Patent Publication Number: US-2021184651-A1

Title: Resonator and resonance device

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of PCT/JP2019/038292 filed Sep. 27, 2019, which claims priority to JP Application No. 2018-183874, filed Sep. 28, 2018, the entire contents of each of which are incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to a resonator and a resonance device that includes the resonator. 
     BACKGROUND 
     A resonance device, which is a kind of micro-electro-mechanical systems (MEMS), is currently used as a timing device incorporated in electronic apparatuses, such as smartphones. The resonance device includes, for example, a lower cover, an upper cover, and a resonator placed in a cavity defined between the upper cover and the lower cover. The resonator includes, for example, a piezoelectric film, an upper electrode, a lower electrode, and an insulating film. The upper and lower electrodes are laid with the piezoelectric film therebetween. The insulating film is laid between layers in the resonator or is laid on a surface of the resonator. 
     Patent Document 1 (identified below) discloses a specific configuration of such a resonator. The resonator disclosed therein includes a base section, vibration arms, a holding part, and a pair of holding arms. The vibration arms extend from a tip portion of the base section. The holding part holds the base section. Each holding arm of the pair of holding arms forms a connection between the base section and the holding part. One of the holding arms extends toward a rear frame of the holding part from a rear end portion of the base section. The holding arm is bent toward a left frame of the holding part and is further bent toward a front frame of the holding part. Moreover, the holding part is again bent toward the left frame of the holding part and is connected to the left frame. Similarly, the other holding arm is connected to a right frame of the holding part. 
     Patent Document 1: International Publication No. 2017/208568. 
     A conventional resonator involves application of voltage on vibration arms through traces disposed on the holding arms, which in turn have electric fields. When the vibration arms vibrate in the main or primary mode, the holding arms vibrate in the spurious mode. The spurious mode would couple with the main mode when the frequency of the spurious mode becomes close to an integer multiple of the frequency of the main mode. As a result, the drive level dependency (DLD) characteristics will degrade. 
     The resonator disclosed in Patent Document 1 has a high degree of design flexibility in terms of the length of the holding arms, which is one of frequency-determining parameters for the spurious mode. Therefore, there is a likelihood that the frequency of the spurious mode will become close to an integer multiple of the frequency of the main mode. 
     SUMMARY OF THE INVENTION 
     The exemplary embodiments of the present invention therefore are provided in view of such circumstances. Thus, it is an object of the present invention to provide a resonator having favorable DLD characteristics and a resonance device including the resonator. 
     In one exemplary aspect, a resonator is provided that includes a base, at least one vibration arm, a frame, and a holding arm. The at least one vibration arm includes a piezoelectric film, an upper electrode, and a lower electrode. The upper and lower electrodes are disposed on opposite sides with the piezoelectric film therebetween. The at least one vibration arm has a fixed end connected to a front end portion of the base section and an open end located away from the front end portion. The frame holds the base. The holding arm forms a connection between the base and the frame. The inequality Fs/Fm&lt;1.9 or the inequality 2.1&lt;Fs/Fm holds, where Fm is the frequency of a main mode in the at least one vibration arm, and Fs is the frequency of a spurious mode in the holding arm. 
     Moreover, a resonator is provided according to another exemplary aspect that includes a base, at least one vibration arm, a frame, and a holding arm. The at least one vibration arm includes a piezoelectric film, an upper electrode, and a lower electrode. The upper and lower electrodes are disposed on opposite sides with the piezoelectric film therebetween. The at least one vibration arm has a fixed end connected to a front end of the base and an open end located away from the front end. The frame holds the base. The holding arm forms a connection between the base and the frame. The holding arm includes a holding rear arm and a holding side arm. The holding rear arm is connected to a rear end opposite the front end of the base and extends along the rear end. The holding side arm is connected to the holding rear arm and extends along the at least one vibration arm. The inequality Lc/L 5 &lt;0.520 or the inequality 0.550&lt;Lc/L 5  holds, where Lc is the distance between an edge farther than any other edge of the holding rear arm from the holding side arm and an edge farther than any other edge of the holding side arm from the holding rear arm, and L 5  is the length of an imaginary vibration arm of constant width. The imaginary vibration arm is obtained by transforming the at least one vibration arm into a form whose moment of inertia is equal to the moment of inertia of the at least one vibration arm. 
     The exemplary embodiments of the present invention provide a resonator having favorable DLD characteristics and a resonance device including the resonator. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a schematic external perspective view of a resonance device according to a first exemplary embodiment. 
         FIG. 2  is an exploded perspective view of the resonance device according to the first exemplary embodiment, schematically illustrating the structure of the resonance device. 
         FIG. 3  is a plan view of a resonator according to the first exemplary embodiment, schematically illustrating the structure of the resonator. 
         FIG. 4  is a sectional view of the resonance device taken along a line extending in the X-axis direction, schematically illustrating the multilayer structure of the resonance device in  FIG. 1 . 
         FIG. 5  is a sectional view of the resonance device taken along a line extending in the Y-axis direction, schematically illustrating the multilayer structure of the resonance device in  FIG. 1 . 
         FIG. 6  is a graph illustrating the relationship between the frequency ratio and variations in drive level dependency (DLD). 
         FIG. 7  is a plan view of vibration arms and holding arms, schematically illustrating the dimensions of the vibration arms and the dimensions of the holding arms. 
         FIG. 8  is a graph illustrating the relationship between the length of a holding side arm and the length of a holding rear arm in a first example. 
         FIG. 9  is a graph illustrating the relationship between the length of a holding side arm and the length of a holding rear arm in a second example. 
         FIG. 10  is a plan view of a resonator according to a second exemplary embodiment, schematically illustrating the structure of the resonator. 
         FIG. 11  illustrates vibration arms and holding arms for the sake of making explanation of the dimensions of the vibration arms and the dimensions of the holding arms. 
         FIG. 12  is a graph illustrating the relationship between the frequency and the ratio of the dimension of a holding arm to the dimension of a vibration arm. 
         FIG. 13  is a graph with an approximate curve for the frequency ratio. 
         FIG. 14  is a graph illustrating changes in the frequency ratio that result from changes in the shape of the vibration arm. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Hereinafter, exemplary embodiments will be described with reference to the accompanying drawings. In the accompanying drawings, the same or like reference signs denote the same or like constituent components. The accompanying drawings are provided merely as examples. The individual components are schematically illustrated in terms of their dimensions and shapes. The following embodiments should not be construed as limiting the technical scope of the present invention. 
     First Exemplary Embodiment 
     The following describes the configuration of a resonance device  1  according to a first exemplary embodiment with reference to  FIGS. 1 and 2 .  FIG. 1  is a schematic external perspective view of a resonance device according to the first embodiment.  FIG. 2  is an exploded perspective view of the resonance device according to the first embodiment, schematically illustrating the structure of the resonance device. 
     Resonance Device  1   
     The resonance device  1  includes a resonator  10 , a lower cover  20 , and an upper cover  30 . The lower cover  20  and the upper cover  30  face each other with the resonator  10  therebetween. The lower cover  20 , the resonator  10 , and the upper cover  30  are stacked on top of each other in the stated order in the Z-axis direction. The resonator  10  and the lower cover  20  are joined to each other, and the resonator  10  and the upper cover  30  are joined to each other. A vibration space for the resonator  10  is defined between the lower cover  20  and the upper cover  30  joined to each other through the resonator  10 . In an exemplary aspect, the resonator  10 , the lower cover  20 , and the upper cover  30  are each made of a semiconductor substrate, a glass substrate, an organic substrate, or any other substrate that may be processed with a micromachining technology. 
     The following describes the individual components of the resonance device  1 . According to these exemplary embodiments, the resonance device  1  is understood to be placed with the upper cover  30  on an upper side (i.e., on a top side) and the lower cover  20  on a lower side (i.e., on a back side). 
     Micro-electro-mechanical systems (MEMS) are used to produce the resonator  10 , which is thus regarded as a MEMS resonator in the exemplary aspect. The resonator  10  includes a vibration part  110 , a holding part  140  (or frame), and a holding arm  150 . The vibration part  110  is held in the vibration space. The vibration part  110  may vibrate out of an X-Y plane, that is, in an out-of-plane bending-vibration mode. In some embodiments, the vibration part  110  may vibrate in the X-Y plane, that is, in an in-plane bending-vibration mode. The holding part  140  is in the form of, for example, a rectangular frame, in which the vibration part  110  is enclosed. The holding arm  150  forms a connection between the vibration part  110  and the holding part  140 . 
     The lower cover  20  includes a bottom plate  22  and a side wall  23 . The bottom plate  22  is in the form of a rectangular flat plate lying in an X-Y plane. The side wall  23  extends in the Z-axis direction from a peripheral edge portion of the bottom plate  22 . The side wall  23  is joined to the holding part  140  of the resonator  10 . The lower cover  20  has a recess  21  on its surface facing the vibration part  110  of the resonator  10 . Moreover, the recess  21  is defined by a top surface of the bottom plate  22  and an inner surface of the side wall  23 . The recess  21  is a cavity having a cuboid shape whose top is open. The vibration space for the resonator  10  is partially defined by the recess  21 . The lower cover  20  includes, on its inner surface, a projection  50 , which projects from the top surface of the bottom plate  22  into the vibration space. 
