Patent Publication Number: US-2021175871-A1

Title: Acoustic wave resonator, filter, multiplexer, and wafer

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2019-221476, filed on Dec. 6, 2019, the entire contents of which are incorporated herein by reference. 
     FIELD 
     A certain aspect of the present embodiments relates to an acoustic wave resonator, a filter, a multiplexer, and a wafer. 
     BACKGROUND 
     Surface acoustic wave resonators have been known as acoustic wave resonators used in communication devices such as smartphones. As disclosed in, for example, Japanese Patent Application Publication No. 2017-34363, it has been known that a piezoelectric layer forming the surface acoustic wave resonator is bonded to a support substrate, and the thickness of the piezoelectric layer is adjusted to be equal to or less than the wavelength of the surface acoustic wave. As disclosed in Japanese Patent Application Publication No. 2015-73331, it has been known that a silicon oxide layer is interposed between the support substrate and the piezoelectric layer. As disclosed in, for example, Japanese Patent Application Publication No. 2017-152868, it has been known that the temperature coefficient of the resonant frequency differs from the temperature coefficient of the antiresonant frequency in the surface acoustic wave resonator. 
     SUMMARY 
     To allow a filter constructed of the acoustic wave resonators to have desired characteristics, it is desired to reduce the difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency in the acoustic wave resonator. 
     An objective of the present invention is to reduce the difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency. 
     According to a first aspect of the present embodiments, there is provided an acoustic wave resonator including: a support substrate; a piezoelectric layer that is disposed on the support substrate and is a rotated Y-cut X-propagation lithium tantalate of which a cut angle is within a range of greater than 50° and less than 150°; and a pair of comb-shaped electrodes disposed on the piezoelectric layer, each of the comb-shaped electrodes including a plurality of electrode fingers, an average pitch of the electrode fingers of one of the comb-shaped electrodes being equal to or greater than ½ of a thickness of the piezoelectric layer. 
     According to a second aspect of the present embodiments, there is provided a filter including the above acoustic wave resonator. 
     According to a third aspect of the present embodiments, there is provided a multiplexer including the above filter. 
     According to a fourth aspect of the present embodiments, there is provided a wafer including: a support substrate; and a piezoelectric layer that is disposed on the support substrate, and is a rotated Y-cut X-propagation lithium tantalate of which a cut angle is within a range of greater than 50° and less than 150°. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a plan view of an acoustic wave resonator in accordance with a first embodiment, and  FIG. 1B  is a cross-sectional view taken along line A-A in in  FIG. 1A ; 
         FIG. 2  is a cross-sectional view for describing an acoustic wave; 
         FIG. 3A  and  FIG. 3B  are cross-sectional views of the acoustic wave resonator in simulations A and B, respectively; 
         FIG. 4A  and  FIG. 4B  are graphs of total displacement distribution versus a position Z in the simulations A and B, respectively; 
         FIG. 5A  to  FIG. 5C  are cross-sectional views for describing a bulk wave; 
         FIG. 6  is a graph of a temperature coefficient of frequency versus the cut angle of a piezoelectric layer; 
         FIG. 7  is a graph of a difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency and an electromechanical coupling coefficient versus the cut angle of the piezoelectric layer; 
         FIG. 8A  presents comparison of the difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency with respect to the cut angle of the piezoelectric layer between the case where an insulating layer is provided and the case where no insulating layer is provided, and  FIG. 8B  presents comparison of the electromechanical coupling coefficient with respect to the cut angle of the piezoelectric layer between the case where an insulating layer is provided and the case where no insulating layer is provided; 
         FIG. 9A  and  FIG. 9B  are graphs of the difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency versus the cut angle of the piezoelectric layer when the thickness of the insulating layer is varied; 
         FIG. 10A  and  FIG. 10B  are graphs of the electromechanical coupling coefficient versus the cut angle of the piezoelectric layer when the thickness of the insulating layer is varied; 
         FIG. 11  is a cross-sectional view of an acoustic wave resonator in accordance with a second embodiment; 
         FIG. 12  is a cross-sectional view of an acoustic wave resonator in accordance with a third embodiment; 
         FIG. 13A  to  FIG. 13C  are cross-sectional views of acoustic wave resonators in accordance with a fourth embodiment, a first variation of the fourth embodiment, and a second variation of the fourth embodiment, respectively; 
         FIG. 14  is a circuit diagram of a filter in accordance with a fifth embodiment; and 
         FIG. 15  is a circuit diagram of a duplexer in accordance with a sixth embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Hereinafter, a description will be given of embodiments of the present invention with reference to the accompanying drawings. 
     First Embodiment 
       FIG. 1A  is a plan view of an acoustic wave resonator in accordance with a first embodiment, and  FIG. 1B  is a cross-sectional view taken along line A-A in  FIG. 1A . The direction in which electrode fingers are arranged (the arrangement direction) is defined as an X direction, the direction in which the electrode fingers extend (the extension direction) is defined as a Y direction, and the direction in which a support substrate and a piezoelectric layer are stacked (the stack direction) is defined as a Z direction. The X direction, the Y direction, and the Z direction do not necessarily correspond to the X-axis orientation and the Y-axis orientation of the crystal orientation of the piezoelectric layer. When the piezoelectric layer is a rotated Y-cut X-propagation layer, the X direction is the X-axis orientation of the crystal orientation. 
