Abstract:
A piezoelectric resonator device comprises five layers. A first layer and a fifth layer include one or more metal electrodes. A second layer and a fourth layer comprise a piezoelectric material. A third layer comprises a metal layer. In a first area of the first layer the first layer metal electrodes include a first layer periodic structure along one dimension comprising one of the one or more first layer metal electrodes and a space with no first layer metal electrodes. In a second area of the fifth layer the fifth layer metal electrodes include a fifth layer periodic structure along the one dimension comprising one of the one or more fifth layer metal electrodes and a space with no fifth layer metal electrodes. The first layer periodic structure and the fifth layer periodic structure are aligned so that the one of the one or more fifth layer metal electrodes are centered under the space with no first layer metal electrodes and the one of the one or more first layer metal electrodes are centered over the space with no fifth layer metal electrodes or are aligned so that the one of the one or more fifth layer metal electrodes are centered under the first layer metal electrodes.

Description:
[0001]    This work was supported by NASA Phase I SBIR NNJ07J04C. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    Micro-electromechanical system (MEMS) filters have advantages in being able to reduce the size, weight, and power required when used as part of electronic systems such as radios. However, MEMS-type filters have limitations. For example, thickness MEMS-type filters (e.g., thickness-extensional mode piezoelectric resonators) are typically limited to a single operating frequency per substrate die. For another example, lithographically-determined operating frequency resonators (e.g., contour-extensional polysilicon resonators) cannot meet low impedance (e.g., 50 Ω) specifications. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0003]    Various embodiments of the invention are disclosed in the following detailed description and the accompanying drawings. 
           [0004]      FIG. 1  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. 
           [0005]      FIG. 2  is a diagram illustrating an embodiment of a top view of top metal electrodes of a piezoelectric resonator with two piezoelectric layers. 
           [0006]      FIG. 3  is a diagram illustrating an embodiment of a top view of bottom metal electrodes of a piezoelectric resonator with two piezoelectric layers. 
           [0007]      FIG. 4  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. 
           [0008]      FIGS. 5A and 5B  are diagrams illustrating embodiments of a via connecting to a middle metal layer. 
           [0009]      FIGS. 6A and 6B  are diagrams illustrating embodiments of a via connecting to top and bottom metal electrode layers. 
           [0010]      FIG. 7  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. 
           [0011]      FIG. 8  is a diagram illustrating an embodiment of a top view including top metal electrodes of a piezoelectric resonator with two piezoelectric layers. 
           [0012]      FIG. 9  is a diagram illustrating an embodiment of a top view of bottom metal electrodes of a piezoelectric resonator with two piezoelectric layers. 
           [0013]      FIG. 10  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. 
           [0014]      FIG. 11  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. 
           [0015]      FIG. 12  is a graph illustrating a frequency response of the resonator structure in one embodiment. 
       
    
    
     DETAILED DESCRIPTION 
       [0016]    The invention can be implemented in numerous ways, including as a process; an apparatus; a system; a composition of matter; a computer program product embodied on a computer readable storage medium; and/or a processor, such as a processor configured to execute instructions stored on and/or provided by a memory coupled to the processor. In this specification, these implementations, or any other form that the invention may take, may be referred to as techniques. In general, the order of the steps of disclosed processes may be altered within the scope of the invention. Unless stated otherwise, a component such as a processor or a memory described as being configured to perform a task may be implemented as a general component that is temporarily configured to perform the task at a given time or a specific component that is manufactured to perform the task. As used herein, the term ‘processor’ refers to one or more devices, circuits, and/or processing cores configured to process data, such as computer program instructions. 
         [0017]    A detailed description of one or more embodiments of the invention is provided below along with accompanying figures that illustrate the principles of the invention. The invention is described in connection with such embodiments, but the invention is not limited to any embodiment. The scope of the invention is limited only by the claims and the invention encompasses numerous alternatives, modifications and equivalents. Numerous specific details are set forth in the following description in order to provide a thorough understanding of the invention. These details are provided for the purpose of example and the invention may be practiced according to the claims without some or all of these specific details. For the purpose of clarity, technical material that is known in the technical fields related to the invention has not been described in detail so that the invention is not unnecessarily obscured. 