     The structure of the upper cover  30  and the structure of the lower cover  20  except for the projection  50  are mirror images of each other with respect to the resonator  10  disposed therebetween. More specifically, the upper cover  30  includes a bottom plate  32  and a side wall  33 . The bottom plate  32  is in the form of a rectangular flat plate lying in an X-Y plane. The side wall  33  extends in the Z-axis direction from a peripheral edge portion of the bottom plate  32 . The side wall  33  is joined to the holding part  140  of the resonator  10 . In addition, the upper cover  30  has a recess  31  on its surface facing the vibration part  110  of the resonator  10 . The recess  31  is a cavity having a cuboid shape whose bottom is open. The vibration space for the resonator  10  is partially defined by the recess  31 . 
     In alternative embodiments, the structure of the lower cover  20  and the structure of the upper cover  30  may not be mirror images of each other. For example, the lower cover  20  or the upper cover  30  may be dome-shaped. The recess  21  of the lower cover  20  and the recess  31  of the upper cover  30  may have different shapes. For example, the recess  21  and the recess  31  may be of different depths. 
     Resonator  10   
     The following describes, in more detail, the vibration part  110 , the holding part  140 , and the holding arm  150  of the resonator  10  according to an embodiment of the present invention with reference to  FIG. 3 .  FIG. 3  is a plan view of a resonator according to the first embodiment, schematically illustrating the structure of the resonator. 
     Vibration Part  110   
     The vibration part  110  is enclosed in the holding part  140  when viewed in plan from the side on which the upper cover  30  is disposed. The vibration part  110  and the holding part  140  are arranged with a predetermined amount of clearance (or space) left therebetween. The vibration part  110  includes a vibration-generating section  120  and a base section  130  (or simply a base). The vibration-generating section  120  includes four vibration arms, which are denoted by  121 A,  121 B,  121 C, and  121 D, respectively. The base section  130  is connected to the vibration-generating section  120 . In alternative aspects, it is not required that four vibration arms be included in the vibration-generating section  120 . The vibration-generating section  120  may include one vibration arm or any other number of vibration arms. Moreover, the vibration-generating section  120  and the base section  130  in the present embodiment can be provided as one member. 
     Vibration Arms  121 A to  121 D 
     The vibration arms  121 A,  121 B,  121 C, and  121 D extend in the Y-axis direction and are arranged side by side in the stated order in the X-axis direction at predetermined spacings. The vibration arm  121 A has a fixed end connected to a front end  131 A of the base section  130  and an open end located away from the front end  131 A of the base section  130 . The base section  130  will be described later. The vibration arm  121 A includes a mass addition portion  122 A (simply a weight or weighted portino) and an arm portion  123 A. The mass addition portion  122 A is provided to the open end. The arm portion  123 A extends from the fixed end and is connected to the mass addition portion  122 A. Similarly, the vibration arms  121 B,  121 C, and  121 D include their respective mass addition portions, which are denoted by  122 B,  122 C, and  122 D, and also include their respective arm portions, which are denoted by  123 B,  123 C, and  123 D. The arm portions  123 A to  123 D each have a width of about 50 μm in the X-axis direction and a length of about 450 μm in the Y-axis direction according to exemplary aspects. 
     Two of the four vibration arms, or more specifically, the vibration arms  121 A and  121 D can be considered outer vibration arms on the outer side in the X-axis direction. The other two, or more specifically, the vibration arms  121 B and  121 C can be considered inner vibration arms on the inner side in the X-axis direction. The width of a clearance between the arm portion  123 B of the inner vibration arm  121 B and the arm portion  123 C of the inner vibration arm  121 C is referred to as a release width W 1 . The width of a clearance between the arm portion  123 A of the outer vibration arm  121 A and the arm portion  123 B of the inner vibration arm  121 B is referred to as a release width W 2 , with the outer vibration arm  121 A and the inner vibration arm  121 B being adjacent to each other in the X-axis direction. The width of a clearance between the arm portion  123 D of the outer vibration arm  121 D and the arm portion  123 C of the inner vibration arm  121 C is also referred to as the release width W 2 , with the outer vibration arm  121 D and the inner vibration arm  121 C being adjacent to each other in the X-axis direction. The release width W 1  is greater than the release width W 2 . The device adjusted such that the release width W 1  is greater than the release width W 2  offers improved vibration characteristics and improved durability. It is noted that the release width W 1  and W 2  are not limited to particular values. For example, the release width W 1  may be about 25 μm, and the release width W 2  may be about 10 μm.  FIG. 3  is a non-limiting example; that is, the release width W 1  between the arm portions of the respective inner vibration arms may be smaller than or equal to the release width W 2  between each of the inner vibration arms and the corresponding outer vibration arm. 
     The mass addition portions  122 A to  122 D include, on their surfaces, their respective mass addition films, which are denoted by  125 A to  125 D. The weight per unit length (hereinafter also simply referred to as weight) of each of the mass addition portions  122 A to  122 D in the Y-axis direction is greater than the weight of the corresponding one of the arm portions  123 A to  123 D. The vibration part  110  may thus be small in size and have improved vibration characteristics. In addition to providing the additional weight to tips of the vibration arms  121 A to  121 D, the mass addition films  125 A to  125 D enable adjustment of the resonant frequencies of the vibration arms  121 A to  121 D, or more specifically, the mass addition films  125 A to  125 D may each be partially removed for use as frequency adjustment films. 
     In the present embodiment, the width of each of the mass addition portions  122 A to  122 D in the X-axis direction is greater than the width of the corresponding one of the arm portions  123 A to  123 D in the X-axis direction. Moreover, the mass addition portions  122 A to  122 D having greater width have correspondingly greater weight. Although it is required that the weight of each of the mass addition portions  122 A to  122 D be greater than the weight of the corresponding one of the arm portions  123 A to  123 D, the width of each of the mass addition portions  122 A to  122 D in the X-axis direction is not necessarily as described above. In an alternative aspect, the width of each of the mass addition portions  122 A to  122 D in the X-axis direction may be equal to or smaller than the width of the corresponding one of the arm portions  123 A to  123 D in the X-axis direction. 
     When viewed in plan from the side on which the upper cover  30  is disposed, the mass addition portions  122 A to  122 D each have a substantially rectangular shape with four rounded corners (e.g., radius corners). The arm portions  123 A to  123 D are substantially rectangular, with radius corners being formed at the fixed ends connected to the base section  130  and at junctions connected to the mass addition portions  122 A to  122 D. Moreover, the shape of each of the mass addition portions  122 A to  122 D and the shape of each of the arm portions  123 A to  123 D are not necessarily as described above. For example, the mass addition portions  122 A to  122 D may each be substantially trapezoidal or substantially L-shaped. The arm portions  123 A to  123 D may each be substantially trapezoidal. In addition, blind grooves may be provided in the mass addition portions  122 A to  122 D and the arm portions  123 A to  123 D in such a manner that the top or bottom surfaces of the respective portions have openings, and/or holes may be provided in the mass addition portions  122 A to  122 D and the arm portions  123 A to  123 D in such a manner that the top and bottom surfaces of the respective portions have openings. The grooves and holes may be located away from side surfaces each forming a connection between the corresponding top surface and the corresponding bottom surface. Blind grooves and/or holes may be provided in the mass addition portions  122 A to  122 D and the arm portions  123 A to  123 D in such a manner that the side surfaces of the respective portions have openings. 
     When viewed in plan from the side on which the upper cover  30  is disposed, the arm portion  123 B of the inner vibration arm  121 B and the arm portion  123 C of the inner vibration arm  121 C are arranged side by side with the projection  50  therebetween. The projection  50  projects from the lower cover  20 . The projection  50  extends along the arm portions  123 B and  123 C in the Y-axis direction. According to an exemplary embodiment, the projection  50  is about 240 μm long in the Y-axis direction and is about 15 μm long in the X-axis direction. The projection  50  makes the lower cover  20  less prone to warpage. 
     Base Section  130   
     Referring to  FIG. 3 , the base section  130  (or simply the base) viewed in plan from the side on which the upper cover  30  is disposed includes the front end portion  131 A (or simply the front end), a rear end portion  131 B (or simply the rear end), a left end portion  131 C (or simply the left end or left side), and a right end portion  131 D (or simply the right end or right side). The front end portion  131 A, the rear end portion  131 B, the left end portion  131 C, and the right end portion  131 D are each part of a peripheral region of the base section  130 . Specifically, the front end portion  131 A adjoins the vibration arms  121 A to  121 D and extends in the X-axis direction. The rear end portion  131 B is on the side opposite the vibration arms  121 A to  121 D and extends in the X-axis direction. The left end portion  131 C is in line with the vibration arm  121 A in a distance from the vibration arm  121 D and extends in the Y-axis direction. The right end portion  131 D is in line with the vibration arm  121 D in a distance from the vibration arm  121 A and extends in the Y-axial direction. 
     The left end portion  131 C has an end linked to one end of the front end portion  131 A and has another end linked to one end of the rear end portion  131 B. The right end portion  131 D has an end linked the front end portion  131 A and has another end linked to the rear end portion  131 B. The front end portion  131 A and the rear end portion  131 B are on opposite sides in the Y-axis direction. The left end portion  131 C and the right end portion  131 D are on opposite sides in the X-axis direction. The front end portion  131 A is connected with the vibration arms  121 A to  121 D. 
