c.    
   In another embodiment, resonator coupling sections  104  may be substantially straight beams having different widths and/or lengths. (See, for example,  FIGS. 4D and 4E ). 
   In yet another embodiment, with reference to  FIGS. 4F and 4G , resonator coupling sections  104  includes a design (for example, shape and width) of anchor coupling sections  116  to manage, control, reduce and/or minimize the stress concentration in or at the connection of resonator coupling sections  104  and elongated beams  106 . In this embodiment, resonator coupling sections  104  are filleted to enhance the management of the stresses between resonator coupling section  104  and associated elongated beams  106 . Such a design, however, may tend to increase the loading on elongated beams  106  relative to non-filleted designs. In this regard, by adjusting the shape and width of resonator coupling section  104  in the vicinity of elongated beam  106  (for example by filleting resonator coupling section  104  in the vicinity of elongated beam  106 ), the stress on resonator coupling section  104  and associated elongated beams  106  may be managed, controlled, reduced and/or minimized. In this way, the durability and/or stability of MEMS resonator array  100  may be increased, enhanced and/or optimized while the mode of operation or mode shape remains relatively undisturbed (or any disturbance is acceptable) and thereby the quality of the nodal points (discussed in more detail below), if any, remains relatively undisturbed (or any disturbance is acceptable). In addition thereto, reducing, minimizing and/or limiting the loading on elongated beams  106  may facilitate an adverse impact on the “Q” factor MEMS resonator array  100 . 
   Other designs and/or configurations of resonator coupling section  104  may be employed to, for example, affect the durability and/or stability of MEMS resonator array  100  as well as minimize, reduce or limit any adverse impact on “Q” factor of MEMS resonator array  100 . Indeed, all designs of resonator coupling section  104  whether now known or later developed are intended to fall within the scope of the present invention. For example, with reference to  FIG. 4H and 4I , resonator coupling section  104  may include voids  110 . The voids  110  may of any shape or size and extend partially or entirely through the height/thickness of coupling sections  104 . Implementing voids in one or more of the resonator coupling sections  104  reduces the mass of resonator coupling section  104  which further minimizes, reduces or limits the loading on elongated beam sections  106  and thereby further minimizes, reduces or limits any adverse impact on “Q” factor of MEMS resonator array  100 . Notably, in certain embodiments, resonator coupling sections  104  have small dimensions (for example, the shape, length, width and/or thickness of resonator coupling sections  104 ) to provide a small mass while adding little to no stiffness to elongated beam sections  106  is preferred. 
   With reference to  FIGS. 5A and 5B , MEMS resonator array  100  of the present inventions may employ loading relief mechanisms  112  (for example, springs or spring-like components) within an associated resonator coupling section  104  to manage, control, reduce, eliminate and/or minimize any stress or strain on the associated pair of elongated beams  106  that are mechanically coupled by resonator coupling section  104 . In particular, loading relief mechanism  112  is disposed within resonator coupling section  104  which mechanically couples elongated beam  106   a  of MEMS resonator  102   b  and elongated beam  106   a  of MEMS resonator  102   c.    
   In operation, loading relief mechanisms  112  slightly expand and contract in conjunction with the motion of one, some or all of elongated beam sections  106   a - d  and/or curved sections  108   a - d  in order to reduce, eliminate and/or minimize any stress or strain on an associated the associated elongated beam sections  106   a - d  which are coupled by resonator coupling section  104 . In addition, this coupling technique of MEMS resonator array  100  may further reduce, eliminate and/or minimize loading on the elongated beam sections  106   a - d  thereby decreasing, reducing, minimizing and/or eliminating energy losses of MEMS resonators  102  due to the mechanical coupling to adjacent MEMS resonators. 
   The loading relief mechanisms  112  may be employed in conjunction with any of the mechanical coupling techniques and/or architectures described and/or illustrated herein. For example, loading relief mechanisms  112  may be implemented within, before and/or after one or more of the one or more resonator coupling section  104  of  FIG. 5A and 5B . 
   Notably, loading relief mechanisms  112  may be well known springs or spring-like components, or may be any mechanism that reduces, eliminates and/or minimizes stress and/or strain on coupled elongated beams  106 . 
   As mentioned above, in operation, the motion of the MEMS resonator is such that the MEMS resonator array and/or the individual MEMS resonators include one or more nodal points (i.e., areas or portions of the resonator structure that do not move, experience little movement, and/or are substantially stationary when the MEMS resonators oscillates). It may be advantageous to anchor the MEMS resonator array and/or the individual MEMS resonators to the substrate through or at one or more of the nodal points of one or more of the individual MEMS resonators of the MEMS resonator array. 
   In one embodiment, the nodal points may be located in or near one or more of curved sections of one or more of the MEMS resonators. For example, with reference to  FIG. 6A , in one embodiment, MEMS resonators  102   a  and  102   b  each include nodal points  114  located on or near an outer area, portion or region of curved sections  108 . The anchor coupling section  116   a  is connected at or near nodal point  114  of MEMS resonator  102   a  to secure, fix and/or connect MEMS resonator  102   a  to the substrate via anchor  118 . Similarly, anchor coupling section  116   b  is connected at or near nodal point  114   c  of curved section  108   c  of MEMS resonator  102   b  to secure, fix and/or connect MEMS resonator  102   b  to the substrate via anchor  118 . In this embodiment, MEMS resonator  102   a  and  102   b  are separately connected to a common substrate anchor  118 . 
   The MEMS resonator array  100  may be anchored to the substrate using a variety of anchoring techniques and/or configurations. In this regard, MEMS resonator  102  of MEMS resonator array  100  may be anchored separately to a common and/or individual anchor. For example, with reference to  FIGS. 6C-6H , one or more of MEMS resonators  102   a - d  are anchored to common anchor  118 . In lieu of a common type anchoring structure, one or more of MEMS resonators  102   a - d  may be anchored separately to individual anchors. (See, for example,  FIGS. 7A-7H ). In this embodiment, MEMS resonator array  100  includes one or more individual anchors  118  that are “dedicated” to an associated MEMS resonator  102  of array  100 . 
   Moreover, the anchoring structure of MEMS resonator array  100  may include combinations or permutations of common and individual anchor techniques. (See, for example,  FIGS. 8A and 8B ). For example, with reference to  FIG. 8A , MEMS resonators  102   a  and  102   c  are anchored separately to individual anchors  118   a  and  118   b  and MEMS resonators  102 B and  102   d  are anchored to a common anchor  118   c . All combinations and permutations of the various anchoring techniques are intended to fall within the scope of the present invention. 
   Notably, in those embodiments where MEMS resonator array  100  employ an anchor technique whereby anchor coupling sections  116  extend outward from one or more curved sections  108 , nodal points  114  may be located on or near an outer region or portion of curved sections  108 . (See, for example,  FIGS. 6A-6H ,  7 A- 7 H,  8 A and  8 B). As such, one or more anchor coupling sections  116  may connect MEMS resonators  102  to one or more substrate anchors  118 , which are located “outside” each of the rounded square shape of MEMS resonators  102   a - d . In this anchoring configuration, outer regions or areas of curved sections  108  are nodal points  114  of MEMS resonators  102 . Thus, by anchoring one or more of MEMS resonators  102   a - d  at or near the outer region or portion of curved section  108  (i.e., at or near one or more nodal points  114 ), the vertical and/or horizontal energy losses of MEMS resonator array  100  and/or MEMS resonator  102  are minimized, limited and/or reduced. 
   In lieu of nodal points located on or near an outer area, portion or region of one or more curved sections  108 , one or more MEMS resonators  102  may include nodal points  114  located on or near an inner area, portion or region of one or more curved sections  108 . (See, for example,  FIGS. 9A-9C ). The anchor coupling sections  116  are connected at or near nodal points  114 , respectively, to secure, fix and/or connect one or more of MEMS resonators  102  of MEMS resonator array  100  to the substrate via one or more anchors  118 . In this way, MEMS resonator array  100  is anchored to the substrate via anchoring one or more of MEMS resonators  102  to the substrate. In this embodiment, at least one MEMS resonator  102  of the MEMS resonator array  100  is anchored according to this technique is coupled to an internal “center” anchor  118 . 