         [0018]    Micro-electromechanical systems (MEMS) piezoelectric resonators with two piezoelectric layers are disclosed. A piezoelectric resonator device comprises a set of layers suspended using tethers. The set of layers comprises two piezoelectric layers separated by a middle metal layer and metal electrode layers adjacent to the outside of the piezoelectric layers (e.g., metal electrode layer, piezoelectric layer, middle metal layer, piezoelectric layer, and metal electrode layer). The metal electrode layers have patterns of electrodes that are correlated with each other. The metal electrodes on the top and bottom layer and middle metal layer are used to apply, sense, or apply and sense an electric potential across each of the two piezoelectric layers. The piezoelectric effect of the piezoelectric layers transduces the electric potential across each layer into mechanical stress in the layer. The inverse piezoelectric effect of the piezoelectric layers transduces the mechanical stress in each piezoelectric layer into an electric potential across the layer. The resonator structure can be operated at mechanical resonance by varying the applied electric field in time at the natural frequency of the device. In various embodiments, the piezoelectric layer is comprised of one of the following: aluminum nitride, zinc oxide, lead zirconate titanate, quartz, gallium arsenide, lithium niobate, or any other appropriate material. In various embodiments, the two piezoelectric layers are comprised of different materials or are comprised of the same materials. The spacing of the electrodes and the connectivity of the electrodes and the middle metal layer determine a frequency response of the resonator structure. 
         [0019]    In various embodiments, the top metal electrodes, the middle metal layer, and/or the bottom metal electrodes is/are comprised of the following: aluminum, platinum, molybdenum, gold, silver, nickel, ruthenium, or any other appropriate metal. In various embodiments, the top metal electrodes, the middle metal layer, and/or the bottom metal electrodes are comprised of the same metals or are comprised of different metals. 
         [0020]    In some embodiments, the two layers of metal electrodes have patterns that alternate areas such that when viewed from the top or bottom of the set of layers, the electrode areas on the top are centered in a space between electrode areas on the bottom and the electrode areas on the bottom are centered in a space between electrode areas on the top. In various embodiments, the areas comprise approximately rectangular areas in an area of the layer, approximately concentric arcs or portions of circles in an area of the layer, or any other appropriate shape. In various embodiments, the electrodes have approximately the same width as the space between electrode areas, are less wide compared to the space between the electrode areas, are wider compared to the space between the electrode areas, or any other appropriate width. The magnitude of the electromechanical coupling factor of the piezoelectric resonator is a function of the relative width of the electrode compared to the width of the space between electrode areas. 
         [0021]    In some embodiments, the top layer of metal electrodes and the bottom metal electrodes are electrically coupled to each other. 
         [0022]    In some embodiments, each of the layers of metal electrodes (e.g., top layer and bottom layer) include two sets of inter-digitated electrodes, where one set of top electrodes is coupled to one set of bottom electrodes and the other set of top electrodes is coupled to the other set of bottom electrodes. In various embodiments, the coupled sets on top and bottom layers are electrically coupled such that the top and bottom set that sit correspondingly above and below each other are coupled or such that the top and bottom set that site correspondingly above and below each other are not coupled. 
         [0023]    In some embodiments, for a given lithographically-defined metal electrode line width, a two layer piezoelectric resonator device enables twice the maximum operating frequency of that achievable by single layer piezoelectric resonator device. A larger line width for a given frequency of operation is desirable as it: 1) reduces lithographic tolerances (e.g., which is favorable because MEMS fabrication equipment is often several generations behind state-of-the-art for CMOS and because lithography of the MEMS device must accommodate wafer topography(e.g., step heights measuring several microns); 2) decreases Ohmic loading in electrodes (e.g., Ohmic loading is associated with electrode resistance, which destroys the Q of low impedance resonators); 3) increases transduction efficiency by allowing electrodes to cover a larger fraction of a lateral strain field (e.g., which is favorable because it includes more charge associated with the motion of a piezoelectric structure of the resonator structure); and 4) makes the forcing function couple more efficiently into a desired mode of vibration and suppresses undesired modes of plate by having the additional transducer layer (e.g., this is especially effective for 2-port topologies). 