     When viewed in plan from the side on which the upper cover  30  is disposed, the base section  130  has a substantially rectangular shape whose long sides, respectively, are the front end portion  131 A and the rear end portion  131 B and whose short sides, respectively, are the left end portion  131 C and the right end portion  131 D. The base section  130  is substantially symmetric with respect to an imaginary plane P, which lies along perpendicular bisectors that respectively bisect the front end portion  131 A and the rear end portion  131 B. In alternative aspects, the base section  130  is not necessarily rectangular as illustrated in  FIG. 3  and may have any other shape that is substantially symmetric with respect to the imaginary plane P. For example, the base section  130  may have a trapezoidal shape two sides of which, respectively, are the front end portion  131 A and the rear end portion  131 B, with either of these portions being longer than the other. At least one of the front end portion  131 A, the rear end portion  131 B, the left end portion  131 C, and the right end portion  131 D may be bent or curved. 
     The imaginary plane P is a plane of symmetry of the entirety of the vibration part  110 . The imaginary plane P may thus be regarded as a plane passing through the center of the vibration arms  121 A to  121 D in the X-axis direction and is located between the inner vibration arms  121 B and  121 C. Specifically, each of the outer vibration arm  121 A and the inner vibration arm  121 B, which are adjacent to each other, and the corresponding one of the outer vibration arm  121 D and the inner vibration arm  121 C, which are adjacent to each other, are arranged symmetrically about the imaginary plane P. 
     The maximum distance between the front end portion  131 A and the rear end portion  131 B of the base section  130  in the Y-axis direction is herein referred to as a base section length and is, for example, about 40 μm. The maximum distance between the left end portion  131 C and the right end portion  131 D of the base section  130  in the X-axis direction is herein referred to as a base section width and is, for example, about 300 μm. As shown in  FIG. 3 , which illustrates a configuration example, the base section length is the length of the left end portion  131 C or the right end portion  131 D, and the base section width is the length of the front end portion  131 A or the rear end portion  131 B. 
     Frame  140   
     The holding part  140  (or simply the frame) is provided such that the vibration part  110  is held in the vibration space defined by the lower cover  20  and the upper cover  30 . The vibration part  110  may, for example, be enclosed in the holding part  140 . Referring to  FIG. 3 , the holding part  140  viewed in plan from the side on which the upper cover  30  is disposed includes a front frame  141 A, a rear frame  141 B, a left frame  141 C, and a right frame  141 D. The front frame  141 A, the rear frame  141 B, the left frame  141 C, and the right frame  141 D are each part of a substantially rectangular frame body in which the vibration part  110  is enclosed. Specifically, the front frame  141 A is located beyond the vibration arms  121 A to  121 D when viewed from the base section  130  and extends in the X-axis direction. The rear frame  141 B is located beyond the base section  130  when viewed from the vibration arms  121 A to  121 D and extends in the X-axis direction. The left frame  141 C is located beyond the vibration arm  121 A when viewed from the vibration arm  121 D and extends in the Y-axis direction. The right frame  141 D is located beyond the vibration arm  121 D when viewed from the vibration arm  121 A and extends in the Y-axis direction. The holding part  140  is symmetric with respect to the imaginary plane P. 
     The left frame  141 C has an end connected to one end of the front frame  141 A and another end connected to one end of the rear frame  141 B. The right frame  141 D has an end connected to the other end of the front frame  141 A and another end connected to the other end of the rear frame  141 B. The front frame  141 A and the rear frame  141 B are on opposite sides in the Y-axis direction with the vibration part  110  therebetween. The left frame  141 C and the right frame  141 D are on opposite sides in the X-axis direction with the vibration part  110  therebetween. It is noted that the holding part  140  is not necessarily in the form of a frame extending continuously in the circumferential direction. However, the holding part  140  preferably should extend along at least part of the periphery of the vibration part  110 . 
     Holding Arm  150   
     The holding arm  150  is disposed on the inner side with respect to the holding part  140  and forms a connection between the base section  130  and the holding part  140 . Referring to  FIG. 3 , the holding arm  150  viewed in plan from the side on which the upper cover  30  is disposed includes a left holding arm  151 A and a right holding arm  151 B. The left holding arm  151 A forms a connection between the rear end portion  131 B of the base section  130  and the left frame  141 C of the holding part  140 . The right holding arm  151 B forms a connection between the rear end portion  131 B of the base section  130  and the right frame  141 D of the holding part  140 . The left holding arm  151 A includes a holding rear arm  152 A and a holding side arm  153 A, and the right holding arm  151 B includes a holding rear arm  152 B and a holding side arm  153 B. As shown, the holding arm  150  is symmetric with respect to the imaginary plane P. 
     As further shown, the holding rear arms  152 A and  152 B extend from the rear end portion  131 B of the base section  130  and lie between and the rear end portion  131 B of the base section  130  and the holding part  140 . Specifically, the holding rear arm  152 A extends toward the rear frame  141 B from the rear end portion  131 B of the base section  130  and is bent to extend toward the left frame  141 C. Moreover, the holding rear arm  152 B extends toward the rear frame  141 B from the rear end portion  131 B of the base section  130  and is bent to extend toward the right frame  141 D. 
     The holding side arm  153 A extends along the outer vibration arm  121 A and lies between the outer vibration arm  121 A and the holding part  140 . The holding side arm  153 B extends along the outer vibration arm  121 D and lies between the outer vibration arm  121 D and the holding part  140 . Specifically, the holding side arm  153 A extends toward the front frame  141 A from an end portion of the holding rear arm  152 A adjacent to the left frame  141 C and is bent to be connected to the left frame  141 C. The holding side arm  153 B extends toward the front frame  141 A from an end portion of the holding rear arm  152 B adjacent to the right frame  141 D and is bent to be connected to the right frame  141 D. 
     The holding side arms  153 A and  153 B, respectively, are connected to the left frame  141 C and the right frame  141 D in such a manner that the junction between the holding side arm  153 A and the left frame  141 C and the junction between the holding side arm  153 B and the right frame  141 D each face the arm portions  123 A to  123 D in the X-axis direction. In other words, the junction between the holding side arm  153 A and the left frame  141 C and the junction between the holding side arm  153 B and the right frame  141 D are closer than the fixed ends of the vibration arm  121 A to  121 D to the mass addition portions  122 A to  122 D and are closer than the mass addition portions  122 A to  122 D to the fixed ends. The mass addition portions  122 A and  122 D, each of which is wider than the corresponding one of the arm portions  123 A and  123 D, are not aligned with the holding side arms  153 A and  153 B in the X-axis direction. This layout and configuration enables a reduction in the dimension of the resonator  10  in the X-axis direction. 
     It is noted that the configuration of holding arm  150  is not necessarily as described above. For example, the holding arm  150  may be connected to the left end portion  131 C and the right end portion  131 D of the base section  130 . Moreover, the holding arm  150  may be connected to the front frame  141 A of the holding part  140  in another exemplary aspect. 
     Multilayer Structure 
     The following describes the multilayer structure and actions of the resonance device  1  according to the first embodiment with reference to  FIGS. 4 and 5 .  FIG. 4  is a sectional view of the resonance device taken along a line extending in the X-axis direction, schematically illustrating the multilayer structure of the resonance device in  FIG. 1 .  FIG. 5  is a sectional view of the resonance device taken along a line extending in the Y-axis direction, schematically illustrating the multilayer structure of the resonance device in  FIG. 1 . To facilitate understanding of the exemplary embodiment of the multilayer structure of the resonance device  1 ,  FIG. 4  schematically illustrates, for example, the arm portions  123 A to  123 D, an extended line C 2 , an extended line C 3 , a through-via electrode V 2 , and a through-via electrode V 3  that are viewed in section. However, this does not necessarily mean that their cross sections are on the same plane. For example, the through-via electrodes V 2  and V 3  may be parallel to a Z-X plane defined by the Z-axis and the X-axis and may be located away, in the Y-axis direction, from the cross sections of the arm portions  123 A to  123 D. The same holds for  FIG. 5 ; that is, to be of assistance in describing the multilayer structure of the resonance device  1 ,  FIG. 5  schematically illustrates the mass addition portion  122 A, the arm portion  123 A, an extended line C 1 , the extended line C 2 , a through-via electrode V 1 , and the through-via electrode V 2  that are viewed in section; however, this does not necessarily mean that their cross sections are on the same plane. 
     The holding part  140  of the resonator  10  of the resonance device  1  is disposed on and joined to the side wall  23  of the lower cover  20 . The holding part  140  of the resonator  10  is also joined to the side wall  33  of the upper cover  30 . The resonator  10  is held between the lower cover  20  and the upper cover  30 . The lower cover  20 , the upper cover  30 , and the holding part  140  of the resonator  10  define the vibration space in which the vibration part  110  vibrates. The resonator  10 , the lower cover  20 , and the upper cover  30  are each formed by using, for example, a silicon substrate (hereinafter referred to as an Si substrate). In some embodiments, the resonator  10 , the lower cover  20 , and the upper cover  30  may each be formed by using a silicon-on-insulator (SOI) substrate, which is a silicon layer with a silicon oxide film laid thereon. 
     Resonator  10   
     The vibration part  110 , the holding part  140 , and the holding arm  150  of the resonator  10  are integrally formed in the same process according to the exemplary embodiment. The resonator  10  includes an Si substrate F 2  and a metal film E 1 . The Si substrate F 2  is an example of the substrate, and the metal film E 1  is stacked on top of the Si substrate F 2 . The metal film E 1  is overlaid with a piezoelectric film F 3 , and a metal film E 2  is stacked on top of the piezoelectric film F 3 . The metal film E 2  is overlaid with a protective film F 5 . Each of the mass addition portions  122 A to  122 D includes the corresponding one of the mass addition films  125 A to  125 D on the protective film F 5 . The vibration part  110 , the holding part  140 , and the holding arm  150  each have a geometry obtained by patterning the multilayer body including mainly the Si substrate F 2 , the metal film E 1 , the piezoelectric film F 3 , the metal film E 2 , and the protective film F 5 . The multilayer body may be patterned by dry etching in which the multilayer body is exposed to argon (Ar) ion beams for removal processing. 