   In addition to nodal points located on or near an outer area, portion or region of one or more curved sections  108 , MEMS resonators  102  may include nodal points  114  located on or near an inner area, portion or region of one or more curved sections  108 . (See, for example,  FIGS. 10A and 10B ). The anchor coupling sections  116  are connected at or near nodal points  114  of one or more MEMS resonators  102  to secure, fix and/or connect MEMS resonator array  100  to the substrate. Thus, in this embodiment, MEMS resonator array  100  employs both common anchoring and internal “center” anchoring techniques. 
   Notably, MEMS resonator array  100  may be anchored to the substrate by anchoring one or more—but not all—of MEMS resonators  102  to the substrate. (See, for example,  FIGS. 6G ,  6 H,  7 C-H,  9 C and  10 B). For example, with reference to  FIGS. 6G , MEMS resonators  102   b ,  102   d ,  102   f  and  102   h  are indirectly anchored to substrate anchor  118  via one, some or all of MEMS resonators  102   a ,  102   c ,  102   e  and  102   g , which are directly connected to anchor  118  via anchor coupling sections  116 . Thus, in these embodiments, one or more MEMS resonators  102  are directly anchored to the substrate and one or more MEMS resonators  102  are indirectly anchored to the substrate. The one or more MEMS resonators that are directly anchored to the substrate may be anchored to a “common” type anchor (see, for example,  FIGS. 6G and 6H ) or an “individual” type anchor (see, for example,  FIGS. 7C-7H ,  9 C), or both (see,  FIGS. 8A ,  8 B and  10 B). 
   With reference to  FIGS. 11A ,  11 B,  12 A- 12 C and  13 A- 13 C, MEMS resonator array  100  of the present inventions may employ stress/strain relief mechanisms  120  (for example, springs or spring-like components) to manage, control, reduce, eliminate and/or minimize any stress or strain on the substrate at the location of the anchor  118  which is caused by the motion of one, some or all of points at which MEMS resonator array  100  is anchored through or at the substrate. For example, with reference to  FIGS. 11A and 11B , curved portions  108  of MEMS resonator  102   a  is mechanically coupled to stress/strain relief mechanism  120  via anchor coupling section  116 . 
   With reference to  FIGS. 12A-12C  and  13 A- 13 C, in operation, stress/strain relief mechanisms  120  expand and contract in conjunction with the motion of one, some or all of elongated beam sections  106   a - d  and curved sections  108   a - d  of MEMS resonators  102   a - d  in order to reduce, eliminate and/or minimize any stress or strain on the substrate and/or to compensate for small remaining movements of the anchoring point due to small asymmetries from manufacturing, material properties may change thereby resulting in a non-100% optimized design (even where Finite Element Modeling (also known as Finite Element Analysis, “FEA” or “F E Analysis”) is employed). In this way, the anchoring architecture of MEMS resonator array  100  may be relatively stress-free and/or strain-free which may significantly decrease, reduce, minimize and/or eliminate any anchor energy loss and thereby increase, enhance, maximize the Q (and output signal) of MEMS resonators  102  and anchor stress will have little to no effect on the resonating frequency of MEMS resonators  102 . Notably, stress/strain relief mechanism  120  and anchor coupling section  116 , in addition to decreasing, reducing, minimizing and/or eliminating anchor energy losses, suspend MEMS resonators  102  (including elongated beam sections  106  and curved sections  108 ) of MEMS resonator array  100  above the substrate. 
   The stress/strain relief mechanisms  120  may be employed within one or more of the one or more anchor coupling section  116 . It may be advantageous to implement stress/strain relief mechanisms  120  in those situations where the point at which MEMS resonator array  100  is anchored through or at the substrate is not sufficiently or adequately motionless (i.e., where there is undesirable movement of the curved section  108  or coupling section  116  which may originate from or be caused by one or more MEMS resonators  102  or the substrate) or where additional de-coupling from the substrate is desired. For example, it may also be advantageous to employ stress/strain relief mechanisms  120  to reduce, eliminate and/or minimize communication of energy between one or more MEMS resonators  102  and the substrate (for example, in those situations where there is an impedance mismatch to a curved section  108  or where “noise” originates in the substrate and is communicated to one or more MEMS resonator  102 ). 
   The stress/strain relief mechanisms  120  may be employed in conjunction with any of the anchoring techniques and/or architectures described and/or illustrated herein. For example, stress/strain relief mechanisms  120  may be implemented within one or more of the one or more anchor coupling section  116  of  FIG. 12A-12C  and/or  FIG. 13A-13C . 
   The stress/strain relief mechanisms  120  may be well known springs or spring-like components, or may be any mechanism that reduces, eliminates and/or minimizes: (i) stress and/or strain on the substrate at the location of the anchor which is caused by the motion of one, some or all of points at which one or more MEMS resonators  102  are anchored through or at the substrate, and/or (ii) communication of energy between one or more MEMS resonators  102  and the substrate. 
   Notably, MEMS resonators  102  need not be anchored at every nodal point or area but may be anchored at one or more locations, preferably at one or more nodal locations (areas or locations of the resonator that do not move, experience little movement, and/or are substantially stationary when the resonator oscillates). For example, with reference to  FIGS. 7A-7F , MEMS resonator array  100 , may be anchored at one point, two points and/or three areas or portions of MEMS resonators  102  (preferably, for example, at or near nodal points  106  of one or more MEMS resonators  102 ). In this regard, one or more anchor coupling sections  116  connect(s) elongated beam sections  106  and curved section  108  of MEMS resonator(s)  102  to corresponding anchors  118 . 
   A finite element analysis and simulation engine may also be employed to design, determine and/or define the location(s) of one or more nodal points at which MEMS resonator  102  may be anchored to the substrate with predetermined, minimal and/or reduced energy loss (among other things). In this regard, beam sections  108  of MEMS resonator  102 , when induced during operation, move in an elongating (or breathing-like) manner and a bending manner. As such, the length of elongated beam sections  106  and the radii of curved sections  108  may determine the location of nodal points on or in the resonator structure whereby there is little, no or reduced rotation movement due to the elongating (breathing-like) mode, as well as little, no or reduced radial movement due to the bending-like mode. The finite analysis engine may be employed to design, determine and assess the location of such nodal points in or on MEMS resonator  102  using a given length of elongated beam sections  106 , and the shape and/or the radii of curved sections  108  of MEMS resonator  102 . In this way, areas or portions in or on curved sections  108  of MEMS resonator  102  that exhibit acceptable, predetermined, and/or little or no movement (radial, lateral and/or otherwise) for anchoring MEMS resonator  102  may be rapidly determined and/or identified. 
   Notably, a finite element analysis and simulation engine may also be employed to design, determine, assess and/or define the location(s) of one or more nodal points of MEMS resonators  102  when implemented in MEMS resonator array  100 . In addition, an empirical approach may also be employed (in addition to or in lieu of a finite element analysis and simulation engine (or the like)) to design, determine, assess and/or define the location(s) of one or more nodal points of MEMS resonators  102  when implemented in MEMS resonator array  100 . Indeed, the entire discussion above regarding finite element analysis and simulation engine is pertinent to the design, analysis and response of MEMS resonator array  100  having a plurality of MEMS resonators  102 . For the sake of brevity those discussions will not be repeated. 
   The MEMS resonator array of the present invention employ any anchor structure and technique whether now known or later developed. Indeed, all structures and techniques are intended to fall within the scope of the present invention. For example, the present invention may employ the anchoring structures and techniques described and illustrated in non-provisional patent application entitled “Anchors for Microelectromechanical Systems Having an SOI Substrate, and Method for Fabricating Same”, which was filed on Jul. 25, 2003 and assigned Ser. No. 10/627,237 (hereinafter “Anchors for Microelectromechanical Systems Patent Application”). It is expressly noted that the entire contents of the Anchors for Microelectromechanical Systems Patent Application, including, for example, the features, attributes, alternatives, materials, techniques and advantages of all of the embodiments and/or inventions, are incorporated by reference herein. 
   In those embodiments where one or more of MEMS resonators  102  are anchored to a “center” anchor  118  (see, for example,  9 A- 9 C,  10 A and  10 B), the design (for example, the shape and width) of anchor coupling sections  116  may impact the inner radii of curved sections  108  and thereby (i) the location of nodal points (if any) in or on MEMS resonator  102  as well as (ii) the resonant frequency of MEMS resonator  102 . In addition to impacting the inner radii of curved sections  108 , the design of anchor coupling section  1   16  may also affect the durability and/or stability of MEMS resonator  102 . In this regard, by adjusting the shape and width of the anchor coupling section  116  in the vicinity of curved section  108  (for example by filleting anchor coupling section  116  in the vicinity of curved section  108  as shown in  FIGS. 14 and 15 ), the stress on MEMS resonator  102  may be managed, controlled, reduced and/or minimized. 