     As illustrated in  FIG. 1A  and  FIG. 1B , in an acoustic wave resonator  100  of the first embodiment, an insulating layer  11  is bonded on a support substrate  10 . A piezoelectric layer  12  is bonded on the insulating layer  11 . The insulating layer  11  may be directly bonded on the support substrate  10 , or may be indirectly bonded on the support substrate  10  with a bonding layer or other layers interposed therebetween. The piezoelectric layer  12  may be directly bonded on the insulating layer  11 , or may be indirectly bonded on the insulating layer  11  with a bonding layer or other layers interposed therebetween. An acoustic wave element  20  is disposed on the piezoelectric layer  12 . The acoustic wave element  20  includes an interdigital transducer (IDT)  22  and reflectors  24 . The reflectors  24  are disposed at both sides of the IDT  22  in the X direction. The IDT  22  and the reflectors  24  are formed of a metal film  14  on the piezoelectric layer  12 . 
     The IDT  22  includes a pair of comb-shaped electrodes  18  facing each other. Each comb-shaped electrode  18  includes a plurality of electrode fingers  15  and a bus bar  16  to which the electrode fingers  15  are connected. The region where the electrode fingers  15  of one of the comb-shaped electrodes  18  overlap the electrode fingers  15  of the other of the comb-shaped electrodes  18  is an overlap region  25 . The length of the overlap region  25  is an aperture length. A pair of the comb-shaped electrodes  18  is arranged so as to face each other in a manner such that the electrode fingers  15  of one of the comb-shaped electrodes  18  and the electrode fingers  15  of the other of the comb-shaped electrodes  18  are substantially alternated in at least a part of the overlap region  25 . The acoustic wave excited by the electrode fingers  15  in the overlap region  25  propagates mainly in the X direction. The pitch of the electrode fingers  15  of one of the comb-shaped electrodes  18  is substantially equal to the wavelength λ of the acoustic wave. That is, the wavelength λ of the acoustic wave is substantially equal to two times the pitch of the electrode fingers  15  of a pair of the comb-shaped electrodes  18 . The reflectors  24  reflect the acoustic wave (a surface acoustic wave) excited by the electrode fingers  15  of the IDT  22 . Thus, the acoustic wave is confined in the overlap region  25  of the IDT  22 . 
     The piezoelectric layer  12  is a monocrystal lithium tantalate (LiTaO 3 ) layer, and is a rotated Y-cut X-propagation lithium tantalate layer. The insulating layer  11  is an insulating layer containing silicon oxide (SiO). The insulating layer  11  may be a silicon dioxide (SiO 2 ) layer, or may be mainly composed of silicon oxide and contain impurities such as fluorine or nitrogen (for example, a SiOF layer or a SiON layer). The temperature coefficient of the elastic constant of the insulating layer  11  is opposite in sign to the temperature coefficient of the elastic constant of the piezoelectric layer  12 . When temperature coefficient of the elastic constant of the insulating layer  11  is adjusted to be opposite in sign to the temperature coefficient of the elastic constant of the piezoelectric layer  12 , the small temperature coefficient of frequency (TCF) of the acoustic wave resonator is achieved. 
     The support substrate  10  has a linear expansion coefficient smaller than the linear expansion coefficient in the X direction of the piezoelectric layer  12 . Use of the material having a linear expansion coefficient smaller than the linear expansion coefficient of the piezoelectric layer  12  as the support substrate  10  reduces the variation in the pitch of the electrode fingers  15  due to the variation in temperature, and thereby the TCF of the acoustic wave resonator can be reduced. The support substrate  10  is, for example, a sapphire substrate, an alumina substrate, a silicon substrate, a spinel substrate, a crystal substrate, a quartz substrate, or a silicon carbide substrate. The sapphire substrate is a substrate containing monocrystal Al 2 O 3  as a main component. The alumina substrate is a substrate containing polycrystalline Al 2 O 3  as a main component. The silicon substrate is a substrate containing monocrystal Si or polycrystalline Si as a main component. The spinel substrate is a substrate containing monocrystal MgAl 2 O 4  or polycrystalline MgAl 2 O 4  as a main component. The crystal substrate is a substrate containing monocrystal SiO 2  as a main component. The quartz substrate is a substrate containing amorphous SiO 2  as a main component. The silicon carbide substrate is a substrate containing monocrystal SiC or polycrystalline SiC as a main component. The main component is a component of which the total concentration of atoms in the substrate is 50 atomic % or greater, or the component of which the total concentration of atoms is 80 atomic % or greater. 
     The metal film  14  is a film containing, for example, aluminum, copper, or molybdenum as a main component, and is, for example, an aluminum film, a copper film, or a molybdenum film. An adhesion film such as a titanium film or a chrome film may be interposed between the electrode fingers  15  and the piezoelectric layer  12 . The adhesion film is thinner than the electrode finger  15 . An insulating film may be provided so as to cover the electrode fingers  15 . The insulating film may act as a protective film or a temperature compensation film. 
