Abstract:
An acoustic wave element comprising a lithium tantalate substrate having the Euler angles (φ, θ, ψ), a first component φ satisfying 10°≦φ≦50°; and an electrode disposed on the lithium tantalate substrate and configured to excite a main acoustic wave of wavelength λ, the electrode having a density ρM satisfying ρM≧ρTi where ρTi represents a density of titanium (Ti), and a thickness hM of the electrode satisfies 0.141×exp(0.075ρM)λ≦hM≦0.134λ Embodiments of the present disclosure minimize a thickness of the electrode and suppress a spurious Rayleigh wave signal.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority under 35 U.S.C. §119(e) to U.S. Provisional Patent Application Ser. No. 62/132,046, titled “ACOUSTIC WAVE ELEMENTS, ANTENNA DUPLEXERS AND ELECTRONIC DEVICES,” filed on Mar. 12, 2015, which is hereby incorporated herein by reference in its entirety. 
     
    
     TECHNICAL FIELD 
       [0002]    Aspects and embodiments of the present disclosure generally relate to an acoustic wave element, an antenna duplexer or a diplexer including the acoustic wave element, and an electronic device including the acoustic wave element, the antenna duplexer, or the diplexer. 
       BACKGROUND 
       [0003]    Conventionally, an acoustic wave element in which non-leaky shear horizontal (SH) waves propagate along a lithium tantalate (LiTaO 3 ) substrate has been used as a filter and an antenna duplexer for a mobile communication device and the like. In such an acoustic wave element, the SH waves are excited by configuring the velocities of the SH waves to be less than those of the slow shear waves (SSW); see for example, US 2007/0090898(A1). 
       SUMMARY OF THE INVENTION 
       [0004]    However, the acoustic wave element in which the non-leaky SH waves propagate along the lithium tantalate substrate requires an increased thickness of an interdigitated transducer (IDT) electrode. Specifically, the thickness of the IDT electrode is more than double that of an IDT electrode of an acoustic wave element that allows leaky SH waves to propagate. For example, when molybdenum (Mo) is used for the material of an IDT electrode, the ratio h M /λ of the thickness h M  of the IDT electrode to the wavelength λ of the SH wave may amount to greater than 9.5%. Such an increased thickness of the IDT electrode would lead to processing limitations in lithography and other processes and cause difficulties in manufacturing. Further, a cut angle for the rotational Y axis of lithium tantalate should be suppressed to less than common 42° in order to ensure a certain electromechanical coupling coefficient k 2 . Such a smaller cut angle would cause a spurious signal due to a Rayleigh wave. 
         [0005]    In view of the aforementioned circumstances, embodiments of the present disclosure provide an acoustic wave element in which non-leaky SH waves propagate along a lithium tantalate substrate, but the thickness of an IDT electrode can be minimized and a spurious Rayleigh wave signal may be suppressed, as well as an antenna duplexer or a diplexer, and an electronic device using the acoustic wave element, the antenna duplexer, or the diplexer. 
         [0006]    In accordance with an embodiment of the present disclosure, the acoustic wave element comprises a lithium tantalate substrate having Euler angles (φ, θ, ψ), the first component ψ satisfying 10°≦φ≦═°; and an electrode disposed on the lithium tantalate substrate and configured to excite a main acoustic wave having a wavelength λ the electrode having a density ρ M  satisfying ρ M ≧ρ Ti  represents a density of titanium and having a thickness h M  satisfying 0.141×exp(−0.075ρ M )λ≦h M  ≦0.134λ. 
         [0007]    In one embodiment, angle θ may satisfy −90°−0.5 ×(−0.2234ρ M   2 +6.9119ρ M −8.928)°≦θ≦−90°+0.5 ×(−0.2234ρ M   2 +6.9119ρ M −8.928) °. Angle ψ may satisfy −16°≦ψ≦−2.5°. 
         [0008]    In accordance with embodiments of the present disclosure, the lithium tantalate substrate and the electrode may be covered thereabove with an insulation layer having a temperature coefficient opposite to that of the lithium tantalate substrate. The insulation layer may consist of silicon dioxide. A thickness h s  of the insulation layer may satisfy 0.08λ≦h s ≦0.55λ. 
         [0009]    In accordance with some embodiments, the insulation layer may have a protrusion thereabove in a cross section taken along a direction perpendicular to an extending direction of electrode fingers of the electrode. A height h T  of the protrusion in the insulation layer may also satisfy 0≦h T ≦h M , where h M  is a thickness of the electrode. In accordance with some embodiments, angle ψ may satisfy (−371.81h S   2 +36.92h S +3.53)°≦ψ≦(−371.81h S   2 +36.92h S +13.53)°. 