   For example, with reference to  FIGS. 14 and 15 , the width of anchor coupling section  116  may be increased (see, for example,  FIG. 15  relative to  FIG. 14 ) to manage, control, reduce and/or minimize the stress concentration in or at nodal points  114 . In this way, the durability and/or stability of MEMS resonator  102  may be increased, enhanced and/or optimized. 
   Other designs and/or configurations of anchor coupling sections  116  may be employed to, for example, affect the durability and/or stability of MEMS resonator  101  as well as impact the inner radii of curved sections  108  and the location of nodal points (if any) and the resonant frequency of MEMS resonator  102 . (See, for example,  FIGS. 16-21 ). Indeed, all designs of anchor coupling sections  116  whether now known or later developed are intended to fall within the scope of the present invention. 
   Notably, the shape and/or width of elongated beam section  106  in the vicinity of curved section  108  may also impact the durability and/or stability of MEMS resonator  102  (and in particular, the stress in curved sections  108  which are employed as anchoring locations) as well as impact the inner radii of curved sections  108  and the location of nodal points (if any) and the resonant frequency of MEMS resonator  102 . In this regard, by widening elongated beam section  106  in the vicinity of curved section  108  and/or filleting elongated beam section  106  in the vicinity of curved section  108 , the stress on the resonator may be reduced and/or minimized. 
   Thus, in one embodiment, by controlling the shape and width of elongated beam sections  106  and/or anchor coupling section  116 , the inner radii of curved sections is defined thereby defining the relationship between the whether and how curved sections  108  move relative to elongated beam sections  106 . In addition to determining the inner radii of curved sections  108  and, as such, the locations of nodal points  114 , the shape of elongating beam sections  106  and/or anchor coupling section  116  in the vicinity of curved section  108  may affect the durability and stability of MEMS resonator  102 . In this regard, by widening elongated beam section  106  in the vicinity of curved section  108  and/or widening (or filleting) the anchor coupling section  116 , the stress on MEMS resonator  102  may be managed, controlled, reduced, minimized and/or optimized. 
   Notably, as mentioned above, the curvature and/or shape of curved sections  108  may be selected and/or designed to include one or more nodal points or areas in or in the vicinity of curved sections  108 . For example, where curved section  108  moves out-of-phase with elongated beam section  106  connected thereto, the radius of a particular curved section  108  may be too small. Conversely, if the radius of a particular curved section  108  is too large, curved section  108  may move in-phase with beam sections  106  that are connected to curved section  108 . In each instance, the particular curved section  108  may or may not include a nodal point that minimizes or reduces energy loss and/or substrate stress. 
   The aforementioned relationship is discussed in detail in “Microelectromechanical Resonator Structure, and Method of Designing, Operating and Using Same”, filed May 19, 2005, and assigned U.S. patent application Ser. No. 11/132,941. The inventions described and illustrated in the aforementioned patent application may be employed to design; implement, and/or fabricate one or more of the MEMS resonators of the MEMS resonator array of the present invention. For the sake of brevity, those discussions will not be repeated. It is expressly noted, however, that the entire contents of the patent application, including, for example, the features, attributes, alternatives, materials, techniques and/or advantages of all of the inventions/embodiments, are incorporated by reference herein. 
   In operation, beam sections  106  of each MEMS resonator  102  of array  100  oscillate or vibrate at the same frequency. In this regard, beam sections  106  oscillate in an elongating (or breathing) motion or mode (for example, like that of a ring oscillator; see ring oscillator  1000  of  FIG. 22A  (expanding motion—ring oscillator  1000 ′) and  FIG. 22B  (contracting motion—ring oscillator  1000 ″)) as well as a bending motion or mode. Focusing on one MEMS resonator  102  of MEMS resonator array  100 , in one embodiment, during operation, beam sections  106   a - d  of rounded square shaped MEMS resonator  102  oscillate between a first deflected state (see,  FIG. 23A ) and a second deflected state (see,  FIG. 23B ). Each deflected state in  FIGS. 23A and 23B  is superimposed over (or illustrated relative to) the stationary state of beam sections  106  and curved sections  108  of MEMS resonator  102 . 
   Notably, when in the first deflected state, in addition to bending, beam sections  106   a - d  elongate by an amount of ΔL 1 . Similarly, in the second deflected state, beam sections  106   a - d  elongate by an amount of ΔL 2  and bend in the opposite direction to that of the first deflected state. The amount of elongation (i.e., ΔL 1  and ΔL 2 ) may or may not be equal. 
   Moreover, with continued reference to  FIGS. 23A and 23B , nodal points  114   a  - d  in or on curved sections  108   a - d  experience little to no movement during operation. That is, as MEMS resonator  102  oscillates between the first deflected state and the second deflected state, the areas or portions of curved sections  108   a - d  which are connect to anchor coupling sections  116  are relatively stationary. The anchors are not illustrated. 
   Notably, each MEMS resonator  102  of MEMS resonator array  100  may oscillate in an inherently or substantially linear mode. As such, the considerations and requirements of the drive and sense circuitry, discussed below, to provide a linear resonator/oscillator may be less stringent and/or complex because there may be no need to very precisely or very accurately control the resonant amplitude of beam sections  106 . In this regard, some resonator structures (for example, resonators having double-clamped beams, such as double-clamped tuning forks) have modes that are non-linear wherein the output frequency is a function of the resonant amplitude. This effect is evident when a beam transitions from a bending mode transitions to a tensile (elongating) mode. A double-clamped beam, in a primary mode, may exhibit this behavior because at smaller amplitudes the “restring” forces are dominated by bending stress and, at larger amplitudes, the resorting force is dominated by tensile stress. Under this situation, to maintain a constant frequency in such a case the resonant amplitude of the beam may need to be carefully regulated, which may be difficult and likely introduces additional complexity. 
   Focusing now on MEMS resonator array  100 , with reference to  FIGS. 24A and 24B , in one embodiment, during operation, beam sections  106   a - d  of each rounded square shaped MEMS resonator  102  oscillate between the first deflected state and the second deflected state—but in an opposite direction relative to beam sections  106   a - d  of an adjacent MEMS resonator  102 . In this regard, opposing beam sections  106  of adjacent MEMS resonators  102  oscillate, in relation to the other, in-phase—but in opposite directions—between the first deflected state and the second deflected state. That is, when beam section  106   b  of MEMS resonator  102   a  is in a first deflected state, beam section  106   d  of MEMS resonator  102   b  (i.e., the beam section opposing beam section  106   b  of MEMS resonator  102   a ) is in a second deflected state. (See,  FIG. 24A ). Similarly, when beam section  106   b  of MEMS resonator  102   a  is in a second deflected state, beam section  106   d  of MEMS resonator  102   b  is in a first deflected state. (See,  FIG. 24B ). In this way, beams sections  106   a - d  of MEMS resonators  102   a - d  of array  100  oscillate or vibrate at the same or substantially the same frequency. Moreover, resonator coupling sections  104  experience relative little to no expansion or contraction as the beams oscillate between the first and second deflected states. 
   Notably, the deflected states in  FIG. 24A and 24B  is superimposed over (or illustrated relative to) the stationary state of beam sections  106  and curved sections  108  of MEMS resonators  102   a - d.    
   The sense and drive electrodes and circuitry may be configured to provide a single-ended output signal or differential output signals. With reference to  FIG. 25 , in one exemplary embodiment of a single-ended output signal configuration, drive electrodes  122  (which are electrically connected to drive circuitry  124 ) are juxtaposed to beam sections  106   a - d  of MEMS resonators  102   b  and  102   d  to induce beam sections  106   a - d  of resonators  102   b  and  102   d  to oscillate or vibrate wherein the oscillation or vibration has one or more resonant frequencies. The sense circuitry  126 , in conjunction with sense electrodes  128  which are also juxtaposed to beam sections  106   a - d  of MEMS resonators  102   a  and  102   c , sense, sample and/or detect a signal having the one or more resonant frequencies. In this regard, sense electrodes  128  are disposed adjacent to beam sections  106  to provide a signal (for example, resulting from a change in capacitance between beam sections  106  and sense electrodes  128  due to the oscillating motion of each MEMS resonator structure) which is representative of the oscillation or vibration to sense circuitry  126 . The sense circuitry  126  receives the signal and, in response thereto, may output a signal, for example, a clock signal having a resonant frequency. Typically the sense signal output is connected to the drive circuit  124  to close the electronic oscillator loop. In this regard, the phase of the drive signal should be appropriate to stimulate/drive the desired mode. 