     The support substrate  10  has a thickness of, for example, 50 μm to 500 μm. The thickness T 1  of the insulating layer  11  is, for example, 0.1 μm to 10 μm, and is equal to or less than, for example, the wavelength λ of the acoustic wave. The thickness T 2  of the piezoelectric layer  12  is, for example, 0.1 μm to 10 μm, and is equal to or less than, for example, the wavelength λ of the acoustic wave. The wavelength λ of the acoustic wave is, for example, 1 μm to 6 μm. When two electrode fingers  15  are defined as one pair of the electrode fingers  15 , the number of pairs of the electrode fingers  15  is 20 pairs to 300 pairs. The duty ratio of the IDT  22  is calculated by dividing the width of the electrode finger  15  by the pitch of the electrode fingers  15 , and is, for example, 30% to 80%. The aperture length of the IDT  22  is, for example, 10λ to 50λ. 
     Manufacturing Method 
     A description will be given of a manufacturing method of the acoustic wave resonator  100 . The insulating layer  11  is formed on the support substrate  10  using the chemical vapor deposition (CVD) method. After a piezoelectric substrate is bonded on the insulating layer  11  using the surface activation method, the piezoelectric substrate is thinned by chemical mechanical polishing (CMP) to form the piezoelectric layer  12 . The IDT  22  and the reflectors  24  are formed on the piezoelectric layer  12 . 
     Description of the Thicknesses of the Insulating Layer  11  and the Piezoelectric Layer  12   
     A description will be given of the thickness T 1  of the insulating layer  11  and the thickness T 2  of the piezoelectric layer  12 .  FIG. 2  is a cross-sectional view for describing the acoustic wave. As illustrated in  FIG. 2 , the electrode fingers  15  of the IDT  22  excites an acoustic wave  50 . The acoustic wave  50  in the drawing illustrates an image of displacement, and is different from the actual displacement of the acoustic wave. When the piezoelectric layer  12  is a rotated Y-cut X-propagation lithium tantalate layer, the IDT  22  excites mainly a shear horizontal (SH) wave. The SH wave is a wave that is displaced in the direction that is parallel to the surface of the piezoelectric layer  12  and orthogonal to the propagation direction of the SH wave. To reduce the TCF of the acoustic wave resonator, the displacement of the surface acoustic wave is required to be distributed in the insulating layer  11 . 
     Thus, the total displacement distribution at the resonant frequency was simulated.  FIG. 3A  and  FIG. 3B  are cross-sectional views of acoustic wave resonators in simulations A and B, respectively. As illustrated in  FIG. 3A , in the simulation A, a piezoelectric substrate  12 ′ made of 42° rotated Y-cut X-propagation lithium tantalate was used. As illustrated in  FIG. 3B , in the simulation B, used was a substrate in which the piezoelectric layer  12  made of 42° rotated Y-cut X-propagation lithium tantalate was disposed on the support substrate  10  made of sapphire. The thickness T 1  of the piezoelectric layer  12  was set at approximately 0.7λ. In both the simulations A and B, the surfaces of the piezoelectric substrate  12 ′ and the piezoelectric layer  12  with which the electrode fingers  15  are in contact were defined as 0, and the position in the depth direction of the substrate was defined as a position Z. 
       FIG. 4A  and  FIG. 4B  are graphs of the total displacement distribution versus the position Z in the simulations A and B, respectively. In the top part of  FIG. 4A  and  FIG. 4B , arrows indicating the regions of the piezoelectric substrate  12 ′, the support substrate  10 , and the piezoelectric layer  12  are illustrated. As presented in  FIG. 4A , in the simulation A, most of displacements are distributed within a range of Z/λ of 2 or less. In particular, most of displacements are within a range of Z/λ of 1.5 or less. This indicates that the surface acoustic wave propagates through the region from the surface of the piezoelectric substrate  12 ′ to 2λ (particularly, 1.5λ) in depth. As presented in  FIG. 4B , in the simulation B, most of displacements are distributed within a range of Z/λ of 1 or less. In particular, displacements are distributed very little within the support substrate  10 . This is because the phase velocity of the support substrate  10  is large. 
     As clear from the above simulation results, the surface acoustic wave propagates through the region from the surface of the piezoelectric substrate  12 ′ to 2λ (in particular, 1.5λ) in depth. This reveals that the insulating layer  11  is required to exist within the region from the upper surface of the piezoelectric layer  12  to 2λ (particularly, 1.5λ) in depth to allow the insulating layer  11  to have an ability to reduce the TCF in the first embodiment. 
     Next, a bulk wave will be described.  FIG. 5A  to  FIG. 5C  are cross-sectional views for describing a bulk wave. As illustrated in  FIG. 5A , when the piezoelectric substrate  12 ′ is used, the IDT  22  excites a surface acoustic wave  52  such as an SH wave on the surface of the piezoelectric substrate  12 ′. The thickness T 4  within which the displacement of the surface acoustic wave  52  exists is approximately 2λ. When the IDT  22  excites the surface acoustic wave  52 , the IDT  22  emits a bulk wave  54  within the piezoelectric substrate  12 ′. The magnitude of the bulk wave  54  is approximately 1/10 of the magnitude of the surface acoustic wave  52  in the primary mode. The thickness T 5  of the region where the bulk wave  54  exists is approximately 10λ. When the bulk wave  54  propagates through the piezoelectric substrate  12 ′, the energy of the surface acoustic wave  52  is lost as the bulk wave  54 . Thus, the loss of the acoustic wave resonator increases. 