         [0010]    An antenna duplexer according to embodiments of the present disclosure includes a reception filter and a transmission filter, at least one of which includes the acoustic wave element. A first frequency and a second frequency may pass respectively through the reception filter and the transmission filter. A diplexer according to embodiments of the present disclosure includes a first reception filter and a second reception filter, at least one of which includes the acoustic wave element. The first reception first reception filter can be configured to receive a first frequency band and the second reception filter may be configured to a second frequency band that is different than the first frequency band. An electronic device according to embodiments of the present disclosure includes the acoustic wave element, a semiconductor element connected to the acoustic wave element, and a reproduction device connected to the semiconductor element. 
         [0011]    According to embodiments of the present disclosure, it is possible to minimize a thickness of an IDT electrode and also to suppress a spurious Rayleigh wave signal. Further, it is possible to improve the frequency characteristic and to downsize the device. 
         [0012]    Still other aspects, embodiments, and advantages of these exemplary aspects and embodiments are discussed in detail below. Embodiments disclosed herein may be combined with other embodiments in any manner consistent with at least one of the principles disclosed herein, and references to “an embodiment,” “some embodiments,” “an alternate embodiment,” “various embodiments,” “one embodiment” or the like are not necessarily mutually exclusive and are intended to indicate that a particular feature, structure, or characteristic described may be included in at least one embodiment. The appearances of such terms herein are not necessarily all referring to the same embodiment. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0013]    Various aspects of at least one embodiment are discussed below with reference to the accompanying figures, which are not intended to be drawn to scale. The figures are included to provide illustration and a further understanding of the various aspects and embodiments, and are incorporated in and constitute a part of this specification, but are not intended as a definition of the limits of the invention. In the figures, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every figure. 
           [0014]      FIGS. 1A and 1B  show a schematic structure of an acoustic wave element according to an embodiment of the present disclosure. 
           [0015]      FIG. 2  is a graph showing a relationship between a lower limit of a normalized thickness and a density of an IDT electrode. 
           [0016]      FIG. 3A  is a graph showing relationships between a phase velocity and angle φ for a slow shear wave (SSW) and a shear horizontal (SH) wave, and  FIGS. 3B and 3C  are graphs showing frequency characteristics of an acoustic wave element according to the embodiment of  FIGS. 1A and 1B . 
           [0017]      FIG. 4  shows a relationship between a normalized electromechanical coupling coefficient k 2  and angle θ. 
           [0018]      FIG. 5  is a graph showing a relationship between an angular difference θ high −θ low  and a density of an IDT electrode. 
           [0019]      FIGS. 6A and 6B  are graphs respectively showing dependencies of angle ψ on angle θ when resonant and antiresonant power flow angles are zero. 
           [0020]      FIG. 7A  is a block diagram schematically showing a configuration of an antenna duplexer that includes an acoustic wave element such as described with respect to  FIGS. 1A and 1B , and  FIG. 7B  is a block diagram schematically showing a configuration of a diplexer that includes an acoustic wave element such as described with respect to  FIGS. 1A and 1B  . 
           [0021]      FIG. 8  is a block diagram schematically showing a configuration of an electronic device that includes an acoustic wave element such as described with respect to  FIGS. 1A and 1B . 
           [0022]      FIGS. 9A and 9B  show a schematic structure of an acoustic wave element according to another embodiment of the present disclosure. 
           [0023]      FIG. 10A  is a graph showing a comparison result between an acoustic wave element according to the embodiment of  FIGS. 9A and 9B  and an acoustic wave element of the prior art in a relationship between a normalized thickness and a thermal coefficient of frequency of a silicon dioxide film, and  FIGS. 10B and 10C  show schematic structures of acoustic wave elements according to two variations of the embodiment of  FIGS. 9A and 9B . 
           [0024]      FIG. 11  is a graph showing a frequency characteristic of an acoustic wave element according to the embodiment of  FIGS. 9A and 9B . 
           [0025]      FIG. 12  is a graph showing a relationship between a spurious Rayleigh wave signal frequency and a thickness of a silicon dioxide film. 
           [0026]      FIG. 13A  is a graph showing a relationship between angle ψ and an intensity of a spurious Rayleigh wave signal and  FIG. 13B  is a graph showing a frequency characteristic of an acoustic wave element according to the embodiment of  FIGS. 9A and 9B . 
           [0027]      FIG. 14  is a graph showing a relationship between a thickness of a silicon dioxide film and a central value for a specific range of angle ψ at which a spurious Rayleigh wave signal is suppressed. 
           [0028]      FIG. 15A  is a block diagram of one example of a module including an embodiment of the acoustic wave element according to aspects of the present disclosure,  FIG. 15B  is a block diagram of one example of a module including an embodiment of a duplexer according to aspects of the present disclosure, and  FIG. 15C  is a block diagram of one example of a module including an embodiment of a diplexer according to aspects of the present disclosure. 
           [0029]      FIG. 16  is a block diagram of one example of an electronic device including an embodiment of the acoustic wave element according to aspects of the present disclosure. 