         [0024]    In some embodiments, as compared to a two layer unpatterned electrode structure, a two layer patterned resonator structure has the following advantages: 1) forcing the resonator structure with periodic and/or alternating polarity potentials couples more efficiently into a desired mode of vibration and suppresses undesired modes of the resonator structure; 2) allows the impedance of the resonator structure to be scaled down by increasing number of electrodes; 3) increases the frequency setting accuracy and/or decreases need for trimming, because variations in structure width have 1/n times the effect on the device frequency as compared to a device operating in a fundamental width extensional mode of the resonator structure, where n is the number of half-wavelength periods (e.g., number of electrodes) on the surfaces of the resonator; and 4) the thickness-extensional response, that appears in characteristic response of all lateral-extensional mode devices, is less pronounced. 
         [0025]      FIG. 1  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. In the example shown, resonator structure  100  is suspended over a cavity using tether  102  and tether  104 . The lateral dimensions of resonator structure  100  are typically on the order of a few hundred microns by a few hundred microns for a device designed to operate around 1 GHz and up to a few 1000 microns by a few 1000 microns for a device designed to operate at 10 MHz; each piezoelectric layer is typically 0.5 to 3 microns thick. The tether  104  is defined in the same piezoelectric layers that make up the resonator structure  100 , and are typically 5 microns wide and designed such that their length (in the direction connecting the resonator structure to its surroundings) is an integer number of resonant quarter wavelengths. Resonator structure  100  comprises a set of layers including a top metal electrode layer, a top piezoelectric layer beneath the top metal electrode layer, a middle metal layer beneath the top piezoelectric layer, a bottom piezoelectric layer beneath the middle metal layer, and a bottom metal electrode layer beneath the bottom piezoelectric layer. The middle metal layer is coupled electrically through tether  104  to via  106  and to contact strip  110  and contact strip  114 . The top metal electrode layer and the bottom metal electrode layer are coupled electrically through tether  102  to via  108  and to contact strip  112 . Contact strip  110 /contact strip  114  and contact strip  112  can be used as 1-port connections to resonator structure  100 , where for example, contact strip  110 /contact strip  114  are coupled to ground and contact strip  112  is coupled to a signal input. Resonator structure  100  includes a pattern of metal electrodes on the top and bottom surfaces that when provided an input signal has a vibrational response that coupled to resonator structures electrical response. The vibrational response is a vibrational oscillation mode along an axis perpendicular to the axis between tether  102  and tether  104 . The top and bottom metal electrodes have a periodic structure in along the axis perpendicular to the axis between tether  102  and tether  104  in an area of the surface of resonator structure  100  that includes all but the bus connector strips at either end of resonator structure  100 . The resonant frequency response of resonator structure  100  is controlled by selecting the periodicity of the periodic structure on the surface. The frequency of resonance is proportional to 1/(perdiod of the electrodes) and is related to the speed of elastic wave propagation in the piezoelectric material(s) in the top piezoelectric layer and the bottom piezoelectric layer. For example, an electrode period of 10 microns corresponds to a resonant frequency of approximately 1 GHz if the resonator structure is made of aluminum nitride. At resonance, the elastic wave propagating in the piezoelectric layer has a half-wavelength that is equal to the period of the patterned electrodes. The structure can be driven into resonance by applying a harmonic electric potential that varies in time at the structure resonant frequency across the patterned metal layers. The layout and interconnectivity of the periodic electrodes preferentially transduces the desired mode of vibration while suppressing the response of undesired spurious modes of vibration of the structure. For example, a specific higher order vibrational mode can be transduced without substantially transducing other modes. Compared to its response to a constant DC electric potential, the amplitude of the mechanical response of the resonator is multiplied by the quality factor (the typical quality factor is on the order of 500 to 5000). Selection of the length of resonator structure  100  along the axis between tether  102  and tether  104  and the number of electrode periods provides control over the impedance of resonator structure  100  by scaling the amount of charge generated by the motion of the piezoelectric material. 