     As illustrated in  FIG. 5B , the insulating layer  11  of which the temperature coefficient of the elastic constant is opposite in sign to that of the piezoelectric layer  12  is disposed on the support substrate  10 , and the piezoelectric layer  12  is disposed on the insulating layer  11 . The thickness T 1  of the piezoelectric layer  12  is less than the thickness T 4 . Thus, the displacement of the surface acoustic wave  52  is distributed in both the piezoelectric layer  12  and the insulating layer  11 . Therefore, the temperature coefficient of frequency is reduced. When the thickness T 2  of the insulating layer  11  is large, the bulk wave  54  propagates through the insulating layer  11 . Thus, the energy of the surface acoustic wave  52  is lost as the bulk wave  54 . Therefore, the loss of the acoustic wave resonator increases. As indicated by arrows  56 , no bulk wave propagates through the support substrate  10 . 
     As illustrated in  FIG. 5C , in the first embodiment, the insulating layer  11  is thinned, and the sum of the thicknesses T 1  and T 2  (i.e., the distance between the upper surface of the piezoelectric layer  12  and the lower surface of the insulating layer  11 ) is reduced to T 4  or less. The support substrate  10  has a phase velocity (an acoustic velocity) greater than the phase velocities of the insulating layers  11  and the piezoelectric layers  12 . For example, the phase velocity of the faster lateral wave of lithium tantalate is approximately 4211 m/s, the phase velocity of the faster lateral wave of silicon dioxide is approximately 5840 m/s, and the phase velocity of the faster lateral wave of sapphire is approximately 6761 m/s. Thus, as indicated by the arrows  56 , it is difficult for the bulk wave to propagate through the support substrate  10 . The surface acoustic wave  52  and the bulk wave  54  are both confined in the piezoelectric layer  12  and the insulating layer  11 . Therefore, the loss of the acoustic wave resonator is reduced. In addition, spurious emissions due to the bulk wave are reduced. 
     Description of the Cut Angle of the Piezoelectric Layer  12   
     A description will be given of a simulation in which the cut angle of the piezoelectric layer  12  was varied and the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa and the electromechanical coupling coefficient (k 2 ) were calculated. The simulation was conducted under the following conditions using the structure illustrated in  FIG. 1A  and  FIG. 1B . 
     Support substrate  10 : Sapphire substrate 
     Insulating layer  11 : Silicon dioxide layer with a thickness of 600 nm (0.4λ) 
     Piezoelectric layer  12 : Rotated Y-cut X-propagation lithium tantalate layer with a thickness of 600 nm (0.4λ) 
     Metal film  14 : Aluminum film with a thickness of 150 nm 
     Pitch of the electrode fingers  15 ×2: 1500 nm (the wavelength λ of the acoustic wave) 
     Width of the electrode fingers  15 : 375 nm 
       FIG. 6  is a graph of the TCF versus the cut angle of the piezoelectric layer. The horizontal axis represents the cut angle of the piezoelectric layer  12 , and the vertical axis represents the TCF. Dots represent simulated points, and the curve line is an approximate curve (the same applies to the similar drawings, hereinafter). The simulation results of the resonant frequency fr are indicated by a bold line, and the simulation results of the antiresonant frequency fa are indicated by a thin line. The TCF was obtained from the difference between the frequency when the temperature of the acoustic wave resonator is 25° C. and the frequency when the temperature of the acoustic wave resonator is 85° C. 
     As illustrated in  FIG. 6 , in a range of the cut angle of the piezoelectric layer  12  of 0° to 80°, the temperature coefficient of the antiresonant frequency fa is smaller than the temperature coefficient of the resonant frequency fr. The difference between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa decreases as the cut angle of the piezoelectric layer  12  increases from 0°, and is very little at 80°. In a range of the cut angle of the piezoelectric layer  12  between 80° to 120°, the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa follow a similar trajectory while maintaining substantially no difference. When the cut angle of the piezoelectric layer  12  exceeds 120°, the difference between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa starts increasing, and the temperature coefficient of the antiresonant frequency fa is smaller than the temperature coefficient of the resonant frequency fr. 
       FIG. 7  is a graph of the difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency and the electromechanical coupling coefficient versus the cut angle of the piezoelectric layer. The horizontal axis represents the cut angle of the piezoelectric layer  12 , the left-hand vertical axis represents the difference (fr−fa) between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa, and the right-hand vertical axis represents the electromechanical coupling coefficient. The difference in TCF is indicated by a bold line, and the electromechanical coupling coefficient is indicated by a thin line. 
     As illustrated in  FIG. 7 , in a range of the cut angle of the piezoelectric layer  12  of greater than 50° and less than 150°, the absolute value of the difference between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa is maintained low, 10 ppm/K or less. In particular, in a range of the cut angle of the piezoelectric layer  12  of 80° or greater and 120° or less, the difference between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa is approximately zero. On the other hand, the electromechanical coupling coefficient is the largest when the cut angle of the piezoelectric layer  12  is 20°, is approximately zero when the cut angle is 120°, and decreases as the cut angle increases when the cut angle is between 20° and 120°. 