     The Si substrate F 2  is, for example, a degenerate n-type silicon (Si) semiconductor having a thickness of about 6 μm and doped with n-type dopants such as phosphorus (P), arsenic (As), and antimony (Sb). The electrical resistivity of the degenerate silicon (Si) for use as the Si substrate F 2  may, for example, be less than 16 mΩ·cm and is more preferably not more than 1.2 mΩ·cm. On a lower surface of the Si substrate F 2  is a temperature characteristics correction layer F 21 , which is formed from silicon oxide such as SiO 2 . 
     The temperature characteristics correction layer F 21  enables, at least at or near room temperatures, a reduction in the temperature coefficient of the resonant frequency of the resonator  10 , that is, a reduction in the rate of change in resonant frequency per unit temperature. The temperature characteristics correction layer F 21  included in the vibration part  110  enables the resonator  10  to exhibit improved temperature characteristics. The vibration part  110  may include a temperature characteristics correction layer provided on an upper surface of the Si substrate F 2  or may include temperature characteristics correction layers respectively provided on the upper and lower surfaces of the Si substrate F 2 . 
     The temperature characteristics correction layer F 21  on the mass addition portions  122 A to  122 D desirably has a uniform thickness. It is noted that for purposes of this disclosure, the uniform thickness herein means that variations within a range of ±20% from the thickness mean value of the temperature characteristics correction layer F 21  are tolerated. 
     The metal films E 1  and E 2  each include a vibration-generating electrode and an extended electrode. The vibration-generating electrode causes the vibration arms  121 A to  121 D to vibrate. The extended electrode electrically connects the vibration-generating electrode to an external power source or the ground potential. In the arm portions  123 A to  123 D of the vibration arms  121 A to  121 D, regions being part of the metal film E 1  and functioning as the vibration-generating electrode are opposite to regions being part of the metal film E 2  and functioning as the vibration-generating electrode, with the piezoelectric film F 3  being located between the metal films E 1  and E 2 . Regions functioning as the extended electrodes of the metal films E 1  and E 2  may, for example, extend out from the base section  130  to the holding part  140  through the holding arm  150 . The metal film E 1  is electrically continuous throughout the resonator  10 . Regions of the metal film E 2  that are included in the outer vibration arms  121 A and  121 D are electrically isolated from regions of the metal film E 2  that are included in the inner vibration arms  121 B and  121 C. The metal film E 1  is a lower electrode, and the metal film E 2  is an upper electrode. 
     The thickness of each of the metal films E 1  and E 2  is, for example, not less than about 0.1 μm and not more than about 0.2 μm. After being formed, the metal films E 1  and E 2  undergo removal processing (e.g., etching) and are pattered into mainly the vibration-generating electrodes and the extended electrodes. The metal films E 1  and E 2  are formed from, for example, metallic materials whose crystal structure is a body-centered cubic structure. Specifically, the metal films E 1  and E 2  are each formed from, for example, molybdenum (Mo) or tungsten (W). 
     According to the exemplary aspect, the piezoelectric film F 3  is a thin film formed from a piezoelectric material for converting between electrical energy and mechanical energy. In operation, the piezoelectric film F 3  expands and contracts in the Y-axis direction in an X-Y plane in accordance with the electric field generated in the piezoelectric film F 3  by the metal films E 1  and E 2 . Through the expansion and contraction of the piezoelectric film F 3 , the open ends of the vibration arms  121 A to  121 D undergo displacement toward the bottom plate  22  of the lower cover  20  and displacement toward the bottom plate  32  of the upper cover  30 . This means that the resonator  10  vibrates in the out-of-plane bending-vibration mode. 
     The piezoelectric film F 3  is formed from a material having a wurtzite hexagonal crystal structure. For example, the piezoelectric film F 3  includes, as a principal component, a nitride or an oxide, and more specifically, aluminum nitride (AlN), scandium aluminum nitride (ScAlN), zinc oxide (ZnO), gallium nitride (GaN), or indium nitride (InN). Scandium aluminum nitride is obtained by substituting part of aluminum in aluminum nitride with scandium. Instead of being substituted with scandium, part of aluminum in aluminum nitride may be substituted with magnesium (Mg) and niobium (Nb), with magnesium (Mg) and zirconium (Zr), or with any other two elements, for example. The piezoelectric film F 3  has a thickness of about 1 μm. In some embodiments, the thickness of the piezoelectric film F 3  may be in the range of about 0.2 μm to about 2 μm in alternative aspects. 
     The protective film F 5  protects the metal film E 2  from oxidation. Although it is required that the protective film F 5  be on a surface of the metal film E 2  that is closer than another surface of the metal film E 2  to the upper cover  30 , the protective film F 5  does not necessarily lie open to the bottom plate  32  of the upper cover  30 . The protective film F 5  may be overlaid with, for example, a parasitic capacitance reduction film that reduces the capacitance of wiring of the resonator  10 . The protective film F 5  is a nitride film formed from aluminum nitride (AlN) or silicon nitride (SiN X ) or is an oxide film formed from aluminum oxide (Al 2 O 3 ), tantalum pentoxide (Ta 2 O 5 ), or silicon oxide (SiO X ). 
     The protective film F 5  on the mass addition portions  122 A to  122 D desirably has a uniform thickness. For purposes of this disclosure, the uniform thickness herein means that variations within a range of ±20% from the thickness mean value of the protective film F 5  are tolerated. 
     Each of the mass addition films  125 A to  125 D is a surface of the corresponding one of the mass addition portions  122 A to  122 D and faces the upper cover  30 . Moreover, each of the mass addition films  125 A to  125 D is a frequency adjustment film for the corresponding one of the vibration arms  121 A to  121 D. The mass addition films  125 A to  125 D are partially removed through trimming processing for adjustment of the frequency of the resonator  10 . With a view to enhancing the efficiency of frequency adjustment, it is desired that the mass addition films  125 A to  125 D be formed from a material whose mass reduction rate at the time of etching is faster than the mass reduction rate of the protective film F 5 . The mass reduction rate is obtained by multiplying the etching rate by the density. The etching rate refers to the thickness removed per unit time. Although the relationship between the mass reduction rate of the protective film F 5  and the mass reduction rate of the mass addition films  125 A to  125 D should be as noted above, the magnitude relationship between the etching rate of the protective film F 5  and the etching rate of the mass addition films  125 A to  125 D may be adjusted as desired. With a view to increasing the weight of the mass addition portions  122 A to  122 D efficiently, it is preferred that the mass addition films  125 A to  125 D be formed from a material of high specific gravity. For these reasons, the mass addition films  125 A to  125 D are formed from a metallic material such as molybdenum (Mo), tungsten (W), gold (Au), platinum (Pt), nickel (Ni), or titanium (Ti). 
     The process of adjusting the frequency involves trimming processing in which upper surfaces of the mass addition films  125 A to  125 D are partially removed. The shape of the mass addition films  125 A to  125 D will be specifically described later. As the trimming processing, dry etching involving radiation of argon (Ar) ion beams may be applied to the mass addition films  125 A to  125 D. A wide area may be radiated with ion beams, which thus provide a high degree of processing efficiency. Meanwhile, the mass addition films  125 A to  125 D would be electrically charged by ion beams bearing electrical charges. The vibration orbit of the vibration arms  121 A to  121 D would be changed due to the coulomb interaction in the electrically charged mass addition films  125 A to  125 D, and the vibration characteristics of the resonator  10  would degrade accordingly. To avoid such a defective condition, the mass addition films  125 A to  125 D are desirably grounded in the exemplary aspect. 
     The extended lines C 1 , C 2 , and C 3  are provided on the protective film F 5  on the holding part  140 . As shown, the extended line C 1  is electrically connected to the metal film E 1  through a through-hole extending through both the piezoelectric film F 3  and the protective film F 5 . The extended line C 2  is electrically connected, through a through-hole in the protective film F 5 , to portions of the metal film E 2  that are included in the outer vibration arms  121 A and  121 D. The extended line C 3  is electrically connected, through a through-hole in the protective film F 5 , to portions of the metal film E 2  that are included in the inner vibration arms  121 B and  121 C. The extended lines C 1  to C 3  are each formed from a metallic material such as aluminum (Al), germanium (Ge), gold (Au), or tin (Sn). 
     Lower Cover  20   
     According to the exemplary embodiment, the bottom plate  22  and the side wall  23  of the lower cover  20  are integrally formed of an Si substrate P 10 . The Si substrate P 10  is formed from nondegenerate silicon and has an electrical resistivity of, for example, 10 Ω·cm or more.  The Si substrate P 10  lies open to the inside of the recess  21  the lower cover  20 . The projection  50  has an upper surface covered with the temperature characteristics correction layer F 21 . With a view to inhibiting electrification of the projection  50 , the Si substrate P 10 , whose electrical resistivity is lower than the electrical resistivity of the temperature characteristics correction layer F 21 , may be exposed at the upper surface of the projection  50 , or an electrically conductive layer may be provided on the upper surface of the projection  50 . 
     The thickness of the lower cover  20  in the Z-axis direction is about 150 μm, for example. Moreover, the depth of the recess  21  in the Z-axis direction is denoted by D 1  and is about 100 μm. The amplitude of the vibration arms  121 A to  121 D is limited by the depth D 1 ; that is, their maximum amplitude on the inner side of the lower cover  20  is about 100 μm. 