           [0030]      FIG. 17  is a block diagram of another example of an electronic device including an embodiment of the antenna duplexer of  FIG. 7A  according to aspects of the present disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0031]    Aspects and embodiments of acoustic wave elements are described below with reference to the accompanying drawings. It is to be appreciated that embodiments of the methods and apparatuses discussed herein are not limited in application to the details of construction and the arrangement of components set forth in the following description or illustrated in the accompanying drawings. The methods and apparatuses are capable of implementation in other embodiments and of being practiced or of being carried out in various ways. Examples of specific implementations are provided herein for illustrative purposes only and are not intended to be limiting. Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use herein of “including,” “comprising,” “having,” “containing,” “involving,” and variations thereof is meant to encompass the items listed thereafter and equivalents thereof as well as additional items. References to “or” may be construed as inclusive so that any terms described using “or” may indicate any of a single, more than one, and all of the described terms. Any references to front and back, left and right, top and bottom, upper and lower, and vertical and horizontal are intended for convenience of description, not to limit the present systems and methods or their components to any one positional or spatial orientation. 
         [0032]      FIGS. 1A and 1B  generally show a schematic structure of an acoustic wave element  10  according to an embodiment of the present disclosure.  FIG. 1A  shows a top view of the acoustic wave element  10 , and  FIG. 1B  shows a cross-sectional view of the acoustic wave element  10  taken along line PP′ in  FIG. 1A . 
         [0033]    The acoustic wave element  10  includes a substrate  11  fabricated by cutting a single crystal of lithium tantalate (LiTaO 3 ) along a plane defined by the Euler angles (ψ, θ, φ). A non-leaky SH wave may propagate on the cut surface of the substrate  11  as a main acoustic wave. The surface of the substrate  11  is provided with interdigitated transducer (IDT) electrodes  12  thereon. Each of the IDT electrodes  12  includes electrode fingers made of a certain metal material and extending in a certain direction for exciting a shear horizontal (SH) wave having a wavelength λ. 
         [0034]    The embodiment illustrated in  FIGS. 1A and 1B  uses a metal material for the IDT electrodes  12  having a density of metal ρ M  greater than the density of titanium ρ T . Therefore, the density of metal ρ M  of the IDT electrodes  12  may be defined by the following equation (1) with respect to the density of titanium ρ Ti : 
         [0000]      ρ M ≧ρ Ti   (1)
 
         [0035]    The IDT electrodes  12  being made of a metal having a density that is greater than or equal to the density of titanium ρ Ti  enables one to reduce a thickness h M  thereof. Therefore, the IDT electrodes  12  may be formed by lithography and other processes. 
         [0036]    In accordance with the embodiment illustrated in  FIGS. 1A and 1B , the thickness h M  of the IDT electrodes  12  may be defined by the following equation (2) with respect to the wavelength λ of an ST wave excited by the IDT electrodes  12 : 
         [0000]      0.141×exp(−0.075ρ M ≦0.134λ  (2)
 
         [0037]    If the thickness h M  of the IDT electrodes  12  is normalized by the wavelength λ of an SH wave, then a normalized thickness h would be introduced as the following equation (3): 
         [0000]        h=h   M /λ  (3)
 
         [0038]    Using the normalized thickness h, the aforementioned equation (2) would be expressed as the following equation (4): 
         [0000]      0.141×exp(−0.075)≦ h≦ 0.134   (4)
 
         [0039]    Equation (4) is obtained as follows. 
         [0040]    The embodiment illustrated in  FIGS. 1A and 1B  may set the upper limit as defined in equation (4), i.e., h≦0.134, to the normalized thickness h. The upper limit setting may ensure the processing of the IDT electrodes  12  to be formed. Therefore, processing defects such as insufficient etching would be prevented in the forming processes. 
         [0041]      FIG. 2  is a graph showing a relationship between a lower limit of a normalized thickness h of the IDT electrodes  12  required to excite an SH wave and a density of metal ρ M  constituting the IDT electrodes  12  for respective metal materials. Here, because the slow shear wave (SSW) propagates fastest, the thickness of the IDT electrode  12  would be least when an upper limit of the velocity of the shear horizontal (SH) wave to be excited is maximized In order to realize the aforementioned least thickness, an excitation condition for the SH wave can be obtained where angle θ of the second component of the Euler angles is −90° and angle φ of the third component is 30°. 
         [0042]    In  FIG. 2 , the values of normalized thickness h are plotted respectively for titanium (Ti), molybdenum (Mo), tungsten (W) and platinum (Pt) when the metal materials are used for the IDT electrodes  12  under the excitation condition. The relationship between the density of metal ρ M  for the IDT electrodes  12  and the lower limit of a normalized thickness h necessary for the IDT electrodes  12  to excite the SH wave can be obtained from the values for the respective materials as the following equation (5): 
         [0043]    y=14.102×exp(−0.075×) (5), where variables x and y correspond respectively to the horizontal axis and the vertical axis as shown in  FIG. 2 . 