     As seen above, when the total thickness of the insulating layer  11  and the piezoelectric layer  12  is 2λ or less such as 0.8λ (i.e., the distance between the upper surface of the piezoelectric layer  12  and the lower surface of the insulating layer  11  is 2λ or less), the absolute value of the difference in TCF is maintained low, 10 ppm/K or less by adjusting the cut angle of the piezoelectric layer  12  to be greater than 50° and less than 150°. In particular, when the cut angle of the piezoelectric layer  12  is adjusted to be 80° or greater and 120° or less, the difference in TCF becomes approximately zero. 
     Next, the difference between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa and the electromechanical coupling coefficient with respect to the cut angle of the piezoelectric layer  12  are compared between the case where the insulating layer  11  is provided and the case where no insulating layer  11  is provided. The simulation for the case where the insulating layer  11  is provided was conducted under the conditions identical to those in  FIG. 6  and  FIG. 7 . The simulation for the case where no insulating layer  11  is provided was conducted under the conditions identical to those in  FIG. 6  and  FIG. 7  except in that no insulating layer  11  was provided. 
       FIG. 8A  illustrates comparison of the difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency with respect to the cut angle of the piezoelectric layer between the case where the insulating layer is provided and the case where no insulating layer is provided, and  FIG. 8B  illustrates comparison of the electromechanical coupling coefficient with respect to the cut angle of the piezoelectric layer between the case where the insulating layer is provided and the case where no insulating layer is provided. In  FIG. 8A  and  FIG. 8B , the horizontal axis represents the cut angle of the piezoelectric layer  12 . The vertical axis in  FIG. 8A  represents the difference (fr−fa) between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa, and the vertical axis in  FIG. 8B  represents the electromechanical coupling coefficient. The case where the insulating layer  11  is provided is indicated by a bold line, while the case where no insulating layer  11  is provided is indicated by a thin line. 
     As presented in  FIG. 8A , in the entire range of the cut angle of the piezoelectric layer  12 , the absolute value of the difference in TCF when the insulating layer  11  is provided is less than the absolute value of the difference in TCF when no insulating layer  11  is provided. In a range of the cut angle of the piezoelectric layer  12  of 80° or greater and 120° or less, the difference in TCF is approximately zero in the case where the insulating layer  11  is provided, while the difference in TCF is not zero but is approximately 2 to 5 ppm/Ka in the case where no insulating layer  11  is provided. 
     As presented in  FIG. 8B , in the entire range of the cut angle of the piezoelectric layer  12 , the electromechanical coupling coefficient when the insulating layer  11  is provided is less than the electromechanical coupling coefficient when no insulating layer  11  is provided, but in a range of the cut angle of the piezoelectric layer  12  of 30° or greater and 90° or less, the difference in electromechanical coupling coefficient is small. 
     As described above, in the first embodiment, to achieve the small difference in TCF and reduce the loss and spurious emissions, the distance between the upper surface of the piezoelectric layer  12  and the lower surface of the insulating layer  11  is adjusted to be 2λ or less. In other words, the average pitch of the electrode fingers  15  of one of the comb-shaped electrodes  18  is adjusted to be equal to or greater than ½ of the distance between the upper surface of the piezoelectric layer  12  and the lower surface of the insulating layer  11 . The average pitch of the electrode fingers  15  of one of the comb-shaped electrodes  18  can be calculated by dividing the length in the X direction of one of the comb-shaped electrodes  18  by the number of the electrode fingers  15  of one of the comb-shaped electrodes  18 . The average pitch of the electrode fingers  15  of one of the comb-shaped electrodes  18  may be the value calculated by dividing the length in the X direction of the IDT  22  by the number of pairs of the electrode fingers  15  (½ of the number of the electrode fingers  15 ). In such a case, as illustrated in  FIG. 7 , the cut angle of the piezoelectric layer  12  is adjusted to be within a range of greater than 50° and less than 150°. This reduces the absolute value of the difference between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa of the acoustic wave resonator. As illustrated in  FIG. 8A , by providing the insulating layer  11 , the absolute value of the difference in TCF is adjusted to be less than the absolute value of the difference in TCF when no insulating layer  11  is provided. To cause the surface acoustic wave to propagate through the piezoelectric layer  12  and the insulating layer  11 , the average pitch of the electrode fingers  15  of one of the comb-shaped electrodes  18  is preferably equal to or less than 1/0.1 times the total thickness of the insulating layer  11  and the piezoelectric layer  12 , more preferably equal to or less than 1/0.5 times the total thickness of the insulating layer  11  and the piezoelectric layer  12 , further preferably equal to or less than 1 times the total thickness of the insulating layer  11  and the piezoelectric layer  12 . 
     The cut angle of the piezoelectric layer  12  expressed in an Euler&#39;s angle representation is (φ, θ, ω)=(0°, 140° to 240°, 0°), and may be (φ, θ, ψ)=(0°±5°, 140° to 240°, 0°±10°) in consideration of the production errors. 