     The lower cover  20  may be regarded as part of the SOI substrate. With the resonator  10  and the lower cover  20  being regarded as a MEMS substrate integrally formed of the SOI substrate, the Si substrate P 10  of the lower cover  20  is a support substrate of the SOI substrate, the temperature characteristics correction layer F 21  of the resonator  10  is a buried oxide (BOX) layer, and the Si substrate F 2  of the resonator  10  is an active layer of the SOI substrate. Part of the continuous MEMS substrate may be used to form various types of semiconductor elements and circuits on the outer portion of the resonance device  1 . 
     Upper Cover  30   
     The bottom plate  32  and the side wall  33  of the upper cover  30  are integrally formed of an Si substrate Q 10  in the exemplary embodiment. It is preferred that a top surface and a back surface of the upper cover  30  and inner surfaces of the through-holes be covered with a silicon oxide film Q 11 . The silicon oxide film Q 11  is formed on a top surface of the Si substrate Q 10  by, for example, oxidation of the Si substrate Q 10  or chemical vapor deposition (CVD). The Si substrate Q 10  lies open to the inside of the recess  31  of the upper cover  30 . In addition, a getter layer may be provided on a surface of the recess  31  of the upper cover  30  in a manner so as to face the resonator  10 . The getter layer is formed from, for example, titanium (Ti) or any other material with a high affinity with hydrogen and oxygen. The getter layer adsorbs gas removed from the Si substrate P 10 , the Si substrate Q 10 , and a joint part H. The decrease in the degree of vacuum in the vibration space may be minimized accordingly. Alternatively, the getter layer may be provided on a surface of the recess  21  of the lower cover  20  in a manner so as to face the resonator  10 . In some embodiments, two getter layers may respectively be formed on the surface of the recess  21  of the lower cover  20  and the surface of the recess  31  of the upper cover  30  in a manner so as to face the resonator  10 . 
     The thickness of the upper cover  30  in the Z-axis direction is about 150 μm, for example. The depth of the recess  31  in the Z-axis direction is denoted by D 2  and is about 100 μm. The amplitude of the vibration arms  121 A to  121 D is limited by the depth D 2 ; that is, their maximum amplitude on the inner side of the upper cover  30  is about 100 μm. 
     As further shown, terminals are provided on the upper surface of the upper cover  30  (i.e., the surface opposite the surface facing the resonator  10 ) and are denoted by T 1 , T 2 , and T 3 , respectively. The terminal T 1  is a mounting terminal that forms a connection between the metal film E 1  and the ground. The terminal T 2  is a mounting terminal that forms an electrical connection between the external power source and regions of the metal film E 2  that are included in the outer vibration arms  121 A and  121 D. The terminal T 3  is a mounting terminal that forms an electrical connection between the external power source and regions of the metal film E 2  that are included in the inner vibration arms  121 B and  121 C. The terminals T 1  to T 3  are formed of a metallized (base) layer of, for example, chromium (Cr), tungsten (W), or nickel (Ni) and are plated with, for example, nickel (Ni), gold (Au), silver (Ag), or copper (Cu). With a view to achieving balanced parasitic capacitance and balanced mechanical strength, a dummy terminal electrically isolated from the resonator  10  may be provided on the top surface of the upper cover  30 . 
     The through-via electrodes V 1 , V 2 , and V 3  are provided in the side wall  33  of the upper cover  30 . The through-via electrode V 1  forms an electrical connection between the terminal T 1  and the extended line C 1 . The through-via electrode V 2  forms an electrical connection between the terminal T 2  and the extended line C 2 . The through-via electrode V 3  forms an electrical connection between the terminal T 3  and the extended line C 3 . The through-via electrodes V 1  to V 3  are through-holes extending in the Z-axis direction through the side wall  33  of the upper cover  30  and filled with an electrically conductive material. The electrically conductive material is, for example, polycrystalline silicon (Poly-Si), copper (Cu), or gold (Au). 
     Yet further, the joint part H (or simply joint) is provided between the side wall  33  of the upper cover  30  and the holding part  140  such that the side wall  33  of the upper cover  30  is joined to the holding part  140  of the resonator  10 . The joint part H is in the form of a closed loop surrounding the vibration part  110  in the X-Y plane such that the vibration space for the resonator  10  is sealed airtight and maintained under vacuum. The joint part H is formed of a metal film obtained by eutectic bonding of, for example, an aluminum (Al) film, a germanium (Ge) film, and an aluminum (Al) film that are stacked on top of each other in the stated order. Alternatively, the joint part H may be formed of a combination of films selected, as appropriate, mainly from a gold (Au) film, a tin (Sn) film, a copper film (Cu), a titanium (Ti) film, and a silicon (Si) film. The joint part H may contain metallic compounds such as titanium nitride (TiN) and tantalum nitride (TaN) to offer enhanced adhesion. 
     Operations 
     In operation of the present embodiment, the terminal T 1  is grounded, and alternating voltages opposite in phase are applied to the terminals T 2  and T 3 , respectively. The electric field generated in the piezoelectric film F 3  in the outer vibration arms  121 A and  121 D and the electric field generated in the piezoelectric film F 3  in inner vibration arms  121 B and  121 C are thus opposite in phase. Consequently, the vibration of the outer vibration arms  121 A and  121 D and the vibration of the inner vibration arms  121 B and  121 C are opposite in phase. When, for example, the mass addition portion  122 A of the outer vibration arm  121 A and the mass addition portion  122 D of the outer vibration arm  121 D undergo displacement toward an inner surface of the upper cover  30 , the mass addition portion  122 B of the inner vibration arm  121 B and the mass addition portion  122 C of the vibration arm  121 C undergo displacement toward the inner surface of the lower cover  20 . That is, the vibration arm  121 A and the vibration arm  121 B, which are adjacent to each other, vibrate vertically in mutually opposite directions about a central axis r 1  extending in the Y-axis direction between the vibration arm  121 A and the vibration arm  121 B. Similarly, the vibration arm  121 C and the vibration arm  121 D, which are adjacent to each other, vibrate vertically in mutually opposite directions about a central axis r 2  extending in the Y-axis direction between the vibration arm  121 C and the vibration arm  121 D. With the direction of twisting moment on the central axis r 1  and the direction of twisting moment on the central axis r 2  being opposite to each other, the base section  130  is bent and vibrates. The vibration arms  121 A to  121 D vibrate in a range of about 100 μm at the maximum and vibrate in a range of about 10 μm under normal driving conditions. 
     When the resonance device  1  is in operation, the vibration arms  121 A to  121 D vibrate in the main or primary mode, whereas the left holding arm  151 A and the right holding arm  151 B vibrate in the spurious mode. The reason is that the left holding arm  151 A and the right holding arm  151 B are each provided with extended electrodes being part of the metal films E 1  and E 2 , and electric fields are thus formed in the left holding arm  151 A and the right holding arm  151 B when the resonance device  1  is in action. For example, the piezoelectric film F 3  in the holding arm  150  extends and contracts in accordance with the electric fields applied to the extended electrodes such that the holding arm  150  vibrates in the spurious mode. Fm denotes the frequency of the main mode in the vibration arms  121 A to  121 D, and Fs denotes the frequency of the spurious mode in the holding arm  150 . Thus, the holding arm  150  can be configured to vibrate in the spurious mode without application of electric fields on the piezoelectric film in the holding arm  150 . For example, vibration of the vibration part  110  can propagate to the holding arm  150 , which in turn vibrates in the spurious mode. 
     Frequency Ratio 
     The following describes, with reference to  FIG. 6 , the relationship between the ratio of the spurious-mode frequency to the main-mode frequency and the drive level dependency (DLD) characteristics.  FIG. 6  is a graph illustrating the relationship between the frequency ratio and variations in DLD. Referring to  FIG. 6 , values (Fs/Fm) obtained by dividing the spurious-mode frequency by the main-mode frequency are plotted along the horizontal axis of the graph, and values (DLD Slope3σ) reflecting variations in DLD are plotted along the vertical axis of the graph. 
     As the value of Fs/Fm becomes closer to 2, the DLD Slope3σ, the DLD slope3σ significantly increases and takes a turn for the worse. The DLD Slope3σ within a range where 1.8≤Fs/Fm≤2.2 is greater than the DLD Slope3σ within a range where Fs/Fm&lt;1.8 and is greater than the DLD Slope3σ within a range where 2.2&lt;Fs/Fm. As shown, the DLD Slope 3σ within a range where 1.9≤Fs/Fm≤2.1 is much greater. Referring to  FIG. 6 , the DLD Slope3σ within the range where 1.8≤Fs/Fm≤2.2 is greater than 10 ppm/0.2 μW. The DLD Slope3σ within the range where 1.9 ≤Fs/Fm≤2.1 is greater than 20 ppm/0.2 μW. Fs and Fm desirably satisfy the inequality Fs/Fm&lt;1.9 or the inequality 2.1&lt;Fs/Fm and more desirably satisfy the inequality Fs/Fm&lt;1.8 or the inequality 2.2&lt;Fs/Fm. Where the range in  FIG. 6  is concerned, Fs and Fm desirably satisfy the inequality 1.0&lt;Fs/Fm&lt;1.9 or the inequality 2.1&lt;Fs/Fm&lt;3.0 and more desirably satisfy the inequality 1.0&lt;Fs/Fm&lt;1.8 or the inequality 2.2&lt;Fs/Fm&lt;3.0. As the value of Fs/Fm becomes closer to 1 or 3, where Fs is an integer multiple of Fm, the DLD Slope3σ increases just as is the case with  FIG. 6  in which Fs/Fm is equal to or close to 2. Fs and Fm desirably satisfy the inequality 1.1&lt;Fs/Fm&lt;2.9 and more desirably satisfy the inequality 1.2&lt;Fs/Fm&lt;2.8. 