         [0044]    The embodiment described with respect to  FIGS. 1A and 1B  sets the lower limit to the normalized thickness h of the IDT electrodes  12  according to equation (4), i.e., 0.141×exp(−0.075)≦h, so that the normalized thickness h can be ensured as the lower limit defined by equation (5) or greater. It is to be appreciated that the coefficient indicated in equation (5) is represented in percentage and accordingly is 100 times that of equation (4). Therefore, regardless of the density of the metal ρ M  of the IDT electrodes  12 , it is possible to excite an SH wave when the normalized thickness h is the lower limit or greater according to equation (4). For example, the lower limit of the normalized thickness h would be 1.5% in the IDT electrodes  12  made of molybdenum (Mo). It is to be appreciated that it is possible to further deposit an aluminum (Al) layer on the IDT electrodes  12  as long as the total thickness of the IDT electrodes  12  including the Al layer satisfies the aforementioned equation (4). 
         [0045]    Angle φ of the first component of the Euler angles (φ, θ, ψ) for defining the cut angle according to the embodiment described above with respect to  FIGS. 1A and 1B  is defined by the following equation (6): 
         [0000]      10°≦φ≦50°  (6)
 
         [0046]    Equation (6) is obtained as follows. 
         [0047]      FIG. 3A  is a graph showing relationships between a phase velocity and angle φ for a SSW and a SH wave, where the thickness of the IDT electrodes  12  is set to the upper limit when titanium (Ti) is used for the material thereof. In  FIG. 3A , the curve labelled “a” represents the phase velocity of an SSW whereas the curve labelled “b” represents that of an SH wave. It is to be appreciated that the phase velocity is referred to merely as the velocity hereinafter. 
         [0048]    A non-leaky SH wave is excited when the velocity is less than that of the slow shear wave (SSW). The non-leaky SH wave is excited when the velocity of the SH wave is that of the SSW or less in a range where angle φ satisfies equation (6). In other words, equation (6) is obtained from the condition of the non-leaky SH wave to be excited. 
         [0049]      FIG. 3B  shows a frequency characteristic when angle φ=10°, whereas  FIG. 3C  shows a frequency characteristic when angle φ=30°. The vertical axis in  FIGS. 3B  and C represents a forward admittance Y 21 . When φ=10°, the antiresonant characteristic can be seen as degraded due to a leakage of an SH wave into a bulk wave. Further, a spurious Rayleigh wave signal can be seen near the resonant frequency. 
         [0050]    On the other hand, when φ=30°, sharp resonant and antiresonant characteristics can be seen for the SH wave due to the excitation of a main mode SH wave. The spurious Rayleigh wave signal when φ=10° is suppressed and shifted to the lower frequency side out of the passband. 
         [0051]    Angle θ of the second component of the Euler angles (φ, θ, ψ) according to the above-described embodiment of  FIGS. 1A and 1B  is defined by the following equation (7): 
         [0052]    −90°−0.5×(−0.2234ρ M   2 +6.9119ρ M −8.928)°≦θ≦−90°+0.5×(−0.2234ρ M   2 +6.9119ρ M −8.928)°(7), where ρ M  is a density of metal for the material of the IDT electrodes  12 . 
         [0053]    Equation (7) is obtained as follows. 
         [0054]    For example, when the material of the IDT electrodes  12  is tungsten (W), angle 0 of the second component of the Euler angles is defined as the following equation (8): 
         [0000]      −111°≦θ≦−69°  (8)
 
         [0055]      FIG. 4  shows a relationship between a normalized electromechanical coupling coefficient k 2  and angle 0. The normalized electromechanical coupling coefficient k 2  is derived from the normalization by an electromechanical coupling coefficient for a conventional lithium tantalate substrate that is cut on 42° rotated Y axis and uses aluminum (Al) for the IDT electrodes. 
         [0056]    In  FIG. 4 , the curves labelled “a,” “b,” “c ,” and “d” follow polynomials corresponding respectively to titanium (Ti), molybdenum (Mo), tungsten (W) and platinum (Pt). In  FIG. 4 , the portion of each curve on which the electromechanical coupling coefficient is 1 or greater, i.e., the portion from the lower limit θ low  to the upper limit θ high , is a region for obtaining an electromechanical coupling coefficient that can be equal to or greater than what is achieved by the conventional lithium tantalate substrate. This region is defined by the following equation (9): 
         [0000]      θ low θ high    (9)
 
         [0057]      FIG. 5  is a graph an angular difference θ high −θ low  and a density of metal ρ M  for the IDT electrodes  12 . Specifically, angular differences θ high −θ low  are plotted in  FIG. 5  for titanium (Ti), molybdenum (Mo), tungsten (W) and platinum (Pt) with respect to corresponding densities of metal ρ M  for the IDT electrodes  12 . 