     As illustrated in  FIG. 7 , to ensure the electromechanical coupling coefficient and reduce the difference in TCF, the cut angle of the piezoelectric layer  12  is preferably greater than 50° and 90° or less, more preferably 55° or greater and 90° or less, further preferably greater than 50° and 70° or less, yet further preferably 55° or greater and 70° or less. In addition, to ensure the electromechanical coupling coefficient and further reduce the difference in TCF, the cut angle of the piezoelectric layer  12  may be 60° or greater and 90° or less, may be 60° or greater and 80° or less, or may be 70° or greater and 90° or less. 
       FIG. 9A  and  FIG. 9B  are graphs of the difference between the temperature coefficient of the resonant frequency and the temperature coefficient of the antiresonant frequency versus the cut angle of the piezoelectric layer when the thickness of the insulating layer is varied. The horizontal axis represents the cut angle of the piezoelectric layer  12 , and the vertical axis represents the difference (fr−fa) between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa.  FIG. 9A  presents results of the simulation conducted under the conditions identical to those in  FIG. 6  and  FIG. 7  except in that the thickness of the piezoelectric layer  12  was fixed to 0.3λ, and the thickness of the insulating layer  11  was varied to 0.3λ, 0.5λ, 0.7λ, and 0.9λ.  FIG. 9B  presents results of the simulation conducted under the conditions identical to those in  FIG. 6  and  FIG. 7  except in that the thickness of the piezoelectric layer  12  was fixed to 0.5λ and the thickness of the insulating layer  11  was varied to 0.3λ, 0.5λ, 0.7λ, and 0.9λ. 
       FIG. 9A  and  FIG. 9B  reveal that there is an appropriate range of the cut angle of the piezoelectric layer  12  within which the absolute value of the difference in TCF is small regardless of the thickness of the insulating layer  11 . To reduce the absolute value of the difference in TCF when the cut angle of the piezoelectric layer  12  is greater than 50°, the thickness of the insulating layer  11  is preferably 0.1λ or greater and 0.7λ or less, more preferably 0.3λ or greater and 0.7λ or less, further preferably 0.5λ or greater and 0.7λ or less. In addition, since  FIG. 9A  presents the simulation results when the thickness of the piezoelectric layer  12  is 0.3λ and  FIG. 9B  presents the simulation results when the thickness of the piezoelectric layer  12  is 0.5λ, the thickness of the piezoelectric layer  12  is preferably 0.2λ or greater and 0.6λ or less, more preferably 0.3λ or greater and 0.5λ or less. 
       FIG. 10A  and  FIG. 10B  are graphs of the electromechanical coupling coefficient versus the cut angle of the piezoelectric layer when the thickness of the insulating layer is varied. The horizontal axis represents the cut angle of the piezoelectric layer  12 , and the vertical axis represents the electromechanical coupling coefficient.  FIG. 10A  presents results of the simulation conducted under the conditions identical to those in  FIG. 6  and  FIG. 7  except in that the thickness of the piezoelectric layer  12  was fixed to 0.3λ and the thickness of the insulating layer  11  was varied to 0.3λ, 0.5λ, 0.7λ, and 0.9λ.  FIG. 10B  presents results of the simulation conducted under the conditions identical to those in  FIG. 6  and  FIG. 7  except in that the thickness of the piezoelectric layer  12  was fixed to 0.5λ, and the thickness of the insulating layer  11  was varied to 0.3λ, 0.5λ, 0.7λ, and 0.9λ. 
       FIG. 10A  and  FIG. 10B  reveal that as the thickness of the insulating layer  11  increases, the electromechanical coupling coefficient decreases. Thus, to inhibit reduction in electromechanical coupling coefficient, the thickness of the insulating layer  11  is preferably 0.7λ or less, more preferably 0.5λ or less, further preferably 0.4λ or less. 
     The simulation results presented in  FIG. 6  to  FIG. 10B  reveal that the thickness of the insulating layer  11  is preferably 0.1λ or greater and 0.7λ or less and the thickness of the piezoelectric layer  12  is preferably 0.2λ or greater and 0.6λ or less to ensure the electromechanical coupling coefficient and reduce the difference in TCF. The thickness of the insulating layer  11  is more preferably 0.3λ or greater and 0.7λ or less, and the thickness of the piezoelectric layer  12  is more preferably 0.3λ or greater and 0.5λ or less. In other words, the thickness of the insulating layer  11  is preferably equal to or greater than 0.1 times the average pitch of the electrode fingers  15  of one of the comb-shaped electrodes  18  and equal to or less than 0.7 times the average pitch of the electrode fingers  15  of the one of the comb-shaped electrodes  18 , while the thickness of the piezoelectric layer  12  is preferably equal to or greater than 0.2 times the average pitch of the electrode fingers  15  of the one of the comb-shaped electrodes  18  and equal to or less than 0.6 times the average pitch of the electrode fingers  15  of the one of the comb-shaped electrodes  18 . The thickness of the insulating layer  11  is more preferably equal to or greater than 0.3 times the average pitch of the electrode fingers  15  of the one of the comb-shaped electrodes  18  and equal to or less than 0.7 times the average pitch of the electrode fingers  15  of the one of the comb-shaped electrodes  18 , while the thickness of the piezoelectric layer  12  is more preferably equal to or greater than 0.3 times the average pitch of the electrode fingers  15  of the one of the comb-shaped electrodes  18  and equal to or less than 0.5 times the average pitch of the electrode fingers  15  of the one of the comb-shaped electrodes  18 . 