     The degree of change in the gradient of the approximate curve within a range where 1.8≤Fs/Fm≤2.2 is higher than the degree of change in the gradient of the approximate curve within a range where Fs/Fm&lt;1.8 and is likewise higher than the degree of change in the gradient of the approximate curve within a range where 2.2&lt;Fs/Fm. The degree of change in the gradient of the approximate curve within a range where 1.9≤Fs/Fm≤2.1 is much higher. The gradient of the approximate curve herein for purposes of this disclosure refers to the amount of change in DLD Slope 3σ relative to the amount of change in Fs/Fm. In other words, where 1.8≤Fs/Fm≤2.2, changes of Fm or Fs that are due to device-to-device variations produced by nonuniformity in the thickness of the piezoelectric film F 3  cause a considerable amount of change in DLD Slope3σ. The allowable ranges of Fs and Fm for satisfying the inequality 1.8≤Fs/Fm≤2.2 are narrower than the allowable ranges of Fs and Fm for satisfying the inequality Fs/Fm&lt;1.8 and are narrower than the allowable ranges of Fs and Fm for satisfying the inequality 2.2&lt;Fs/Fm. Thus, Fs and Fm desirably satisfy the inequality Fs/Fm&lt;1.9 or the inequality 2.1&lt;Fs/Fm and more desirably satisfy the inequality Fs/Fm&lt;1.8 or the inequality 2.2&lt;Fs/Fm, thus making the demands on processing accuracy less stringent and minimizing the reduction in yield accordingly. 
     FIRST EXAMPLE 
     The following describes a first example with reference to  FIGS. 7 and 8 .  FIG. 7  is a plan view of vibration arms and holding arms, schematically illustrating the dimensions of the vibration arms and the dimensions of the holding arms.  FIG. 8  is a graph illustrating the relationship between the length of a holding side arm and the length of a holding rear arm in the first example. Referring to  FIG. 8 , values obtained by dividing the length of the holding side arm by the length of the vibration arm are plotted along the horizontal axis of the graph, and values obtained by dividing the length of the holding rear arm by the length of the vibration arm are plotted along the vertical axis of the graph. The vibration arms  121 A to  121 D all have substantially the same dimensions. For this reason, the dimensions of the vibration arm  121 A will be described below, and no mention will be made of the dimensions of each of the vibration arms  121 B to  121 D. The left holding arm  151 A and the right holding arm  151 B have substantially the same dimensions. For this reason, the dimensions of the left holding arm  151 A will be described below, and no mention will be made of the dimensions of the right holding arm  151 B. 
     Referring to  FIG. 7 , the length of the vibration arm  121 A in the Y-axis direction is denoted by  121 L. The width of the mass addition portion  122 A of the vibration arm  121 A in the X-axis direction is denoted by  122 W, the width of the arm portion  123 A of the vibration arm  121 A in the X-axis direction is denoted by  123 W. The length  121 L is the length of the region extending from the fixed end to the open end of the vibration arm  121 A. The width  122 W is the width of the region between two edges being on opposite sides of the mass addition portion  122 A in the X-axis direction and extending in the Y-axis direction. The width  123 W is the width of the region between two edges being on opposite sides of the arm portion  123 A in the X-axis direction and extending in the Y-axis direction. 
     Moreover, the length of the holding rear arm  152 A of the left holding arm  151 A in the X-axis direction is denoted by  152 L, and the width of the holding rear arm  152 A in the Y-axis direction is denoted by  152 W. The length of the holding side arm  153 A of the left holding arm  151 A in the Y-axis direction is denoted by  153 L, and the width of the holding side arm  153 A in the X-axis direction is denoted by  153 W. The length  152 L is the length of the region that extends from an edge to another edge of the holding rear arm  152 A, or more specifically, the length of the region that extends from the edge facing the right frame  141 D to the edge facing the left frame  141 C. The width  152 W is the width of the region being part of the holding rear arm  152 A and extending in the X-axis direction, or more specifically, the width of the region defined by two edges being on opposite sides of the holding rear arm  152 A in the Y-axis direction and extending in the X-axis direction. The length  153 L is the length of the region extending from an edge to another edge of the holding side arm  153 A, or more specifically, the length of the region that extends from the edge facing the front frame  141 A to the edge facing the rear frame  141 B. The width  153 W is the width of the region being part of the holding side arm  153 A and extending in the Y-axis direction, or more specifically, the width of the region between two edges being on opposite sides of the holding side arm  153 A in the X-axis direction and extending in the Y-axis direction. 
     The width  122 W is greater than the width  123 W. The width  152 W is substantially equal to the width  153 W and is smaller than the width  123 W. The length  153 L is greater than the length  152 L and is smaller than the length  121 L. 
     The dimensions in the first example are as follows. The width  122 W is about 70 μm. The width  123 W is about 50 μm. The length  121 L is about 465 μm. The length  152 L is about 35 μm. The length  153 L is about 225 μm. The widths  152 W and  153 W are each about 20 μm. 
     The specifics of the first example are as follows. The temperature characteristics correction layer F 21  is formed from silicon oxide and has a thickness of about 470 nm. Moreover, the Si substrate F 2  has a thickness of about 6,000 nm. The metal film E 1  is formed from Mo and has a thickness of about 200 nm. The Si substrate F 2  and the metal film E 1  (see  FIGS. 4 and 5 ) are laid with an AlN film (not illustrated) therebetween. The AlN film has a thickness of about 20 nm. The piezoelectric film F 3  is formed from AlN. The piezoelectric film F 3  included in the arm portion  123 A of the vibration arm  121 A has a thickness of about 800 nm. The metal film E 2  is formed from Mo and has a thickness of about 100 nm. The protective film F 5  is formed from AlN. The protective film F 5  included in the arm portion  123 A of the vibration arm  121 A has a thickness of about 170 nm. The sum of the thicknesses of the piezoelectric film F 3  and the protective film F 5  that are formed from AlN and included in the mass addition portion  122 A of the vibration arm  121 A is about 1,040 nm. The mass addition film  125 A of the vibration arm  121 A is formed from Mo and has a thickness of about 300 nm. 
     In this case, the inequality 2.2&lt;Fs/Fm holds, and the resonator thus has favorable DLD characteristics. Let L be a variable representing the length  121 L of the vibration arm  121 A, where 465 μm is an example value of the length  121 L. Moreover, La can be a variable representing the length  153 L of the holding side arm  153 A, where 225 μm is an example value of the length  152 L. Lb can be a variable representing the length  152 L of the holding rear arm  152 A, where 35 μm is an example value of the length  152 L. Referring to  FIG. 8 , points at which the equation Fs/Fm=1.8 holds and points at which the equation Fs/Fm=2.2 holds are plotted on the graph whose vertical axis is Lb/L and whose horizontal axis is La/L. According to the approximate curve for the points at which the equation Fs/Fm=1.8 holds and the approximate curve for the points at which the equation Fs/Fm=2.2 holds, Lb/L and La/L can be expressed as follows. 
     The equation Fs/Fm=1.8 holds if Lb/L=−7.3466×(La/L) 2 +5.5467×(La/L)−0.5527. The equation Fs/Fm=2.2 holds if Lb/L=−8.1217×(La/L) 2 +5.6022×(La/L)−0.5361. 
     Therefore, the inequality Fs/Fm&lt;1.8 holds if Lb/L&gt;−7.3466×(La/L) 2 +5.5467×(La/L)−0.5527, and the inequality 2.2&lt;Fs/Fm holds if Lb/L&lt;−8.1217×(La/L) 2 +5.6022×(La/L)−0.5361. 
     SECOND EXAMPLE 
     The following describes a second example with reference to  FIG. 9 .  FIG. 9  is a graph illustrating the relationship between the length of a holding side arm and the length of a holding rear arm in the second example. The horizontal axis and the vertical axis of the graph in  FIG. 9  are identical to the respective axes of the graph in  FIG. 8 . 
     The second example differs from the first example in that the width  122 W of the mass addition portion  122 A and the width  123 W of the arm portion  123 A are both about 50 μm. According to  FIG. 9  with the approximate curve for the points at which the equation Fs/Fm=1.8 holds and the approximate curve for the points at which the equation Fs/Fm=2.2 holds, Lb/L and La/L can be expressed as follows. 
     The equation Fs/Fm=1.8 holds if Lb/L=−8.3441×(La/L) 2 +6.0905×(La/L)−0.6659. The equation Fs/Fm=2.2 holds if Lb/L=−9.8294×(La/L) 2 +6.7517×(La/L)−0.7925. Therefore, the inequality Fs/Fm&lt;1.8 holds if Lb/L&gt;−8.3441×(La/L) 2 +6.0905×(La/L)−0.6659, and the inequality 2.2&lt;Fs/Fm holds if Lb/L&lt;−9.8294×(La/L) 2 +6.7517×(La/L)−0.7925. 
     The inequality Fs/Fm&lt;1.9 or the inequality 2.1&lt;Fs/Fm holds in the present embodiment, which can thus minimize the increase in variations in DLD. The increase in variations in DLD may be further minimized especially when the inequality Fs/Fm&lt;1.8 or the inequality 2.2&lt;Fs/Fm holds. 