         [0058]    The relationship between the angular difference θ high −θ low  and the density of metal ρ M  for the IDT electrodes  12  may exist along a curve according to the following equation (10) that is generally fitted to the values for the respective materials: 
         [0059]    y=−0.2234x 2 +6.9119x−8.928 (10), where the variables x and y correspond respectively to the horizontal axis and the vertical axis as shown in  FIG. 5 . 
         [0060]    Aforementioned equation (7) is derived from the combination of equation (10) with equation (9) considering that the angular difference θ high −θ low  ranges symmetrically with respect to angle θ=−90°. When angle θ is defined by equation (7), an electromechanical coupling coefficient is ensured to be equal to or greater than that of the conventional lithium tantalate substrate that is cut on 42° rotated Y axis and uses aluminum (Al) for the IDT electrodes. 
         [0061]    Angle ψ of the third component of the Euler angles (φ, θ, ψ) according to the above-described embodiment of  FIGS. 1A and 1B  is defined by the following equation (11): 
         [0000]      −16°≦ψ≦−2.5°  (11)
 
         [0000]    Equation (11) is obtained as follows. 
         [0062]      FIGS. 6A and 6B  are graphs respectively showing dependencies of angle ψ on angle θ when resonant and antiresonant power flow angles are zero for the IDT electrodes  12  made of tungsten (W). 
         [0063]      FIG. 6A  shows angle ψ when a resonant power flow angle is zero. Angle φ ranges according to aforementioned equation (6) so that angle φ can range symmetrically with respect to angle φ=30° as clearly shown in the curves labelled “a” and “b” of  FIG. 3A . Accordingly,  FIG. 6A  shows angle ψ when the resonant power flow angle is zero along curves a, b, and c corresponding respectively to angles φ=10°, 20° and 30°. 
         [0064]      FIG. 6B  shows angle ψ when an antiresonant power flow angle is zero. Similar to  FIG. 6A ,  FIG. 6B  also shows angle ψ when the antiresonant power flow angle is zero along curves a, b, and c corresponding respectively to angles φ=10°, 20° and 30°. When the IDT electrodes  12  are made of tungsten (W), the range of angle θ is defined according to aforementioned equation (8). In the range of angle θ, as shown in  FIGS. 6A and 6B , the range of angle ψ where a resonant or antiresonant power flow angle can be zero includes the lower limit of angle ψ=−16° when θ=111° in  FIG. 6A  and the upper limit of angle ψ=−2.5° when θ=−90° in  FIG. 6B . Accordingly, equation (11) is obtained. 
         [0065]    Therefore, defining angle ψ by equation (11) enables at least one of the resonant power flow angle and the antiresonant power flow angle to be zero when the IDT electrodes  12  are made of tungsten (W). This can reduce an energy loss originated from a power flow angle for the acoustic wave element 10. Here, the lower limit of the range of angle ψ where a power flow angle is zero is represented by values of angle w at the lower limit of angle θ as shown in  FIG. 6A . Therefore, considering that angle θ depends on a density of metal ρ M  as shown in aforementioned equation (7), the lower limit of the range of angle w as shown in equation (11) would depend on a density of the IDT electrodes  12 . 
         [0066]      FIG. 7A  shows a configuration of an antenna duplexer that includes an acoustic wave element such as described above with respect to  FIGS. 1A and 1B . The antenna duplexer  60  can be configured to include a reception filter  61  and a transmission filter  62 , each of which can include an acoustic wave element  10  as described above so that a first frequency and a second frequency may pass respectively through the reception filter and the transmission filter. As shown in  FIG. 7A , the duplexer  60  can include a common terminal  67  that can be used as an antenna terminal that can be coupled to an antenna  68 , a first terminal  65  that can be used as a transmission terminal that can be coupled to transmission circuitry  66 , and a second terminal  63  that can be used as a reception terminal that can be coupled to reception circuitry  64 . The transmission circuitry  66  and the reception circuitry may be disposed in the same module, or in the same package as the antenna duplexer  60 , or they may be disposed in a module or package that is external to the antenna duplexer  60 . 
         [0067]    According to another embodiment, the acoustic wave element  10  may be incorporated into a filter device such as a diplexer that is used in separating reception signals having two frequency bands. An example of a diplexer  70  incorporating an example of an acoustic wave element  10  is shown schematically in  FIG. 7B . In this example, the diplexer  70  can include a common terminal  77  that can be connected to an antenna  78  to receive signals having different frequencies. The diplexer  70  includes a first reception filter  71   a  and a second reception filter  71   b , each of which is connected to the common terminal  77 . In accordance with an aspect of the present disclosure, the acoustic wave element  10  described above can be used in the first reception filter  71   a  and/or the second reception filter  71   b . The diplexer  70  further includes a first terminal  73   a  that can be used as a first reception terminal, and a second terminal  73   b  that can be used as a second reception terminal The first terminal  73   a  can be coupled to first reception circuitry  74   a  configured to receive a first frequency band, and the second terminal  73   b  can be coupled to second reception circuitry  74   b  configured to receive a second frequency band, different than the first frequency band. 