     To ensure the electromechanical coupling coefficient and reduce the difference in TCF, the thickness of the insulating layer  11  may be 0.3λ or greater and 0.5λ or less, may be 0.4λ or greater and 0.7λ or less, or may be 0.5λ or greater and 0.7λ or less. The thickness of the piezoelectric layer  12  may be 0.3λ or greater and 0.4λ or less. 
     The simulation results presented in  FIG. 6  to  FIG. 10B  are results in the case where the insulating layer  11  is a silicon dioxide (SiO 2 ) layer, but it is considered that the same results are obtained when the insulating layer  11  is a fluorine-added silicon oxide (SiOF) layer or a nitrogen-added silicon oxide (SiON) layer containing silicon oxide (SiO). Therefore, the insulating layer  11  is not limited to a silicon dioxide layer, and may be any insulating layer containing silicon oxide. A layer containing silicon oxide means a layer containing Si and O at a concentration of 50 atomic % or greater in total, preferably a layer containing Si and O at a concentration of 80 atomic % or greater in total. 
     Second Embodiment 
       FIG. 11  is a cross-sectional view of an acoustic wave resonator in accordance with a second embodiment. As illustrated in  FIG. 11 , in an acoustic wave resonator  200  in accordance with the second embodiment, no insulating layer  11  is provided, and the piezoelectric layer  12  is bonded on the support substrate  10 . The piezoelectric layer  12  may be directly bonded on the support substrate  10  or indirectly bonded on the support substrate  10  with a bonding layer or other layers interposed therebetween. The remaining structure is the same as that of the first embodiment, and the description thereof is thus omitted. The acoustic wave resonator  200  of the second embodiment is formed using the method identical to the method for forming the acoustic wave resonator  100  of the first embodiment except in that a piezoelectric substrate is bonded on the support substrate  10  using the surface activation method. 
     As described in the first embodiment, the surface acoustic wave propagates through the region from the surface of the piezoelectric layer  12  to 2λ in depth. That is, when the thickness of the piezoelectric layer  12  is greater than 2λ, the region deeper than 2λ from the surface of the piezoelectric layer  12  is the region through which the surface acoustic wave does not propagate. Therefore, when no insulating layer  11  is interposed between the support substrate  10  and the piezoelectric layer  12 , the thickness of the piezoelectric layer  12  is adjusted to be 2λ or less to reduce the size of the acoustic wave resonator. In other words, the average pitch of the electrode fingers  15  of one of the comb-shaped electrodes  18  is adjusted to be equal to or greater than ½ of the thickness of the piezoelectric layer  12 . In this case, by adjusting the cut angle of the piezoelectric layer  12  to be within a range of greater than 50° and less than 150° as illustrated in  FIG. 8A , the absolute value of the difference between the temperature coefficient of the resonant frequency fr and the temperature coefficient of the antiresonant frequency fa of the acoustic wave resonator can be reduced. 
     Even when no insulating layer  11  is provided, as in the first embodiment, to ensure the electromechanical coupling coefficient and reduce the difference in TCF, the cut angle of the piezoelectric layer  12  is preferably greater than 50° and 90° or less, more preferably 55° or greater and 90° or less, further preferably greater than 50° and 70° or less, yet further preferably 55° or greater and 70° or less. In addition, to ensure the electromechanical coupling coefficient and further reduce the difference in TCF, the cut angle of the piezoelectric layer  12  may be 60° or greater and 90° or less, may be 60° or greater and 80° or less, or may be 70° or greater and 90° or less. 
     To reduce the size of the acoustic wave resonator, the thickness of the piezoelectric layer  12  may be 1.8λ or less, may be 1.5λ or less, or may be 1.0λ or less. 
     Third Embodiment 
       FIG. 12  is a cross-sectional view of an acoustic wave resonator in accordance with a third embodiment. As illustrated in  FIG. 12 , in an acoustic wave resonator  300  in accordance with the third embodiment, a bonding layer  26  is interposed between the insulating layer  11  and the piezoelectric layer  12 . The bonding layer  26  is formed of, for example, alumina, silicon, or aluminum nitride, and has a thickness of, for example, 1 nm to 100 nm. Recesses and protrusions are formed on a boundary face  60  between the support substrate  10  and the insulating layer  11 . The recesses and protrusions may be formed regularly or formed irregularly. The arithmetic average roughness Ra of the boundary face  60  is, for example, 10 nm or greater and 1000 nm or less, more preferably 50 nm or greater and 500 nm or less, further preferably 100 nm or greater and 300 nm or less. The recesses and protrusions on the boundary face  60  are formed by removing the upper part of the support substrate  10  by etching or sandblasting using a mask layer formed on the support substrate  10  as a mask, and then forming the insulating layer  11  on the support substrate  10 . For example, when the support substrate  10  is a sapphire substrate, the upper part of the support substrate  10  may be removed by dry etching using chlorine-based gas. An etching liquid and an etching gas are appropriately selected according to the material of the support substrate  10 . The remaining structure is the same as that of the first embodiment, and the description thereof is thus omitted. 