     The holding side arms  153 A and  153 B of the holding arm  150 , respectively, are connected to the left frame  141 C and the right frame  141 D of the holding part  140 . The frequency of the spurious mode may thus be adjustable by changes in the design of the holding side arms  153 A and  153 B. The high degree of flexibility in the design of the holding arm  150  increases the possibility that the value of Fs/Fm will become close to 2. Satisfying the relevant conditional expressions for Fs/Fm by means of design is an important factor in providing resonators having favorable DLD characteristics. 
     The holding side arms  153 A and  153 B, respectively, are connected to the left frame  141 C and the right frame  141 D in such a manner that the junction between the holding side arm  153 A and the left frame  141 C and the junction between the holding side arm  153 B and the right frame  141 D are aligned with each other, with the arm portions  123 A to  123 D lying between the junctions. Moreover, the mass addition portions  122 A and  122 D, each of which is wider than the corresponding one of the arm portions  123 A and  123 D, are not aligned with the holding side arms  153 A and  153 B in the X-axis direction. This layout and configuration enables a reduction in the dimension of the resonator  10  in the X-axis direction. 
     The following describes the configuration of resonators according to additional embodiments of the present invention. Description of features common to the first embodiment and these additional embodiments will be omitted, and the following embodiments will be described with regard to only their distinctive features. Specifically, not every embodiment refers to actions and effects caused by similar configurations. 
     Second Exemplary Embodiment 
     The following describes a resonator  210  according to a second exemplary embodiment with reference to  FIG. 10 .  FIG. 10  is a plan view of a resonator according to the second embodiment, schematically illustrating the structure of the resonator. 
     As with the resonator according to the first embodiment, the resonator  210  according to the second embodiment includes vibration arms, a base section, a holding part, and a holding arm. The vibration arms are denoted by  221 A to  221 D. The base section, the holding part, and the holding arm are denoted by  230 ,  240 , and  250 , respectively. The vibration arms  221 A to  221 D include their respective mass addition portions, which are denoted by  222 A to  222 D, and also include their respective arm portions, which are denoted by  223 A to  223 D. The base section  230  includes a front end portion  231 A, a rear end portion  231 B, a left end portion  231 C, and a right end portion  231 D. The holding part  240  includes a front frame  241 A, a rear frame  241 B, a left frame  241 C, and a right frame  241 D. The holding arm  250  includes a left holding arm  251 A and a right holding arm  251 B. 
     The second embodiment differs from the first embodiment in that the left holding arm  251 A and the right holding arm  251 B, respectively, are connected to the left end portion  231 C and the right end portion  231 D of the base section  230 . Specifically, the left holding arm  251 A extends toward the left frame  241 C from the left end portion  231 C, is bent to extend toward the front frame  241 A, and is bent to extend toward the left frame  241 C. The right holding arm  251 B extends toward the right frame  241 D from the right end portion  231 D, is bent to extend toward the front frame  241 A, and is bent to extend toward the right frame  241 D. This means that the left holding arm  251 A and the right holding arm  251 B do not include their respective holding rear arms. The left holding arm  251 A and the right holding arm  251 B include their respective holding side arms, which are denoted by  253 A and  253 B. 
     Effects similar to those of the aforementioned resonator may be attained in the present embodiment. 
     The following describes the relationship between the dimensional ratio and the frequency ratio with reference to  FIGS. 11 to 13 .  FIG. 11  illustrates vibration arms and holding arms for the sake of making explanation of the dimensions of the vibration arms and the dimensions of the holding arms.  FIG. 12  is a graph illustrating the relationship between the frequency and the ratio of the dimension of a holding arm to the dimension of a vibration arm.  FIG. 13  is a graph with an approximate curve for the frequency ratio. The following describes, with reference to  FIG. 11 , the dimensions of the left holding arm  151 A, which is a representative of the holding arms, and describes the dimensions of the vibration arm  121 A, which is a representative of the vibration arms. It should be appreciated that the left holding arm  151 A and the right holding arm  151 B are structurally mirror images of each other and have substantially the same dimensions. For this reason, no mention will be made of the dimensions of the right holding arm  151 B. Similarly, no mention will be made of the dimensions of each of the vibration arms  121 B to  121 D, which are each substantially identical in dimensions to the vibration arm  121 A. 
     Referring to  FIG. 11 , La denotes the length of the holding side arm  153 A, Lb denotes the length of the holding rear arm  152 A, and Lc denotes the distance from an edge to another edge of the left holding arm  151 A. As described with reference to  FIG. 7 , the length La is the length of the region extending from an edge to another edge of the holding side arm  153 A, or more specifically, the length of the region extending from the edge on the +Y side to the edge on the −Y side in the Y-axis direction, that is, the length of the region that extends from the edge facing the front frame  141 A to the edge facing the rear frame  141 B. The length Lb is the length of the region extending from an edge to another edge of the holding rear arm  152 A, or more specifically, the length of the region extending from the edge on the +X side to the edge on the −X side in the X-axis direction, that is, the length of the region that extends from the edge facing the right frame  141 D to the edge facing the left frame  141 C. The distance Lc is the distance between an edge farther than any other edge of the holding rear arm  152 A from the holding side arm  153 A and an edge father than any other edge of the holding side arm  153 A from the holding rear arm  152 A. More specifically, the distance Lc is the distance between two corners. One of the corners is on an edge farther than any other edge of the holding rear arm  152 A from the holding side arm  153 A and is on the +X side in the X-axis direction and on the −Y side in the Y axis direction. The other corner is on an edge farther than any other edge of the holding side arm  153 A from the holding rear arm  152 A and is on the −X side in the X-axis direction and on the +Y side in the Y axis direction. In other words, the distance Lc is the distance between the corner being part of the holding rear arm  152 A and facing both the right frame  141 D and the rear frame  141 B and the corner being part of the holding side arm  153 A and facing both the front frame  141 A and the left frame  141 C. 
     As shown, the holding rear arm  152 A and the holding side arm  153 A in the present embodiment are connected to each other in a manner so as to form a right angle, and the distance Lc can be written as the square root of the sum of the square of the length La and the square of the length Lb and can be expressed by the following equation: 
         Lc =√{square root over ( La   2   +Lb   2 )}
 
     Referring to  FIG. 11 , L denotes the length of the vibration arm  121 A, L 3  denotes the length of the mass addition portion  122 A of the vibration arm  121 A, W 3  denotes the width of the mass addition portion  122 A, L 4  denotes the length of the arm portion  123 A of the vibration arm  121 A, W 4  denotes the width of the arm portion  123 A, and L 5  denotes equivalent vibration arm length. The equivalent vibration arm length L 5  is the length of an imaginary vibration arm of constant width. The imaginary vibration arm is obtained by transforming the vibration arm  121 A into a form whose moment of inertia is equal to the moment of inertia of the vibration arm  121 A. The frequency of the vibration arm is dependent on the moment of inertia. For the sake of comparisons between vibration arms including mass addition portions of different lengths and/or different widths, the vibration arm  121 A is transformed into an imaginary arm whose width is equal to the width W 4  of the arm portion  123 A, and the length of the imaginary vibration arm is herein referred to as the equivalent vibration arm length L 5 . 
     The moment of inertia of a plate can be expressed as follows: 
       I∝∫ 0   L W(x)*x 2 dx
 
     where I is the moment of inertia, x is the distance from the fixed end of the vibration arm, and W(x) is the width of the vibration arm at the distance x. 
     The moment of inertia of the vibration arm  121 A can be expressed as follows: 
         I∝∫   0   L4   W 4 *x   2   dx+∫   L4   L   W 3 *x   2   dx=W 4*⅓*( L 4) 3   +W 3*⅓*{( L   3 −( L 4) 3 )}
 
     The moment of inertia of the imaginary vibration arm of constant width can be expressed with the equivalent vibration arm length L 5  as follows: 
         I∝∫   0   L5   W 4 *x   2   dx=W 4*⅓*( L 5) 3  
 
     The moment of inertia of the vibration arm  121 A is equal to the moment of inertia of the imaginary vibration arm having the equivalent vibration arm length L 5 , and the equivalent vibration arm length L 5  can thus be determined by using the following equation: 
         W 4*⅓*( L 4) 3   +W 3*{( L   3 −( L 4) 3 )}= W 4*⅓*( L 5) 3  
 
     Referring to  FIGS. 12 to 14 , the horizontal axis of each graph represents the ratio of the distance Lc (i.e., the distance from an edge to another edge of the holding arm) to the equivalent vibration arm length L 5 , and this dimensional ratio is expressed as Lc/L 5 . The vertical axis of each graph represents the ratio of the spurious-mode frequency Fs to the main-mode frequency Fm, and this frequency ratio is expressed as Fs/Fm. 
       FIG. 12  illustrates the relationship between the dimensional ratio and the frequency ratio that were determined through simulation conducted on vibration arms with variations in the thickness of the Si substrate F 2 . The length L of each of the vibration arms was adjusted such that the main-mode frequency Fm was consistent across all samples. The samples used in the simulation were made in accordance with the first example of the first embodiment, with L=431 μm for the Si substrate F 2  having a thickness of 5 μm, L=465 μm for the Si substrate F 2  having a thickness of 6 μm, and L=498 μm for the Si substrate F 2  having a thickness of 7 μm. The dimensional ratio Lc/L 5  and the frequency ratio Fs/Fm of the individual samples were calculated and plotted on the graph. The simulation revealed that the correlation between the dimensional ratio Lc/L 5  and the frequency ratio Fs/Fm followed the same pattern irrespective of sample-to-sample variations in the thickness of the Si substrate F 2  and in the length L of the vibration arms. 