   Notably, drive circuitry  124  and sense circuitry  126 , as well as drive electrodes  122  and sense electrodes  128 , may be conventional well-known drive and sense circuitry. Indeed, drive circuitry  124  and sense circuitry  126  may be any MEMS sense and drive circuitry whether now known or later developed. 
   In addition, drive electrodes  122  and sense electrodes  128  may be disposed or positioned relative to beam sections  106  in order to detect one or more selected or predetermined harmonics of beam sections  106  of MEMS resonators  102 . Moreover, the number and length of drive electrodes  122  and sense electrodes  128  may be selected in order to optimize, enhance and/or improve the operation of MEMS resonator array  100  and/or MEMS resonators  102 . Indeed, drive electrodes  122  and sense electrodes  128  may be of any type and/or shape whether now known or later developed. 
   Moreover, drive circuitry  124  and/or sense circuitry  126  may be integrated on the same substrate in which MEMS resonator array  100  resides (or is fabricated in). In addition thereto, or in lieu thereof, drive circuitry  124  and/or sense circuitry  126  may be integrated on a substrate that is physically separate from (and electrically interconnected with) the substrate in which MEMS resonator array  100  resides. 
   In another embodiment, MEMS resonator array  100  is configured to provide a differential output signal. In this embodiment, the sense and drive electrodes and circuitry are configured to provide output signals that are (or are substantially) 180 degrees out of phase. In this way, MEMS resonator array  100  provides a differential output signal pair which includes a relatively large signal to noise relationship due to the summing effects of oscillating beam sections  106  (for example, symmetrical oscillating beam sections) of the plurality of MEMS resonators  102 . 
   With reference to  FIG. 26A , in one exemplary embodiment of a differential output signal configuration, drive electrodes  130  and  132  (which are electrically connected to differential drive circuitry  138 ) are juxtaposed to beam sections  106   a - d  of MEMS resonator  102   a  and  102   b  to induce beam sections  106   a - d  of MEMS resonator  102   a  and  102   b  to oscillate or vibrate. In this regard, each MEMS resonator  102  vibrates or resonates, in-plane, to generate output signals that are (or are substantially) 180 degrees out of phase. The sense electrodes  134  and  136  are disposed adjacent to beam sections  106   a - d  of MEMS resonator  102   c  and  102   d  to provide a signal (for example, resulting from a change in capacitance between beam sections  106  and sense electrodes  134  and  136  due to the oscillating motion of the resonator structure) which is representative of the oscillation or vibration to differential sense circuitry  140  which senses, samples and/or detects a signal having the one or more resonant frequencies. The differential sense circuitry  140  receives the signal and, in response thereto, may output a differential signal pair, for example, a differential clock signal having a resonant frequency. 
   The differential drive circuitry  138  and differential sense circuitry  140  may be conventional well-known circuitry. Indeed, differential drive circuitry  138  and differential sense circuitry  140  may be any type of circuitry (whether or not integrated (or fabricated) on the same substrate in which the MEMS resonator structure resides), and all such circuitry, whether now known or later developed, are intended to fall within the scope of the present invention. 
   In addition, drive electrodes  130  and  132 , and sense electrodes  134  and  136 , may be of a conventional, well known type or may be any type and/or shaped electrode whether now known or later developed. Further, the physical electrode mechanisms may include, for example, capacitive, piezoresistive, piezoelectric, inductive, magnetorestrictive and thermal. Indeed, all physical electrode mechanisms whether now known or later developed are intended to fall within the scope of the present invention. 
   In addition, drive electrodes  130 / 132  and sense electrodes  134 / 136  may be disposed or positioned relative to beam sections  106  of MEMS resonators  102  in order to detect one or more selected or predetermined harmonics of beam sections  106 . Moreover, the number and length of drive electrodes  130 / 132  and sense electrodes  134 / 136  may be selected in order to optimize, enhance and/or improve the operation of the MEMS resonator. 
   Notably, differential drive circuitry  138  and differential sense circuitry  140  maybe integrated on the same substrate in which the MEMS resonator structure resides (or is fabricated in). In addition thereto, or in lieu thereof, differential drive circuitry  138  and differential sense circuitry  140  may be integrated on a substrate that is physically separate from (and electrically interconnected with) the substrate in which the MEMS resonator structure resides. 
   In the embodiment of  FIG. 26A , drive electrodes  130 / 132  and sense electrodes  134 / 136 , are symmetrically configured, which in conjunction with the symmetrical structures of MEMS resonators  102 , manage the stress on resonator coupling sections  104 , beam sections  106 , curved sections  108 , anchor coupling sections  116 , anchors  118  and/or the substrate. In this way, resonator coupling sections  104  and/or anchor coupling sections  116  may be a low stress point which may manage, minimize and/or reduce energy loss of one, some or all of MEMS resonator  102  of MEMS resonator array  100 . 
   Notably, the differential and single-ended output signal configurations maybe implemented in MEMS resonator arrays  100  having less than or greater than four MEMS resonators  102 . (See, for example, the differential output signal configuration of  FIG. 26B ). Indeed, all of the features, embodiments and alternatives discussed herein with respect to MEMS resonator array  100  in the context of sensing and driving the array are applicable to arrays of any size (for example, an array having 2, 3, 4, 5, 6, 7 and 8 MEMS resonators  102 ) and/or configuration (for example, arrays comprised of the same or different geometric shapes of MEMS resonators  102  such as rounded squares, rounded hexagons or rounded triangles). For the sake of brevity, those discussions will not be repeated. 
   Further, it should be noted that there are many other configurations and/or architectures of the sense and drive electrodes that cause or induce beam sections  106  to resonate and thereby generate and/or produce output signals that are (or are substantially) 180 degrees out of phase. The MEMS resonator array  100  of the present invention may employ any sense and drive structure, technique, configurations and/or architectures whether now known or later developed. For example, the drive and sense electrodes may be of a conventional type or may be any type and/or shape. (See, for example,  FIGS. 27A and 27B ). The number and design of drive and/or sense electrodes may be selected to provide addition drive signal and/or sense signal. For example, in one embodiment, the number of sense electrodes, and the cross-sectional sense electrode-beam section interface, is increased in order to increase the signal provided to sense circuitry (for example, the differential sense circuitry). (See, for example,  FIG. 28A ). In one embodiment, sense electrodes are disposed on the inner and outer perimeters of one or more of MEMS resonators  102 . (See, for example,  FIG. 28B ). Thus, MEMS resonator array  100  of the present invention may employ any sense and drive electrode structure and configuration whether now known or later developed. (See, for example,  FIGS. 29A-29F ). 
   Moreover, implementing a differential signal configuration may facilitate canceling, limiting, reducing and/or minimizing the effect of capacitive coupling from the drive electrodes to the sense electrodes. In addition, a fully differential signaling configuration may also significantly decrease any sensitivity to electrical and/or mechanical noise coupled from the substrate. Further, implementing MEMS resonator array  100  in a differential signaling configuration may also eliminate, minimize and/or reduce charge flow through the anchor to and from the structure. As such, a voltage drop between the substrate anchor and drive and sense electrodes may be avoided. Notably, this voltage drop could degrade or adversely impact the electric transfer function of the MEMS resonators of the array especially at higher frequencies (for example, frequencies greater than 100 MHz). 
   In one embodiment of the present invention, MEMS resonator array  100  employs temperature management techniques in order to manage and/or control the Q factor of MEMS resonators  102 . In this regard, when beam sections  106  and/or curved sections  108  bend, one side of the section is stretched thereby causing a slight cooling in the area of the stretching, and the other side is compressed, thereby causing a slight heating in the area of the compression. The heat gradient causes diffusion from the “hotter” side to the “cooler” side. The diffusion of heat (“heat flow”) results in a loss of energy, which may impact (for example, reduce) the Q factor of MEMS resonator  102 . This effect is often referred to as Thermoelastic Dissipation (“TED”), which may be a dominate limit of the Q factor of a resonant structure. As such, is may be advantageous to implement temperature management techniques in order to manage, control, limit, minimize and/or reduce TED. 