     As in the third embodiment, the bond strength between the insulating layer  11  and the piezoelectric layer  12  is enhanced by providing the bonding layer  26  between the insulating layer  11  and the piezoelectric layer  12 . Formation of the recesses and protrusions on the boundary face  60  between the support substrate  10  and the insulating layer  11  allows the acoustic wave excited by the IDT  22  to be easily confined in the piezoelectric layer  12  and the insulating layer  11 . In addition, spurious emissions due to the bulk wave are reduced. 
     Fourth Embodiment 
       FIG. 13A  to  FIG. 13C  are cross-sectional views of acoustic wave resonators in accordance with a fourth embodiment, a first variation of the fourth embodiment, and a second variation of the fourth embodiment, respectively. As illustrated in  FIG. 13A , in an acoustic wave resonator  400  of the fourth embodiment, the bonding layer  26  is interposed between the insulating layer  11  and the piezoelectric layer  12 , and a boundary layer  27  is interposed between the support substrate  10  and the insulating layer  11 . The acoustic velocity of the bulk wave in the boundary layer  27  is greater than that of the bulk wave in the piezoelectric layer  12  and the insulating layer  11 . The boundary layer  27  is formed of, for example, alumina, aluminum nitride, silicon, silicon nitride, or silicon carbide. The thickness of the boundary layer  27  is, for example, 0.5λ or greater, further preferably 1.5λ or greater. The remaining structure is the same as that of the first embodiment, and the description thereof is thus omitted. 
     As in the fourth embodiment, when the boundary layer  27  that is a high acoustic velocity layer is interposed between the support substrate  10  and the insulating layer  11 , the acoustic wave excited by the IDT  22  is easily confined within the piezoelectric layer  12  and the insulating layer  11 . 
     As illustrated in  FIG. 13B , in an acoustic wave resonator  410  of the first variation of the fourth embodiment, recesses and protrusions are formed on a boundary face  62  between the support substrate  10  and the boundary layer  27 . The recesses and protrusions may be formed regularly or may be formed irregularly. The arithmetic average roughness Ra of the boundary face  62  is, for example, 10 nm or greater and 1000 nm or less, preferably 50 nm or greater and 500 nm or less, further preferably 100 nm or greater and 300 nm or less. The remaining structure is the same as that of the fourth embodiment, and the description thereof is thus omitted. Formation of recesses and protrusions on the boundary face  62  between the support substrate  10  and the boundary layer  27  reduces spurious emissions due to the bulk wave. 
     As illustrated in  FIG. 13C , in an acoustic wave resonator  420  in accordance with the second variation of the fourth embodiment, recesses and protrusions are formed on the boundary face  62  between the support substrate  10  and the boundary layer  27 , and recesses and protrusions are formed on the boundary face  64  between the boundary layer  27  and the insulating layer  11 . The recesses and protrusions may be formed regularly or may be formed irregularly. The arithmetic average roughness Ra of each of the boundary faces  62  and  64  is, for example, 10 nm or greater and 1000 nm or less, preferably 50 nm or greater and 500 nm or less, further preferably 100 nm or greater and 300 nm or less. The remaining structure is the same as that of the fourth embodiment, and the description thereof is thus omitted. Formation of recesses and protrusions on the boundary face  64  between the boundary layer  27  and the insulating layer  11  allows the acoustic wave excited by the IDT  22  to be easily confined in the piezoelectric layer  12  and the insulating layer  11 . In addition, spurious emissions due to the bulk wave are reduced. 
     Fifth Embodiment 
       FIG. 14  is a circuit diagram of a filter in accordance with a fifth embodiment. As illustrated in  FIG. 14 , a filter  500  of the fifth embodiment includes one or more series resonators S 1  to S 3  connected in series between an input terminal Tin and an output terminal Tout. One or more parallel resonators P 1  and P 2  are connected in parallel between the input terminal Tin and the output terminal Tout. Any one of the acoustic wave resonators of the first embodiment to the second variation of the fourth embodiment may be used for at least one of one or more series resonators S 1  to S 3  and one or more parallel resonators P 1  and P 2 . The number of resonators of the ladder-type filter may be appropriately selected. 
     Sixth Embodiment 
       FIG. 15  is a circuit diagram of a duplexer in accordance with a sixth embodiment. As illustrated in  FIG. 15 , in a duplexer  600  of the sixth embodiment, a first end of a transmit filter  70  is connected to a common terminal Ant, and a second end of the transmit filter  70  is connected to the transmit terminal Tx. A first end of a receive filter  72  is connected to the common terminal Ant, and a second end of the receive filter  72  is connected to a receive terminal Rx. The transmit filter  70  transmits signals in the transmit band to the common terminal Ant as a transmission signal among high-frequency signals input from the transmit terminal Tx, and suppresses signals with other frequencies. The receive filter  72  transmits signals in the receive band to the receive terminal Rx as a reception signal among high-frequency signals input from the common terminal Ant, and suppresses signals with other frequencies. At least one of the transmit filter  70  and the receive filter  72  may be the filter of the fifth embodiment. A duplexer is described as an example of the multiplexer, but the multiplexer may be a triplexer or a quadplexer. 
     Although the embodiments of the present invention have been described in detail, the present invention is not limited to such a specific embodiment, and it is to be understood that the various change, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.