         [0026]      FIG. 2  is a diagram illustrating an embodiment of a top view of top metal electrodes of a piezoelectric resonator with two piezoelectric layers. In some embodiments, the metal electrodes of  FIG. 2  are used to implement the metal electrodes associated with the resonator of  FIG. 1 . In the example shown, a resonator structure has electrically coupled bus  200 , periodic stripes  210 , and bus  208  as part of the top layer metal electrode seen from a top view. Bus  200 , periodic stripes  210 , and bus  208  are electrically coupled to via  204  using connector  202 . Connector  202  crosses tether  214 , which is used to suspend the resonator structure. The resonator structure is surrounded by space  206  and is coupled to the structure surrounding the resonator structure using tether  214  and tether  216 . Periodic stripes  210  are periodic along a direction perpendicular to an axis that would run between tether  214  and tether  216 —for example, along axis  212 . Periodic stripes  210  have alternating areas of metal and areas without metal along the direction associated with line  212 . In various embodiments, the areas of metal and the areas without metal have the same width, the areas of metal are wider than the areas without metal, the areas of metal are narrower than the areas without metal, or any other appropriate relation between the widths. The widths of the areas with and without metal electrodes are typically on the order of 5 microns each for resonators designed to operate at 1 GHz; the metal electrode thickness is typically on the order of 100 to 300 nanometers. The magnitude of the electromechanical coupling factor of the piezoelectric resonator is a function of the relative width of the electrode compared to the width of the space between electrode areas. In some embodiments, the resonator electromechanical coupling is a nonlinear function of the electrode/space width ratio. In some embodiments, the width of the metal electrodes is not a width corresponding to the full half period of the electrode spacing (e.g., equal to the full half period of the electrode spacing) because the resonator electromechanical coupling is more efficient when the metal electrode width is less than the full half period of the electrode spacing. In some embodiments, the optimal value of the electrode/space width ratio is approximately 74%. Periodic stripes  210  are arranged in such a manner that the areas without metal are directly over a set of periodic stripes in the bottom metal electrode. 
         [0027]      FIG. 3  is a diagram illustrating an embodiment of a top view of bottom metal electrodes of a piezoelectric resonator with two piezoelectric layers. In some embodiments, the metal electrodes of  FIG. 3  are used to implement the metal electrodes associated with the resonator of  FIG. 1 . In the example shown, a resonator structure has electrically coupled bus  300 , periodic stripes  310 , and bus  308  as part of the bottom layer metal electrode seen from a top view. Bus  300 , periodic stripes  310 , and bus  308  are electrically coupled to via contact  304  using connector  302 . Connector  302  crosses tether  314 , which is used to suspend the resonator structure. The resonator structure is surrounded by space  306  and is coupled to the structure surrounding the resonator structure using tether  314  and tether  316 . Periodic stripes  310  are periodic along a direction perpendicular to an axis that would run between tether  314  and tether  316 —for example, along axis  312 . Periodic stripes  310  have alternating areas of metal and areas without metal along the direction associated with line  312 . In various embodiments, the areas of metal and the areas without metal have the same width, the areas of metal are wider than the areas without metal, the areas of metal are narrower than the areas without metal, or any other appropriate relation between the widths. The widths of the areas with and without metal electrodes are typically on the order of 5 microns each for resonators designed to operate at 1 GHz; the metal electrode thickness is typically on the order of 100 to 300 nanometers. The magnitude of the electromechanical coupling factor of the piezoelectric resonator is a function of the relative width of the electrode compared to the width of the space between electrode areas. Periodic stripes  310  are arranged in such a manner that the areas without metal are directly underneath periodic stripes  210 . 