     The curve drawn on  FIG. 13  is based on an approximate equation obtained from all the plots on the graph in  FIG. 12 . The approximate equation is shown in the graph, where y represents the vertical axis and x represents the horizontal axis. As can be seen from the graph, the inequality Fs/Fm&lt;1.9 holds if 0.550&lt;Lc/L 5 , and the inequality 2.1&lt;Fs/Fm holds if Lc/L 5 &lt;0.520. The inequality Fs/Fm&lt;1.8 holds if 0.566&lt;Lc/L 5 , and the inequality 2.2&lt;Fs/Fm holds if Lc/L 5 &lt;0.506. For minimization of the increase in variations in DLD, Lc and L 5  desirably satisfy the inequality Lc/L 5 &lt;0.520 or the inequality 0.550&lt;Lc/L 5  and more desirably satisfy the inequality Lc/L 5 &lt;0.506 or the inequality 0.566&lt;Lc/L 5 . 
     The preferable ranges for Lc/L 5  remain unchanged irrespective of any difference in the dimensions of the arm portions and the dimensions of the mass addition portions. This can be seen from the graph in  FIG. 14 .  FIG. 14  illustrates the relationship between the dimensional ratio Lc/L 5  and the frequency ratio Fs/Fm that were determined through simulation conducted on vibration arms with variations in the dimensions of the arm portions and in the dimensions of the mass addition portions. The length L of each of the vibration arms was adjusted such that the main-mode frequency Fm was consistent across all samples. The annotation saying “L500 W50 Wh50” refers to results obtained from simulation conducted on a sample that included vibration arms each having a length of 500 μm and including a 50-μm-wide arm portion and a 50-μm-wide mass addition portion. The annotation saying “L465 W50 Wh70” refers to results obtained from simulation conducted on a sample that included vibration arms each having a length of 465μm and including a 50-μm-wide arm portion and a 70-μm-wide mass addition portion. The annotation saying “L396 W30 Wh70” refers to results obtained from simulation conducted on a sample that included vibration arms each having a length of 396μm and a 30-μm-wide arm portion and a 70-μm-wide mass addition portion. Under these conditions, the ratio of the length of the mass addition portion to the length of the vibration arm stood at 36%. 
     Plots under the condition “L500 W50 Wh50” and plots under the condition “L465 W50 Wh70” can be approximated by substantially the same curve. This means that the correlation between the dimensional ratio Lc/L 5  and the frequency ratio Fs/Fm followed substantially the same pattern irrespective of sample-to-sample variations in the width of the mass addition portion. The plots under the condition “L465 W50 Wh70”, the plots under the condition “L500 W50 Wh50”, and plots under the condition “L396 W30 Wh70” can be approximated by substantially the same curve. This means that the correlation between the dimensional ratio Lc/L 5  and the frequency ratio Fs/Fm followed substantially the same pattern irrespective of sample-to-sample variations in the width of the arm portion. 
     In general, exemplary embodiments of the present invention will be described, in part or in whole, as follows. The following description should not be construed as limiting the scope of the present invention. 
     An exemplary aspect provides a resonator that includes a base, at least one vibration arm, a frame, and a holding arm. The at least one vibration arm includes a piezoelectric film, an upper electrode, and a lower electrode. The upper and lower electrodes are laid on opposite sides with the piezoelectric film therebetween. The at least one vibration arm has a fixed end connected to a front end of the base and an open end located away from the front end. The frame holds the base section. The holding arm forms a connection between the base and the frame. The inequality Fs/Fm&lt;1.9 or the inequality 2.1&lt;Fs/Fm holds, where Fm is the frequency of a main mode in the at least one vibration arm, and Fs is the frequency of a spurious mode in the holding arm. 
     This configuration enables minimization of the increase in variations in DLD. 
     In another aspect, the inequality Fs/Fm&lt;1.8 or the inequality 2.2&lt;Fs/Fm holds for the resonator. 
     This configuration enables further minimization of the increase in variations in DLD. 
     Still another aspect is as follows. The holding arm includes a holding rear arm and a holding side arm. The holding rear arm is connected to a rear end portion opposite the front end of the base and extends along the rear end. The holding side arm is connected to the holding rear arm and extends along the at least one vibration arm. The inequality Lc/L 5 &lt;0.520 or the inequality 0.550&lt;Lc/L 5  holds, where Lc is the distance between an edge farther than any other edge of the holding rear arm from the holding side arm and an edge farther than any other edge of the holding side arm from the holding rear arm, and L 5  is the length of an imaginary vibration arm of constant width. The imaginary vibration arm is obtained by transforming the at least one vibration arm into a form whose moment of inertia is equal to the moment of inertia of the at least one vibration arm. 
     In still another aspect, the inequality Lc/L 5 &lt;0.506 or the inequality 0.566&lt;Lc/L 5  holds for the resonator. 
     Still another aspect is as follows. The frame includes a left frame and a right frame. The left and right frames extend along the at least one vibration arm and are disposed on opposite sides with the at least one vibration arm being disposed between the left frame and the right frame. The holding arm includes a holding side arm extending along the at least one vibration arm. The holding side arm is connected to the left frame or the right frame of the holding part. 
     The frequency of the spurious mode may thus be adjustable by changes in the design of the holding side arm. The high degree of flexibility in the design of the holding arm increases the possibility that the value of Fs/Fm will become close to 2. Satisfying the relevant conditional expressions for Fs/Fm by means of design is an important factor in providing resonators having favorable DLD characteristics. 
     Still another aspect is as follows. The at least one vibration arm includes an arm portion and a mass addition portion. The arm portion extends from the front end of the base. The mass addition portion is connected to a tip of the arm portion and is heavier in weight per unit length than the arm portion. The holding side arm is connected to the left frame or the right frame of the holding part in such a manner that the junction between the holding side arm and the left or right frame faces the arm portion. 
     The holding side arm and the mass addition portion wider than the arm portion do not lie side by side. This layout enables a reduction in the dimension of the resonator. 
     Still another aspect of the present invention provides a resonator including a base, at least one vibration arm, a frame, and a holding arm. The at least one vibration arm includes a piezoelectric film, an upper electrode, and a lower electrode. The upper and lower electrodes are disposed on opposite sides with the piezoelectric film therebetween. The at least one vibration arm has a fixed end connected to a front end of the base and an open end located away from the front end. The frame holds the base section. The holding arm forms a connection between the base and the frame. The holding arm includes a holding rear arm and a holding side arm. The holding rear arm is connected to a rear end opposite the front end of the base and extends along the rear end. The holding side arm is connected to the holding rear arm and extends along the at least one vibration arm. The inequality Lc/L 5 &lt;0.520 or the inequality 0.550&lt;Lc/L 5  holds, where Lc is the distance between an edge farther than any other edge of the holding rear arm from the holding side arm and an edge farther than any other edge of the holding side arm from the holding rear arm, and L 5  is the length of an imaginary vibration arm of constant width. It is noted that the imaginary vibration arm is obtained by transforming the at least one vibration arm into a form whose moment of inertia is equal to the moment of inertia of the at least one vibration arm. 
     In still another aspect, the inequality Lc/L 5 &lt;0.506 or the inequality 0.566&lt;Lc/L 5  holds for the resonator. 
     Still another exemplary aspect of the present invention provides a resonance device including the resonator, a lower cover, and an upper cover. The lower cover is joined to the resonator. The upper cover is joined to the lower cover with the resonator therebetween. A vibration space in which the at least one vibration arm vibrates is defined between the upper cover and the lower cover. 
     As has been described so far, a resonator having favorable DLD characteristics and a resonance device including the resonator are provided in accordance with the aspects of the present invention. 
     The exemplary embodiments above have been described to facilitate the understanding of the present invention and should not be construed as limiting the scope of the present invention. The present invention may be altered and/or improved as would be appreciated to one skilled in the art without departing from the spirit of the present invention and embraces equivalents thereof. That is, the embodiments with design changes made as appropriate by those skilled in the art fall within the scope of the present invention as long as the features of the present invention are involved. For example, components in the embodiments above and the arrangement, materials, conditions, shapes, and sizes of the components are not limited to those mentioned in the description and may be changed as appropriate. Varying combinations of the components of the embodiments may be devised as long as they are technically possible, and these combinations also fall within the scope of the present invention as long as the features of the present invention are involved. 
     REFERENCE SIGNS LIST 
       1  resonance device 
       10  resonator 
       20  lower cover 
       30  upper cover 
       21 ,  31  recess 
       22 ,  32  bottom plate 
       23 ,  33  side wall 
       110  vibration part 
       121 A to  121 D vibration arm 
       122 A to  122 D mass addition portion 
       123 A to  123 D arm portion 
       125 A to  125 D mass addition film 
       130  base section 
       131 A front end portion 
       131 B rear end portion 
       131 C left end portion 
       131 D right end portion 
       140  holding part 
       141 A front frame 
       141 B rear frame 
       141 C left frame 
       141 D right frame 
       150  holding arm 
       151 A left holding arm 
       151 B right holding arm 
       152 A,  152 B holding rear arm 
       153 A,  153 B holding side arm 
     F 2  Si substrate 
     F 21  temperature characteristics correction layer 
     F 3  piezoelectric film 
     F 5  protective film 
     E 1  metal film (lower electrode) 
     E 2  metal film (upper electrode)