         [0068]      FIG. 8  shows a configuration of an electronic device that includes an acoustic wave element such as described above with respect to  FIGS. 1A and 1B . As shown, the electronic device  100  includes an acoustic wave element  10  as described above with respect to  FIGS. 1A and 1B , a semiconductor element  80  connected to the acoustic wave element  10 , and a reproduction device  90  connected to the semiconductor element  80 . 
         [0069]    As discussed above, in accordance with the embodiment described above with respect to  FIGS. 1A and 1B , it is possible to excite a non-leaky SH wave while minimizing the thickness h M  of the IDT electrodes  12 , the range of which is defined by aforementioned equation (2). Therefore, one can ensure the formation of the IDT electrodes  12  using conventional lithographic and other semiconductor processing techniques. Further, it is possible to suppress and shift out of the pass band a spurious Rayleigh wave signal because angle φ is defined by aforementioned equation (6). Therefore, an improved filtering function can be achieved in the acoustic wave element  10 , the reception filter  61  and/or the transmission filter  62  of the antenna duplexer  60 , the first reception filter  71   a  or the second reception filter  71   b  of the diplexer  70 , or the electronic device  100 . 
         [0070]    In accordance with a further aspect of the present disclosure, an acoustic wave element, an antenna duplexer, a diplexer, and an electronic device according to another embodiment is now described. 
         [0071]      FIGS. 9A and 9B  generally show a schematic structure of an acoustic wave element  20  according to another embodiment of the present disclosure. Specifically,  FIG. 9A  shows a top view of the acoustic wave element  20  and  FIG. 9B  shows a cross-sectional view of the acoustic wave element  20  taken along line PP′ in  FIG. 9A . 
         [0072]    The acoustic wave element  20  is configured similarly to the acoustic wave element  10  described above with respect to  FIGS. 1A and 1B  in that a substrate  21  is fabricated by cutting a single crystal of lithium tantalate along a plane defined by the Euler angles (φ, θ, ψ), and IDT electrodes  22 , each of which may excite an acoustic wave having a wavelength λ, are provided on a surface of the substrate  21 . However, in contrast to the acoustic wave element  10  described above, the acoustic wave element  20  of embodiment  2  further includes a silicon dioxide (SiO 2 ) film  25  having a thickness h S  that is formed on the surface of the substrate  21 , on which the IDT electrodes  22  are formed. It is to be appreciated that because the silicon dioxide film  25  may be transparent, the silicon dioxide film  25  is not shown in the top view of  FIG. 9A , whereas it is shown in the cross-sectional view of  FIG. 9B . 
         [0073]    The silicon dioxide film  25  has a thickness h s  generally greater than a thickness h M  of the IDT electrodes  22  from the surface of the substrate  21 . Further, the silicon dioxide film  25  includes protrusions  27  formed on a surface  26  thereof. The protrusions  27  have a height h T  and are disposed immediately above the electrode fingers of the IDT electrodes  22  having the thickness h S . The protrusions  27  protrude above electrode fingers of the IDT electrode  22 . As shown in  FIG. 9B , the protrusions  27  appear in the cross-section in a direction perpendicular to the direction in which the electrode fingers of the IDT electrode  22  extend. 
         [0074]    In accordance with an aspect of the present disclosure, the thickness h S  of the silicon dioxide film  25  is defined by the following equation (12) using a wavelength λ of the IDT electrodes  22 : 
         [0000]      0.08λ≦h S ≦0.55λ  (12)
 
         [0000]    Equation (12) is derived as follows. 
         [0075]      FIG. 10A  is a graph showing a comparison result between an acoustic wave element according to the embodiment illustrated in  FIGS. 9A and 9B  and that of the prior art in a relationship between a normalized thickness h S /λ, which is derived from the thickness h S  of the silicon dioxide film  25  normalized by the wavelength λ, and a temperature coefficient of frequency (TCF). In  FIG. 10A , the curve labelled “a” shows a TCF when the thickness h T  of the protrusions  27  is configured to be zero as shown in  FIG. 10B . The curve labelled “b” shows a TCF when the thickness h T  of the protrusions  27  is configured to be the thickness h M  of the IDT electrodes  22  as shown in  FIG. 10C . The line labelled “c” shows a conventional typical value of TCF, i.e., −33 ppm/°C., when the IDT electrodes  22  are made of aluminum (Al). 
         [0076]    As clearly shown in  FIG. 10A , the TCF ranges from −33 ppm/°C. at 8% of h S /λ to 0 ppm/°C. at 55% of h S /λ so that the thickness h S  of the silicon dioxide film  25  can be defined by equation (12). Therefore, according to equation (12), a temperature characteristic of frequency better than that of the prior art can be realized. 