         [0028]      FIG. 4  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. In some embodiments, the resonator of  FIG. 4  is used to implement the resonator of  FIG. 1 . In the example shown, periodic stripes  420  are coupled electrically to bus  418  which in turn is electrically coupled to connector  414  and via  412 . Connector  414  crosses tether  416 . Tether  416  suspends a resonator structure over cavity  410  within substrate  408 . The resonator structure comprises: 1) a top layer of metal electrodes including periodic stripes  420  and bus  418 ; 2) top piezoelectric layer  404 ; 3) middle metal layer  406 ; 4) bottom piezoelectric layer  402 ; and 5) bottom layer of metal electrodes including periodic stripes  400 . Top layer periodic stripes  420  are centered over spaces between bottom layer periodic stripes  400 . Similarly, spaces between top layer periodic stripes  420  are centered over bottom layer periodic stripes  400 . 
         [0029]      FIGS. 5A and 5B  are diagrams illustrating embodiments of a via connecting to a middle metal layer. In some embodiments, the vias of  FIGS. 5A and 5B  are used to implement the vias associated with the resonator of  FIG. 1 . In the example shown, a resonator structure is coupled to substrate using tether  502 . The resonator structure is separated from the substrate by space  500 . Metal connector  504  is coupled electrically to a middle metal layer in the resonator structure and crosses to the resonator structure on tether  502 . Contact strip  506  is coupled electrically with metal connector  504  using a via. In the cross section shown in  FIG. 5B , metal connector  514  is coupled electrically to contact strip  516 . In some embodiments, metal connector  514  corresponds to metal connector  504  and contact strip  516  corresponds to contact strip  506 . 
         [0030]      FIGS. 6A and 6B  are diagrams illustrating embodiments of a via connecting to top and bottom metal electrode layers. In some embodiments, the vias of  FIGS. 6A and 6B  are used to implement the vias associated with the resonator of  FIG. 1 . In the example shown, a resonator structure is coupled to substrate using tether  602 . The resonator structure is separated from the substrate by space  600 . Metal connector  604  is coupled electrically to a top metal electrode layer in the resonator structure and crosses to the resonator structure on tether  602 . Metal connector  606  is coupled electrically to a bottom metal electrode layer in the resonator structure and crosses to the resonator structure on tether  602 . Contact strip  608  is coupled electrically with metal connector  604  and metal connector  606  using a via. In the cross section shown in  FIG. 6B , metal connector  614  is coupled electrically to metal connector  616  and contact strip  618 . In some embodiments, metal connector  614  corresponds to metal connector  604 , metal connector  616  corresponds to metal connector  606 , and contact strip  618  corresponds to contact strip  608 . 
         [0031]      FIG. 7  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. In the example shown, resonator structure  700  is suspended over a cavity using tether  702  and tether  704 . The lateral dimensions of resonator structure  700  are typically on the order of a few hundred microns by a few hundred microns for a device designed to operate around 1 GHz and up to a few 1000 microns by a few 1000 microns for a device designed to operate at 10 MHz; each piezoelectric layer is typically 0.5 to 3 microns thick. The tether  704  is defined in the same piezoelectric layers that make up the resonator structure  700 , and are typically 5 microns wide and designed such that their length (in the direction connecting the resonator structure to its surroundings) is an integer number of resonant quarter wavelengths. Resonator structure  700  comprises a set of layers including a top metal electrode layer, a top piezoelectric layer beneath the top metal electrode layer, a middle metal layer beneath the top piezoelectric layer, a bottom piezoelectric layer beneath the middle metal layer, and a bottom metal electrode layer beneath the bottom piezoelectric layer. The middle metal layer is coupled electrically through tether  702  and tether  704  to via  716 , via  718 , via  720 , and via  722  and to contact strip  710  and contact strip  714 . A set of electrodes on the top metal layer and a set of electrodes on the bottom metal layer are coupled electrically through tether  702  to via  708  and to contact strip  712 . Another set of electrodes on the top metal layer and another set of electrodes on the bottom metal layer are coupled electrically through tether  704  to via  706  and to contact strip  724 . Contact strip  710 /contact strip  714  and contact strip  712  and contact strip  724  can be used as 2-port connections to resonator structure  700 , where for example, contact strip  710 /contact strip  714  are coupled to ground and contact strip  712  is coupled to a signal input/output and contact strip  724  is coupled to another signal input/output. Resonator structure  700  includes a pattern of metal electrodes on the top and bottom surfaces that when provided