   In one temperature management embodiment, with reference to  FIGS. 30A and 30B , slots  142  are formed in one or more of beam sections  106   a - d  and curved sections  108   a - d  of MEMS resonator  102 . The slots  142  suppress/reduce heat flow between the sides of beam sections  106   a - d  and the sides of curved sections  108   a - d  as beam sections  106   a - d  and curved sections  108   a - d  stretch and compress during operation. The suppression/reduction of heat transfer within the beam sections  106   a - d  and curved sections  108   a - d  may lead to a higher Q factor for MEMS resonator  102  and MEMS resonator array  100 . It has to be noted that the methods of temperature management by using slots affects the optimization of the zero movement at the anchoring point and has to be considered by the design (for example, FEA). 
   The temperature management techniques may be employed in one or more beam sections  106  or one or more curved sections  108  of one or more MEMS resonators  102  (see, for example,  FIGS. 31 ,  34 ,  38  and  41 ), or both (see, for example,  FIGS. 32 ,  33 ,  35 ,  37  and  42 ). In addition thereto, or in lieu thereof, the temperature management techniques may also be implemented in anchor coupling sections  116 . (See, for example,  FIGS. 36 ,  41  and  42 ). The slots  142  may be any shape including, for example, square, rectangle, circular, elliptical and/or oval. Indeed, slots  142  of any shape, whether geometric or otherwise, may be incorporated into beam sections  106 , curved sections  108  and/or anchoring coupling sections  116 . 
   Notably, slots  142  may also change the stiffness of the beam sections  106 , curved sections  108  and/or anchoring coupling sections  116 . 
   There are many inventions described and illustrated herein. While certain embodiments, features, materials, configurations, attributes and advantages of the inventions have been described and illustrated, it should be understood that many other, as well as different and/or similar embodiments, features, materials, configurations, attributes, structures and advantages of the present inventions that are apparent from the description, illustration and claims. As such, the embodiments, features, materials, configurations, attributes, structures and advantages of the inventions described and illustrated herein are not exhaustive and it should be understood that such other, similar, as well as different, embodiments, features, materials, configurations, attributes, structures and advantages of the present inventions are within the scope of the present invention. 
   Notably, although a significant portion of the description of the present inventions was set forth in the context of a MEMS resonator array including a plurality of rounded square shaped MEMS resonators, a MEMS resonator array according to the present invention may include MEMS resonators of any geometric shaped resonator architecture or structure including a plurality of elongated beam sections that are connected by curved or rounded sections. For example, as mentioned above, in one embodiment, the MEMS resonator array of the present inventions may include three elongated beam sections that are connected together via curved sections to form a rounded triangle shape, as illustrated in  FIG. 3A . In another embodiment, the MEMS resonator array of the present invention may include six beam sections and six curved sections as illustrated in  FIG. 3C . All of the features, embodiments and alternatives discussed herein with respect to a MEMS resonator having a rounded square shape are applicable to MEMS resonators, according to the present invention, which have other shapes. (See, for example,  FIGS. 43A and 43B ). Moreover, all of the features, embodiments and alternatives discussed herein with respect to MEMS resonator array  100  having a plurality of rounded square shaped resonators are applicable to MEMS resonators, according to the present invention, which have other shapes. For the sake of brevity, those discussions will not be repeated. 
   In another embodiment, the MEMS resonator array of the present invention may include a plurality of MEMS resonators  102  having different shapes. For example, with reference to  FIG. 43C , rounded square shaped MEMS resonator  102   a  may be mechanically coupled to rounded triangle shaped MEMS resonator  102   b  ( FIG. 43C ). With reference to  FIG. 43D , in another example, rounded hexagon shaped MEMS resonators  102   a  and  102   c  maybe mechanically coupled to rounded square shaped MEMS resonator  102   b . All of the features, embodiments and alternatives discussed herein with respect to a MEMS resonator array  100  having a plurality of rounded square shaped resonators are applicable to MEMS resonator array including a plurality of MEMS resonators  102  having two or more different shapes. For the sake of brevity, those discussions will not be repeated. 
   Further the MEMS resonator array of the present invention may employ any sense and drive techniques whether now known or later developed. The drive and sense circuitry (whether differential or not) may be integrated on the same substrate in which the MEMS resonators of the array resides (or is fabricated in). In addition thereto, or in lieu thereof, drive and sense circuitry may be integrated on a substrate that is physically separate from (and electrically interconnected with) the substrate in which the MEMS resonators resides. Moreover, the drive and sense electrode may be of a conventional type or may be any type and/or shape whether now known or later developed. 
   Notably, the dimensions, characteristics and/or parameters of the MEMS resonators and MEMS resonator array according to the present inventions may be determined using a variety of techniques including finite element modeling and simulation techniques (for example, a finite element modeling via a computer driven analysis engine such as FemLab (from Consol), ANSYS (from ANSYS INC.), IDEAS and/or ABAKUS and/or empirical data/measurements. For example, a finite element modeling engine, using or based on a set of boundary conditions (for example, the size of the resonator structure), may be employed to design, determine and/or assess the dimensions, characteristics and/or parameters of (i) elongated beam sections  106 , (ii) curved sections  108 , (iii) loading relief mechanisms  112 , (iv) nodal point(s)  114  (if any), (v) anchor coupling sections  116  and/or (vi) stress/strain mechanisms  120 . Indeed, the impact and/or response of MEMS resonator  102 , alone or incorporated into a MEMS resonator array  100 , on or at the anchor and/or substrates may also be observed and/or determined using such a finite element modeling, simulation and analysis engine. 
   As mentioned above, a finite element analysis and simulation engine may also be employed to design and/or determine the location of any nodal points. Such nodal points may provide a suitable location at which MEMS resonator array  100  (and/or one or more of MEMS resonator  102 ) may be anchored to the substrate with predetermined, minimal and/or reduced energy loss (among other things). In this regard, beam sections  106  of MEMS resonator  102 , when induced, move in a breathing-like manner and a bending-like manner. As such, the length of beam sections  106  and the radii of curved sections  108  may determine the location of nodal points of MEMS resonator  102  (when incorporated into the MEMS resonator array  100 ) whereby there is little, no or reduced rotation movement due to the elongating-like (breathing-like) mode, as well as little, no or reduced radial movement due to the bending-like mode. A finite element analysis engine may be employed to design, determine or predict the location of such nodal points based on a given length of beam sections  106  and the radii of curved sections  108  of each MEMS resonator  102  of MEMS resonator array  100 . In this way, locations that exhibit acceptable, predetermined, and/or little or no movement (radial and/or otherwise) for anchoring MEMS resonator array  100  and/or one or more MEMS resonators  102  may be rapidly determined and/or identified. 
   Moreover, an empirical approach may also be employed (in addition to or in lieu of a finite element analysis (or the like) approach) to design, determine, define and/or assess the dimensions, characteristics and/or parameters of (i) elongated beam sections  106 , (ii) curved sections  108 , (iii) loading relief mechanisms  112 , (iv) nodal point(s)  114  (if any), (v) anchor coupling sections  116  and/or (vi) stress/strain mechanisms  120 . Such an empirical approach may be implemented in the context of one or more MEMS resonators  102  and/or MEMS resonator array  100 . 
   As mentioned above, in the context of MEMS resonator array  100 , a finite element analysis  1   5  and simulation engine, using or based on a set of boundary conditions (for example, the size of the resonator structure), may be employed to design, determine and/or assess the dimensions, characteristics and/or parameters of(i) elongated beam sections  106 , (ii) curved sections  108  and/or (iii) nodal point(s)  114  (if any) of the MEMS resonators  102 , and/or (iv) loading relief mechanisms  112 , (v) anchor coupling sections  116  and/or (vi) stress/strain mechanisms  120 . 