         [0077]    In accordance with this embodiment, a height h T  of a protrusion  27  of the silicon dioxide film  25  is defined by following equation (13) using the thickness h M  of the IDT electrode  22 : 
         [0000]      0≦h T ≦h M    (13)
 
         [0078]    When the surface  26  of the silicon dioxide film  25  is configured to be flat, the thickness h T  of a protrusion  27  is minimized to be zero. Further, the height h T  of a protrusion  27  is maximized to the thickness h M  of the IDT electrode  22 . Therefore, the height h T  of a protrusion  27  of the silicon dioxide film  25  is limited to the range defined by equation (13). 
         [0079]    According to the embodiment described with respect to  FIGS. 9A and 9B , angle ψ of the third component of the Euler angles is defined by the following equation (14): 
         [0000]      (−371.81 h   S   2 +36.92 h   S +3.53)°≦ψ≦(−371.81 h   S   2 +36.92 h   S +13.53)°  (14)
 
         [0000]    Equation (14) is obtained as follows. 
         [0080]      FIG. 11  is a graph showing a frequency characteristic of an acoustic wave element  20  according to the embodiment described with respect to  FIGS. 9A and 9B .  FIG. 12  is a graph showing a relationship between a spurious Rayleigh wave signal frequency and a thickness of a silicon dioxide film  25 . As shown in  FIG. 12 , a normalized frequency of a spurious Rayleigh wave signal becomes greater as the thickness h S  of the silicon dioxide film  25  becomes greater. Accordingly, it can be seen from  FIGS. 11 and 12  that a spurious Rayleigh wave signal designated by the letter “a” in  FIG. 11  approaches the resonant frequency and the antiresonant frequency of a main mode SH wave as the thickness h S  of the silicon dioxide film  25  becomes greater. 
         [0081]      FIG. 13A  is a graph showing a relationship between angle ψ and an intensity of a spurious Rayleigh wave signal and  FIG. 13B  is a graph showing a frequency characteristic of an acoustic wave element  20  according to the embodiment of  FIGS. 9A and 9B . As shown in  FIG. 13A , a normalized spurious signal intensity (I/I min ) −1  may be suppressed to zero or close to zero within a specific range of angle w as designated by the letter “a.” Within the specific range as shown in  FIG. 13A , there is no spurious Rayleigh wave signal near the resonant and antiresonant frequencies as shown in the frequency characteristic of  FIG. 13B , where the vertical axis designates an admittance characteristic based on the resonance frequency. 
         [0082]      FIG. 14  is a graph showing a relationship between a thickness h S  of the silicon dioxide film  25  and a central value for a specific range of angle ψ within which a spurious Rayleigh wave signal is suppressed. The specific range corresponds to a specific range as designated by the letter “a” in  FIG. 13A  where the normalized spurious signal intensity may be suppressed to zero or close to zero. Some central values for a thickness h S  of the silicon dioxide film  25  falling within the specific range are plotted in  FIG. 14 . The following equation (15) can be obtained from the values to represent the relationship between a thickness h S  of the silicon dioxide film  25  and a central value within the specific range: 
         [0083]    ψ=−371.81x 2 +36.92x+3.5256 (15), where the variables x and y correspond respectively to the horizontal axis and the vertical axis as shown in  FIG. 14 . 
         [0084]    Setting a certain width to central values within the specific range defined by equation (15) may define a specific angular range for angle ψ. For example, if the certain width is set as ±5°, then the angular range for angle ψ can be obtained as defined by aforementioned equation (14). It should be appreciated that the width is not limited to ±5° and can be appropriately determined based on a specific range, such as a range designated by the letter “a” in  FIG. 13A . Therefore, according to equation (14), suppressing a spurious Rayleigh wave signal near resonant and antiresonant frequencies can be realized so that the frequency characteristic is improved. 
         [0085]    It should be appreciated that an antenna duplexer, a diplexer, and/or an electronic device can be configured using the acoustic wave element  20  described above with respect to the embodiment of  FIGS. 9A and 9B  in the same manner as discussed above with respect to the acoustic wave element  10  of  FIGS. 1A and 1B . The antenna duplexer, the diplexer, and the electronic device are similar to those described above with respect to  FIGS. 7A, 7B, and 8  except for the use of acoustic wave element  20  instead of the acoustic wave element  10 . Further, although the silicon dioxide film  25  is formed on the substrate  21  and the IDT electrode  22  as shown in  FIGS. 9A and 9B , it may be possible to form an insulation layer using another suitable material instead of silicon dioxide. Suitable materials for such an insulation lay may be selected from those having a temperature coefficient of frequency that is opposite to that of the lithium tantalate substrate. 