an input signal has a vibrational response that coupled to resonator structures electrical response. The vibrational response is a vibrational oscillation mode along an axis perpendicular to the axis between tether  702  and tether  704 . The top and bottom metal electrodes have a periodic structure in along the axis perpendicular to the axis between tether  702  and tether  704  in an area of the surface of resonator structure  700  that includes all but the bus connector strips at either end of resonator structure  700 . The resonant frequency response of resonator structure  700  is controlled by selecting the periodicity of the periodic structure on the surface. The frequency of resonance is proportional to 1/(period of the electrodes) and is related to the speed of elastic wave propagation in the piezoelectric material(s) in the top piezoelectric layer and the bottom piezoelectric layer. For example, an electrode period of 10 microns corresponds to a resonant frequency of approximately 1 GHz if the resonator structure is made of aluminum nitride. At resonance, the elastic wave propagating in the piezoelectric layer has a half-wavelength that is equal to the period of the patterned electrodes. The structure can be driven into resonance by applying a harmonic electric potential that varies in time at the structure resonant frequency across the patterned metal layers. The layout and interconnectivity of the periodic electrodes preferentially transduces the desired mode of vibration while suppressing the response of undesired spurious modes of vibration of the structure. For example, a specific higher order vibrational mode can be transduced without substantially transducing other modes. Compared to its response to a constant DC electric potential, the amplitude of the mechanical response of the resonator is multiplied by the quality factor (the typical quality factor is on the order of 500 to 5000). Selection of the length of resonator structure  700  along the axis between tether  702  and tether  704  and the number of electrode periods provides control over the impedance of resonator structure  700  by scaling the amount of charge generated by the motion of the piezoelectric material. 
         [0032]      FIG. 8  is a diagram illustrating an embodiment of a top view including top metal electrodes of a piezoelectric resonator with two piezoelectric layers. In some embodiments, the metal electrodes of  FIG. 8  are used to implement the metal electrodes associated with the resonator of  FIG. 7 . In the example shown, a resonator structure has electrically coupled bus  800 , periodic stripes  810 , periodic stripes  811 , and bus  808  as part of the top layer metal electrode seen from a top view. Bus  800  and periodic stripes  810  are electrically coupled to via  804  and contact strip  816  using connector  802 . Connector  802  crosses tether  814 , which is used to suspend the resonator structure. The resonator structure is surrounded by space  806  and is coupled to the structure surrounding the resonator structure using tether  814  and tether  816 . Periodic stripes  811  and bus  808  are electrically coupled to via  832  and contact strip  828  using connector  834 . 
         [0033]    Periodic stripes  810  and periodic stripes  811  are periodic along a direction perpendicular to an axis that would run between tether  814  and tether  816 —for example, along axis  812 . Periodic stripes  810  and periodic stripes  811  are inter-digitated and have areas of metal separated by an area without metal along the direction associated with line  812 . In various embodiments, the areas of metal and the areas without metal have the same width, the areas of metal are wider than the areas without metal, the areas of metal are narrower than the areas without metal, or any other appropriate relation between the widths. In various embodiments, periodic stripes  810  and periodic stripes  811  have the same width electrodes, have different width electrodes, or any other appropriate width electrodes. The widths of the areas with and without metal electrodes are typically on the order of 3 microns and 2 microns, respectively, for resonators designed to operate at 1 GHz; the metal electrode thickness is typically on the order of 100 to 300 nanometers. Note that every other metal electrode is coupled electrically together so that adjacent metal electrodes are not coupled electrically together. The magnitude of the electromechanical coupling factor of the piezoelectric resonator is a function of the relative width of the electrode compared to the width of the space between electrode areas. Periodic stripes  810  and periodic stripes  811  are arranged in such a manner that the areas with metal are directly over two sets of periodic stripes in the bottom metal electrode. Periodic stripes  810  are over a set of periodic stripes in the bottom metal electrode, where the set of periodic stripes in the bottom metal electrode are electrically coupled to periodic stripes  811 . Periodic stripes  811  are over a set of periodic stripes in the bottom metal electrode, where the set of periodic stripes in the bottom metal electrode are electrically coupled to periodic stripes  810 . 