   Further, a thermo-mechanical finite element analysis engine may be employed to enhance any temperature considerations of beam sections  106 , curved sections  108  and/or anchoring coupling sections  116  during operation. In this regard, thermo-mechanical finite element analysis engine may model the operation of MEMS resonator array  100  and/or MEMS resonators  102  and thereby determine the size, location, dimensions, and number of slots to implement in one or more beam sections  106 , curved sections  108  and/or anchoring coupling sections  116 . In this way, the characteristics of MEMS resonator array  100  and/or MEMS resonators  102 , having temperature management techniques implemented therein, may be enhanced and/or optimized and the TED loss minimized and/or reduced. 
   Thus, as mentioned above, many of the properties of the structures of the present inventions may be optimized with Finite Element Modeling (FEM), which is also known as “FEA” or “FE Analysis”. 
   The beam sections  106  of MEMS resonators  102  may or may not include identical or substantially identical dimensions/designs (i.e., have the same or substantially the same width, thickness, height, length and/or shape). In addition, curved sections  108  may or may not include identical or substantially identical dimensions/designs (i.e., have the same or substantially the same inner radius, width, thickness, height, length, outer radius and/or shape). As such, MEMS resonators  102  of array  100  may include beam sections  106  and/or curved sections  108  having different dimensions, shapes and/or designs. 
   The MEMS resonator array of the present inventions may be fabricated from well-known materials using well-known techniques. For example, the MEMS resonator array (including its constituent parts) may be fabricated from well-known semiconductors such as silicon, germanium, silicon-germanium or gallium-arsenide. Indeed, the MEMS resonator array may be comprised of, for example, materials in column IV of the periodic table, for example silicon, germanium, carbon; also combinations of these, for example, silicon germanium, or silicon carbide; also of III-V compounds for example, gallium phosphide, aluminum gallium phosphide, or other III-V combinations; also combinations of III, IV, V, or VI materials, for example, silicon nitride, silicon oxide, aluminum carbide, or aluminum oxide; also metallic silicides, germanides, and carbides, for example, nickel silicide, cobalt silicide, tungsten carbide, or platinum germanium silicide; also doped variations including phosphorus, arsenic, antimony, boron, or aluminum doped silicon or germanium, carbon, or combinations like silicon germanium; also these materials with various crystal structures, including single crystalline, polycrystalline, nanocrystalline, or amorphous; also with combinations of crystal structures, for instance with regions of single crystalline and polycrystalline structure (whether doped or undoped). 
   Moreover, the MEMS resonator array according to the present inventions may be formed in or on semiconductor on insulator (SOI) substrate using well-known lithographic, etching, deposition and/or doping techniques. For the sake of brevity, such fabrication techniques are not discussed herein. However, all techniques for forming or fabricating the resonator structure of the present invention, whether now known or later developed, are intended to fall within the scope off the present invention (for example, well-known formation, lithographic, etching and/or deposition techniques using a standard or over-sized (“thick”) wafer (not illustrated) and/or bonding techniques (i.e., bonding two standard wafers together where the lower/bottom wafer includes a sacrificial layer (for example, silicon oxide) disposed thereon and the upper/top wafer is thereafter thinned (ground down or back) and polished to receive the mechanical structures in or on). 
   Notably, the SOI substrate may include a first substrate layer (for example, a semiconductor (such as silicon), glass or sapphire), a first sacrificial/insulation layer (for example, silicon dioxide or silicon nitride) and a first semiconductor layer (for example, silicon, gallium arsenide or germanium) disposed on or above the sacrificial/insulation layer. The mechanical structure maybe formed using well-known lithographic, etching, deposition and/or doping techniques in or on the first semiconductor layer (for example, semiconductors such as silicon, germanium, silicon-germanium or gallium-arsenide). 
   In one embodiment, the SOI substrate may be a SIMOX wafer which is fabricated using well-known techniques. In another embodiment, the SOI substrate may be a conventional SOI wafer having a first semiconductor layer. In this regard, SOI substrate, having a relatively thin first semiconductor layer, may be fabricated using a bulk silicon wafer which is implanted and oxidized by oxygen to thereby form a relatively thin SiO 2  beneath or underneath the single or mono crystalline wafer surface. In this embodiment, the first semiconductor layer (i.e., monocrystalline silicon) is disposed on the first sacrificial/insulation layer (i.e. silicon dioxide) which is disposed on a first substrate layer (i.e., monocrystalline silicon in this example). 
   In those instances where the MEMS resonators of the MEMS resonator array are fabricated in or on polycrystalline silicon or monocrystalline silicon, certain geometric shaped MEMS resonator structures according to the present inventions, for example, the rounded square shaped resonator, may maintain structural and material symmetry with polycrystalline silicon or monocrystalline silicon. In particular, a rounded square shape MEMS resonator according to the present inventions may be inherently more compatible with the cubic structure of monocrystalline silicon. In each lateral orthogonal direction on a standard wafer (e.g.  100 ,  010 , or  110 ), the properties of the monocrystalline silicon may be matched to one or more geometric shaped resonators. In this regard, the crystalline properties of monocrystalline silicon may have the same or suitable symmetry as the one or more geometric shaped resonator structure. 
   The MEMS resonator array  100  of the present invention may be packaged using a variety of techniques and materials, for example, thin film techniques, substrate bonding techniques (for example, bonding semiconductor or glass-like substrates) and prefabricated package (for example, a TO-8 “can”). Indeed, any packaging and/or fabricating techniques may be employed, whether now known or later developed; as such, all such fabrication and/or packaging techniques are intended to fall within the scope of the present invention. For example, the systems, devices and/or techniques described and illustrated in the following non-provisional patent applications may be implemented: 
   (1) “Electromechanical System having a Controlled Atmosphere, and Method of Fabricating Same”, which was filed on Mar. 20, 2003 and assigned Ser. No. 10/392,528; 
   (2) “Microelectromechanical Systems, and Method of Encapsulating and Fabricating Same”, which was filed on Jun. 4, 2003 and assigned Ser. No. 10/454,867; and 
   (3) “Microelectromechanical Systems Having Trench Isolated Contacts, and Methods of Fabricating Same”, which was filed on Jun. 4, 2003 and assigned Ser. No. 10/455,555. 
   The inventions described and illustrated in the aforementioned patent applications may be employed to fabricate MEMS resonators and the array of the present inventions. For the sake of brevity, those discussions will not be repeated. It is expressly noted, however, that the entire contents of the aforementioned patent applications, including, for example, the features, attributes, alternatives, materials, techniques and/or advantages of all of the inventions/embodiments, are incorporated by reference herein. 
   Where MEMS resonator  102  implements a rounded square shape resonator structure that is symmetrically anchored (see, for example,  FIG. 30B ), the center of gravity of the structure remains relatively constant or fixed during operation. Notably, the four beam sections of MEMS resonator  102  implementing a rounded square shape resonator structure may statistically average Gaussian process tolerances which may provide better parameter control. 
   As mentioned above, MEMS resonator array  100  may employ any anchoring technique or anchor structure, whether now known or later developed. In addition, the stress/strain management techniques/structures (for example, stress/strain mechanisms  120 ) may be implemented in conjunction with any of the anchoring technique or anchor structure described and illustrated herein and/or, whether now known or later developed. For example, the substrate anchors and/or stress/strain management techniques/structures may be placed at one, some or all of nodal points and/or anchors of one or more of the MEMS resonators  102 . Other substrate anchoring-stress/strain management techniques may also be suitable. (See, for example,  FIGS. 44-48 ). Indeed, MEMS resonator  102  may be coupled to a substrate anchor (and stress/strain mechanism  120 ) at non-nodal points in a symmetrical or non-symmetrical manner (for example, in or around a “center” of MEMS resonator  102 ). Notably, the anchoring-stress/strain management techniques may be implemented in conjunction with any of the embodiments described and illustrated herein. (See, for example,  FIGS. 49-52 ). 
   Further, the loading relief techniques/structures (for example, loading relief mechanisms  12 ) may also be implemented in conjunction with any of the embodiments described and illustrated herein. (See, for example,  FIGS. 52-55 ). 
   In the claims, the term “straight elongated beam section” means (i) a straight or substantially straight elongated beam, and/or (ii) an elongated beam having a longitudinal axis that is straight or substantially straight regardless of variations in thickness and/or width (if any) of the beam, and/or (iii) a beam that is substantially more straight than curved. 
   Further, in the claims, the term “slots” means openings, voids and/or slots (whether extending partially or entirely through the entire height/thickness of the elongated beam section or curved section), of any shape and/or size. Moreover, in the claims, the term “voids” means openings, voids and/or slots (whether extending partially or entirely through the entire height/thickness of the resonator coupling section), of any shape and/or size. 