         [0034]    Contact strip  826  is coupled electrically to a middle metal layer in the resonator structure using via  822  and via  820 . Contact strip  830  is coupled electrically to a middle metal layer in the resonator structure using via  824  and via  818 . 
         [0035]      FIG. 9  is a diagram illustrating an embodiment of a top view of bottom metal electrodes of a piezoelectric resonator with two piezoelectric layers. In some embodiments, the metal electrodes of  FIG. 9  are used to implement the metal electrodes associated with the resonator of  FIG. 7 . In the example shown, a resonator structure has bus  900  electrically coupled to periodic stripes  911  as part of a bottom layer metal electrode as seen from a top view. Bus  900  and periodic stripes  911  are coupled to connector  902  and via contact  904 . Connector  902  crosses tether  914 , which is used to suspend resonator structure. A resonator structure has bus  920  electrically coupled to periodic stripes  910  as part of a bottom layer metal electrode as seen from a top view. Bus  920  and periodic stripes  910  are coupled to connector  922  and via contact  918 . Connector  922  crosses tether  916 , which is used to suspend resonator structure. 
         [0036]    The resonator structure is surrounded by space  906  and is coupled to the structure surrounding the resonator structure using tether  914  and tether  916 . Periodic stripes  910  and periodic stripes  911  are inter-digitated and have areas with metal next to areas without metal along a direction perpendicular to an axis that would run between tether  914  and tether  916 —for example, along axis  912 . In various embodiments, the areas of metal and the areas without metal have the same width, the areas of metal are wider than the areas without metal, the areas of metal are narrower than the areas without metal, or any other appropriate relation between the widths. The widths of the areas with and without metal electrodes are typically on the order of 3 microns and 2 microns, respectively, for resonators designed to operate at 1 GHz; the metal electrode thickness is typically on the order of 100 to 300 nanometers. The magnitude of the electromechanical coupling factor of the piezoelectric resonator is a function of the relative width of the electrode compared to the width of the space between electrode areas. Periodic stripes  911  are arranged in such a manner that the areas metal are directly underneath periodic stripes  811 . Periodic stripes  910  are arranged in such a manner that the areas metal are directly underneath periodic stripes  810 . 
         [0037]      FIG. 10  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. In some embodiments, the resonator of  FIG. 10  is used to implement the resonator of  FIG. 7 . In the example shown, periodic stripes  1021  are coupled electrically to bus  1018  which in turn is electrically coupled to connector  1014  and via  1012 . Connector  1014  crosses tether  1016 . Tether  1016  suspends a resonator structure over cavity  1010  within substrate  1008 . The resonator structure comprises: 1) a top layer of metal electrodes including periodic stripes  1020 , periodic stripes  1021 , and bus  1018 ; 2) top piezoelectric layer  1004 ; 3) middle metal layer  1006 ; 4) bottom piezoelectric layer  1002 ; and 5) bottom layer of metal electrodes including periodic stripes  1000  and periodic stripes  1001 . Top layer periodic stripes  1020  are centered over periodic stripes  1001 . Top layer periodic stripes  1021  are centered over periodic stripes  1000 . Top layer periodic stripes  1020  are electrically coupled to periodic stripes  1000 . Top layer periodic stripes  1021  are electrically coupled to periodic stripes  1001 . 