   The above embodiments of the present inventions of MEMS resonator array  100  are merely exemplary. They are not intended to be exhaustive or to limit the inventions to the precise forms, techniques, materials and/or configurations disclosed. Many modifications and variations are possible in light of the above teaching. It is to be understood that other embodiments may be utilized and operational changes may be made without departing from the scope of the present invention. As such, the foregoing description of the exemplary embodiments of the invention has been presented for the purposes of illustration and description. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention not be limited solely to this detailed description. 
   In another aspect, the present invention is a plurality of square frame resonators arranged in an N×M MEMS frame array structure (where N and M are integers). Each beam section of the individual square shaped MEMS resonator is shared with an adjacent resonator. In this way, the structure provides different and resonant coupling effects. In addition, the structure provides a relatively low energy loss, high Q-factor and flexibility of driving/sensing electrode placement. 
   With reference to  FIGS. 56A and 56B , N×M MEMS frame array structure  200  (in this exemplary embodiment, where N and M are equal to 2) includes square shaped MEMS resonators  202   a - d . The square shaped MEMS resonators  202   a - d  each includes four beam sections  204 . Two of the beam sections  204  of each square shaped MEMS resonators  202   a - d  are shared with adjacent square shaped MEMS resonators  202   a - d . For example, square shaped MEMS resonator  202   a  shares (i) beam section  204   c  with square shaped MEMS resonator  202   b  and (ii) beam section  204   d  with square shaped MEMS resonator  202   d . Similarly, square shaped MEMS resonator  202   c  shares (i) beam section  204   b  with square shaped MEMS resonator  202   b  and (ii) beam section  204   a  with square shaped MEMS resonator  202   d . In this way, each square shaped MEMS resonators  202   a - d  is mechanically coupled to and/or integrated with the other square shaped MEMS resonators  202   a - d . In operation, square shaped MEMS resonators  202   a - d  vibrate in-plane and at the same frequency because of the mechanic coupling. 
   The MEMS frame array structure may be anchored using a number of techniques, including those described above with respect to MEMS frame array structure. Where the corner sections of one or more square shaped MEMS resonators include nodal points, it may be advantageous to anchor MEMS frame array structure to the substrate at the nodal points. In this regard, with reference to  FIGS. 57A and 57B , in one embodiment, MEMS frame array structure  200  includes anchor coupling sections  206  that mechanically couple MEMS frame array structure  200  to anchors  208 . The anchor coupling sections  206  are connected to MEMS frame array structure  200  at or near nodal points  210 . In this way, the vertical and horizontal energy losses due to anchoring will be minimized, reduced and/or limited, which may result or provide a relatively high Q MEMS structure. 
   Notably, the anchoring technique illustrated in  FIGS. 57A and 577B , further provides the benefit whereby no additional or extra mask is necessary for defining the anchor to the substrate. That is, square shaped MEMS resonators  202   a - d  and the anchoring structure may be fabricated 
   contemporaneously. The MEMS frame array structure  200  need not be anchored at every nodal point or area but may be anchored at one or more locations, preferably at one or more nodal locations (areas or locations of the resonator that do not move, experience little movement, and/or are substantially stationary when the resonator oscillates). For example, with reference to  FIGS. 57A ,  58  and  59 , MEMS frame array structure  200 , may be anchored at one point, two points and/or four areas or portions of MEMS frame array structure  200  (preferably, for example, at or near nodal points  210  of one or more square shaped MEMS resonators  202   a - d ). In this regard, one or more anchor coupling sections  206  connect(s) certain comers formed by beam sections  204  to corresponding anchors  208 . 
   Notably, with reference to  FIGS. 60A and 60B , MEMS frame array structure  200  of the present inventions may employ stress/strain relief mechanisms  212  (for example, springs or spring-like components) to manage, control, reduce, eliminate and/or minimize any stress or strain on the substrate at the location of the anchor  208  which is caused by the motion of one, some or all of points at which MEMS frame array structure  200  is anchored through or at the substrate. For example, the corner of square shaped MEMS resonators  202   a  is mechanically coupled to stress/strain relief mechanism  212  via anchor coupling section  206 . 
   Notably, in addition to or in lieu thereof, MEMS frame array structure  200  may be anchored to the substrate via one or more of the internal corners of square shaped MEMS resonators  202  (see, for example, nodal points  210   c  in  FIGS. 60A and 60B ). In this regard, the anchor may be located under one, some or all of the internal corners of square shaped MEMS resonators  202  since all those corners may be designed to include motionless nodes. Where MEMS frame array structure  200  includes a large number of square shaped MEMS resonators  202 , in order to enhance horizontal plane (in-plane) vibration of MEMS frame array structure  200 , it may be advantageous to employ one or more anchor structures in or at internal corners of square shaped MEMS resonators  202 . 
   In operation, each of square shaped MEMS resonators  202  of MEMS frame array structure  200  vibrate in-plane and at the same frequency. The phase difference of any two adjacent square shaped MEMS resonators  202  is or is approximately 180 degrees. In this regard, with reference to  FIGS. 61 and 62 , in one embodiment, when induced square shaped MEMS resonators  202  vibrate approximately 180 degrees out-of-phase relative to adjacent square shaped MEMS resonators  202 . For example, square shaped MEMS resonators  202   a  vibrates approximately 180 degrees out- 1  of-phase relative to square shaped MEMS resonators  202   b  and  202   e . The vibration modes of square shaped MEMS resonators  202  may be the conventional flexural in place modes. As such, it is not necessary to place any sense or drive electrodes “underneath” or “above” square shaped MEMS resonators  202  in order to drive and sense MEMS frame array structure  200 . 
   Notably, with continued reference to  FIG. 62 , nodal points  210  in or on the corners of square shaped MEMS resonators  202  experience little to no movement during operation. That is, as square shaped MEMS resonators  202  oscillate between the first deflected state and the second deflected state, the areas or portions of the corners, particularly those that are connect to anchor coupling sections  210 , are relatively stationary. 
   With reference to  FIGS. 63 and 64 , in operation, the exterior corner sections of square shaped MEMS resonators  202   a  and  202   c  are relatively motionless and stress-free nodes (i.e., nodal points). As such, in this embodiment, MEMS frame array structure  200  includes anchor coupling sections  206  that mechanically couple MEMS frame array structure  200  to anchors  208  thereby minimizing, reducing and/or limiting the vertical and horizontal energy losses due, for example, motional resistance at an anchoring point. 
   The sense and drive electrodes and circuitry may be configured to provide a single-ended output signal or differential output signals. For example, With reference to  FIG. 65 , in one embodiment, MEMS frame array structure  200  is configured to provide a differential output signal. In this embodiment, the sense and drive electrodes and circuitry are configured to provide output signals that are (or are substantially) 180 degrees out of phase. In this way, MEMS frame array structure  200  provides a differential output signal pair which includes a relatively large signal to noise relationship due to the summing effects of oscillating beam sections  204  (for example, symmetrical oscillating beam sections) of the plurality of square shaped MEMS resonators  202 . 
   The differential drive circuitry  222  and differential sense circuitry  224  may be conventional well-known circuitry. Indeed, differential drive circuitry  222  and differential sense circuitry  224  may be any type of circuitry (whether or not integrated (or fabricated) on the same substrate in which the MEMS frame array structure  200  resides), and all such circuitry, whether now known or later developed, are intended to fall within the scope of the present invention. 
   In addition, drive electrodes  214  and  216 , and sense electrodes  218  and  220 , may be of a conventional, well known type or may be any type and/or shaped electrode whether now known or later developed. Further, the physical electrode mechanisms may include, for example, capacitive, piezoresistive, piezoelectric, inductive, magnetorestrictive and thermal. Indeed, all physical electrode mechanisms whether now known or later developed are intended to fall within the scope of the present invention. 
   The drive electrodes  214 / 216  and sense electrodes  218 / 220  may be disposed or positioned relative to beam sections of square shaped MEMS resonators  202  in order to detect one or more selected or predetermined harmonics of beam sections. Moreover, the number and length of drive electrodes  214 / 216  and sense electrodes  218 / 220  may be selected in order to optimize, enhance and/or improve the operation of the MEMS resonator. Further, drive electrodes  214 / 216  and sense electrodes  218 / 220  may be fabricated without an additional or extra mask(s). That is, square shaped MEMS resonators  202   a - d , drive electrodes  214 / 216  and sense electrodes  218 / 220  maybe fabricated contemporaneously. 
   The differential drive circuitry  222  and differential sense circuitry  224  may be integrated on the same substrate in which MEMS frame array structure  200  resides (or is fabricated in). In addition thereto, or in lieu thereof, differential drive circuitry  222  and differential sense circuitry  224  may be integrated on a substrate that is physically separate from (and electrically interconnected with) the substrate in which the MEMS resonator structure resides. 
   It should be noted that there are many other configurations and/or architectures of the sense and drive electrodes that cause or induce beam  204  to resonate and thereby generate and/or produce output signals that are (or are substantially) 180 degrees out of phase. All such configurations and/or architectures are intended to fall within the scope of the present invention. 
   The MEMS frame array structure of the present inventions may be fabricated from well-known materials using well-known techniques. For example, the MEMS frame array structure (including its constituent parts) may be fabricated from well-known semiconductors such as silicon, germanium, silicon-germanium or gallium-arsenide. Indeed, the MEMS frame array structure may be comprised of, for example, materials in column IV of the periodic table, for example silicon, germanium, carbon; also combinations of these, for example, silicon germanium, or silicon carbide; also of III-V compounds for example, gallium phosphide, aluminum gallium phosphide, or other III-V combinations; also combinations of III, IV, V, or VI materials, for example, silicon nitride, silicon oxide, aluminum carbide, or aluminum oxide; also metallic silicides, germanides, and carbides, for example, nickel silicide, cobalt silicide, tungsten carbide, or platinum germanium silicide; also doped variations including phosphorus, arsenic, antimony, boron, or aluminum doped silicon or germanium, carbon, or combinations like silicon germanium; also these materials with various crystal structures, including single crystalline, polycrystalline, nanocrystalline, or amorphous; also with combinations of crystal structures, for instance with regions of single crystalline and polycrystalline structure (whether doped or undoped). 
   Moreover, the MEMS frame array structure according to the present inventions may be formed in or on semiconductor on insulator (SOI) substrate using well-known lithographic, etching, deposition and/or doping techniques. For the sake of brevity, such fabrication techniques are not discussed herein. However, all techniques for forming or fabricating the resonator structure of the present invention, whether now known or later developed, are intended to fall within the scope of the present invention (for example, well-known formation, lithographic, etching and/or deposition techniques using a standard or over-sized (“thick”) wafer (not illustrated) and/or bonding techniques (i.e., bonding two standard wafers together where the lower/bottom wafer includes a sacrificial layer (for example, silicon oxide) disposed thereon and the upper/top wafer is thereafter thinned (ground down or back) and polished to receive the mechanical structures in or on). 
   Notably, the SOI substrate may include a first substrate layer (for example, a semiconductor (such as silicon), glass or sapphire), a first sacrificial/insulation layer (for example, silicon dioxide or silicon nitride) and a first semiconductor layer (for example, silicon, gallium arsenide or germanium) disposed on or above the sacrificial/insulation layer. The mechanical structure maybe formed using well-known lithographic, etching, deposition and/or doping techniques in or on the first semiconductor layer (for example, semiconductors such as silicon, germanium, silicon-germanium or gallium-arsenide). 
   In one embodiment, the SOI substrate may be a SIMOX wafer which is fabricated using well-known techniques. In another embodiment, the SOI substrate maybe a conventional SOI wafer having a first semiconductor layer. In this regard, SOI substrate, having a relatively thin first semiconductor layer, may be fabricated using a bulk silicon wafer which is implanted and oxidized by oxygen to thereby form a relatively thin SiO 2  beneath or underneath the single or mono crystalline wafer surface. In this embodiment, the first semiconductor layer (i.e., monocrystalline silicon) is disposed on the first sacrificial/insulation layer (i.e. silicon dioxide) which is disposed on a first substrate layer (i.e., monocrystalline silicon in this example). 
   In those instances where the plurality of square shaped MEMS resonators of the MEMS frame array structure are fabricated in or on polycrystalline silicon or monocrystalline silicon, certain geometric shaped MEMS resonator structures according to the present inventions, for example, the rounded square shaped MEMS resonator, may maintain structural and material symmetry with polycrystalline silicon or monocrystalline silicon. In particular, a rounded square shape MEMS resonator according to the present inventions may be inherently more compatible with the cubic structure of monocrystalline silicon. In each lateral orthogonal direction on a standard wafer (e.g.  100 ,  010 , or  110 ), the properties of the monocrystalline silicon may be matched to one or more geometric shaped resonators. In this regard, the crystalline properties of monocrystalline silicon may have the same or suitable symmetry as the one or more geometric shaped resonator structure. 
   The MEMS frame array structure of the present invention maybe packaged using a variety of techniques and materials, for example, thin film techniques, substrate bonding techniques (for example, bonding semiconductor or glass-like substrates) and prefabricated package (for example, a TO-8 “can”). Indeed, any packaging and/or fabricating techniques may be employed, whether now known or later developed; as such, all such fabrication and/or packaging techniques are intended to fall within the scope of the present invention. For example, the systems, devices and/or techniques described and illustrated in the following non-provisional patent applications may be implemented: 
   (1) “Electromechanical System having a Controlled Atmosphere, and Method of Fabricating Same”, which was filed on Mar. 20, 2003 and assigned Ser. No. 10/392,528; 
   (2) “Microelectromechanical Systems, and Method of Encapsulating and Fabricating Same”, which was filed on Jun. 4, 2003 and assigned Ser. No. 10/454,867; and 
   (3) “Microelectromechanical Systems Having Trench Isolated Contacts, and Methods of Fabricating Same”, which was filed on Jun. 4, 2003 and assigned Ser. No. 10/455,555. 
   The inventions described and illustrated in the aforementioned patent applications may be employed to fabricate square shaped MEMS resonators and the MEMS frame array structure of the present inventions. For the sake of brevity, those discussions will not be repeated. It is expressly noted, however, that the entire contents of the aforementioned patent applications, including, for example, the features, attributes, alternatives, materials, techniques and/or advantages of all of the inventions/embodiments, are incorporated by reference herein. 
   Notably, the dimensions, characteristics and/or parameters of the square shaped MEMS resonator and the MEMS frame array structure according to the present inventions may be determined using a variety of techniques including finite element modeling and simulation techniques (for example, a finite element modeling via a computer driven analysis engine such as FemLab (from Consol), ANSYS (from ANSYS INC.), IDEAS and/or ABAKUS and/or empirical data/measurements. For example, a finite element modeling engine, using or based on a set of boundary conditions (for example, the size of the resonator structure), may be employed to design, determine and/or assess the dimensions, characteristics and/or parameters of (i) beam sections  204 , (ii) anchor coupling section  206  (ii) nodal point(s)  210  (if any), and/or (vi) stress/strain mechanisms  212 . Indeed, the impact and/or response of square shaped MEMS resonator  202 , alone or incorporated into MEMS frame array structure  200 , on or at the anchor and/or substrates may also be observed and/or determined using such a finite element modeling, simulation and analysis engine. 
   The above embodiments of the present inventions of MEMS frame array structure  200  are merely exemplary. They are not intended to be exhaustive or to limit the inventions to the precise forms, techniques, materials and/or configurations disclosed. Many modifications and variations are possible in light of the above teaching. It is to be understood that other embodiments maybe utilized and operational changes may be made without departing from the scope of the present invention. As such, the foregoing description of the exemplary embodiments of the invention has been presented for the purposes of illustration and description. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention not be limited solely to this detailed description. 
   The MEMS array structure and MEMS frame array structure of the present invention may be implemented in a wide variety of applications including, for example, timing or clock devices or clock alignment circuitry wherein a resonator or oscillator is employed. Indeed, MEMS array structure and MEMS frame array structure of the present invention may be implemented in any system or device where a clock signal or reference clock is employed, for example, in data, satellite and/or wireless communication systems/networks, mobile phone systems/networks, Bluetooth systems/networks, zig bee systems/networks, watches, real time clocks, set top boxes and systems/networks therefor, computer systems (for example, laptops, PCs and/or handheld devices), televisions and systems/networks therefor, consumer electronics (such as DVD player/recorder, MP3, MP2, DIVX or similar audio/video systems).