         [0038]      FIG. 11  is a block diagram illustrating an embodiment of a piezoelectric resonator with two piezoelectric layers. In some embodiments, the resonator of  FIG. 11  is used to implement the resonator of  FIG. 7 . In the example shown, periodic stripes  1100  are coupled electrically to via  1112 . Tether  1116  suspends a resonator structure over cavity  1110  within substrate  1108 . The resonator structure comprises: 1) a top layer of metal electrodes including periodic stripes  1120 ; 2) top piezoelectric layer  1104 ; 3) middle metal layer  1106 ; 4) bottom piezoelectric layer  1102 ; and 5) bottom layer of metal electrodes including periodic stripes  1100 . Top layer periodic stripes  1120  are centered over periodic stripes  1100 . Top layer periodic stripes  1120  are electrically coupled to each other. Bottom layer periodic stripes  1100  are electrically coupled to each other. Periodic stripes  1120  and periodic stripes  1100  each comprise a set of areas of metal and areas without metal. In some embodiments, the width of the top metal electrodes in the dimension along which the electrodes form a periodic structure is approximately 3 μm with a 2 μm space between the metal electrodes for an AIN piezoelectric resonator structure. 
         [0039]      FIG. 12  is a graph illustrating a frequency response of the resonator structure in one embodiment. In the examples shown, the magnitude of the admittance of a one port resonator structure (e.g., a resonator structure similar to  FIG. 1 ,  FIG. 2 ,  FIG. 3 ,  FIG. 4 ,  FIG. 5 , or  FIG. 6 ) associated with electrode pattern  1200  is plotted for frequencies from 500 MHz to 2500 MHz. The resonator structure has a width of 75 μm, a length of 150 μm, and a thickness of 2 μm (each of the two piezoelectric layers is 1 μm thick); the piezoelectric structure is made out of aluminum nitride. Each metal electrode stripe is 2.5 μm wide and 150 nm thick; the electrodes are made out of aluminum. The resonator structure has a width corresponding to 30 half wavelengths. The fundamental frequency is 58 MHz (e.g., the elastic wave propagation associated frequency 1740/30 MHz). The fundamental width extensional mode of the structure and many of its overtones are suppressed. For example, the response of the overtones at 522 MHz, 580 MHz, 638 MHz, 696 MHz, 754 MHz, 812 MHz, 870 MHz, 928 MHz, 986 MHz, 1044 MHz, 1102 MHz, 1160 MHz, 1218 MHz, 1276 MHz, 1334 MHz, 1392 MHz, 1450 MHz, 1508 MHz, 1566 MHz, 1624 MHz, or 1682 MHz are not seen in  FIG. 12 . The small low Q peak near 2.1 GHz is due to a thickness extensional mode. 
         [0040]    In some embodiments, the resonator structure comprises a circle or annular ring, where the periodic electrodes on the top and bottom layers are portions of arcs or circles. 
         [0041]    In some embodiments, the resonator structure comprises a polygon other than a rectangle, where the periodic electrodes on the top and bottom layers are portions of inscribed polygons of the same type as the resonator structure. 
         [0042]    In some embodiments, the resonator structure further comprises a sixth layer adjacent to the outside of the bottom layer of metal electrodes. The sixth layer, which is made of a low acoustic loss material (e.g., silicon, sapphire, nickel, diamond, silicon dioxide, or silicon carbide), acts as a resonant cavity for the mode of vibration and raises the quality factor of the piezoelectric resonator at the expense of lower electromechanical coupling. The sixth layer has the same lateral dimensions as the piezoelectric layers of the resonator structure. The sixth layer is between the bottom layer (e.g., the fifth layer electrode) and the substrate and is part of the layer stack that is suspended by tethers over the released cavity. The sixth layer cavity is between 1 and 100 μm thick. 
         [0043]    Although the foregoing embodiments have been described in some detail for purposes of clarity of understanding, the invention is not limited to the details provided. There are many alternative ways of implementing the invention. The disclosed embodiments are illustrative and not restrictive.