         [0086]    It should be appreciated that embodiments of the present disclosure can be applied to a mobile communication device and the like. For example, embodiments of the acoustic wave element  10  or  20 , a duplexer  60  or a diplexer  70  including the acoustic wave element  10 ,  20  may be incorporated into and packaged as a module that may ultimately be used in an electronic device  100 , such as a wireless communications device, for example.  FIG. 15A  is a block diagram illustrating one example of a module  200  including the acoustic wave element  10 ,  20 . The module  200  further includes connectivity  202  to provide signal interconnections, packaging  204 , such as for example, a package substrate, for packaging of the circuitry, and other circuitry die  206 , such as, for example amplifiers, pre-filters, modulators, demodulators, down converters, and the like, as would be known to one of skill in the art of semiconductor fabrication in view of the disclosure herein.  FIGS. 15B and 15C  are block diagrams illustrating other examples of a module  210 ,  220  including an embodiment of the duplexer  60  or diplexer  70 , respectively, either of which may incorporate an embodiment of the acoustic wave element  10 ,  20 , as discussed above with reference to  FIGS. 7A and 7B . 
         [0087]    The acoustic wave element  10 ,  20 , duplexer  60 , diplexer  70 , or any of the modules  200 ,  210 ,  220 , according to examples and embodiments disclosed herein may be useful in a variety of electronic devices, such as communications or wireless devices (e.g., cell phones, tablets, etc.). 
         [0088]      FIG. 16  is a block diagram illustrating an example of an electronic device  300  that can include acoustic wave elements  10 ,  20  having one or more features discussed herein. For instance, the example electronic device  300  can include an acoustic wave element  10 ,  20 , duplexer  60 , or diplexer  70  in accordance with any of the principles and advantages discussed above with reference to any of  FIGS. 1A-14 . The example electronic device  300  can be a mobile phone, such as a smart phone. The electronic device  300  can include elements that are not illustrated in  FIG. 16  and/or a sub-combination of the illustrated elements. The electronic device  300  depicted in  FIG. 16  can represent a multi-band and/or multi-mode device such as a multi-band/multi-mode mobile phone. By way of example, the electronic device  300  can be a wireless device that communicates in accordance with Long Term Evolution (LTE). In this example, the electronic device  300  can be configured to operate at one or more frequency bands defined by an LTE standard. The electronic device  300  can alternatively or additionally be configured to communicate in accordance with one or more other communication standards, including but not limited to one or more of a Wi-Fi standard, a Bluetooth standard, a 3G standard, a 4G standard or an Advanced LTE standard. In certain embodiments, the electronic device  300  can include a filtering module  310  that includes one or more embodiments of the acoustic wave element  10 ,  20  and which is connected to circuits  320  and  330  via terminals  65  and  63 , respectively. The electronic device can further include an antenna  340  connected to the filtering module  310  via common terminal  67 . The filtering module  310  can include any of the modules  200 ,  210 , or  220  discussed above with reference to  FIGS. 15A-C . The circuits  320  and  330  can be reception or transmission circuits that can generate RF signals for transmission via the antenna  340  or receive incoming signals from the antenna  340 . 
         [0089]    Referring to  FIG. 17 , in one particular example, the filtering module  310  of  FIG. 16  includes the antenna duplexer module  210 . In this example, the electronic device  300  can include the antenna duplexer  60 , the transmission circuitry  66  connected to the antenna duplexer via input terminal  65 , the reception circuitry  64  connected to the antenna duplexer via output terminal  63 , and the antenna  340  connected to the antenna duplexer via antenna terminal  67 . The transmission circuitry  66  and reception circuitry  64  may be part of a transceiver that can generate RF signals for transmission via the antenna  340  and can receive incoming RF signals from the antenna  340 . 
         [0090]    As shown in  FIGS. 16 and 17 , the communication device  300  can further include a controller  350 , at least one computer readable medium  360 , at least one processor  370 , and a battery  380 . 
         [0091]    It will be understood that various functionalities associated with the transmission and receiving of RF signals can be achieved by one or more components that are represented in  FIG. 17  as the transmission circuitry  66  and the reception circuitry  64 . For example, a single component can be configured to provide both transmitting and receiving functionalities. In another example, transmitting and receiving functionalities can be provided by separate components Similarly, it will be understood that various antenna functionalities associated with the transmission and receiving of RF signals can be achieved by one or more components that are collectively represented in  FIGS. 16 and 17  as the antenna  340 . For example, a single antenna can be configured to provide both transmitting and receiving functionalities. In another example, transmitting and receiving functionalities can be provided by separate antennas. In yet another example in which the communication device is a multi-band device, different bands associated with the communication device  300  can be provided with different antennas. 
         [0092]    To facilitate switching between receive and transmit paths, the antenna duplexer  60  can be configured to electrically connect the antenna  340  to a selected transmit or receive path. Thus, the antenna duplexer  60  can provide a number of switching functionalities associated with an operation of the communication device  300 . In addition, as discussed above, the antenna duplexer  60  includes the transmission filter  62  and reception filter  61  (see  FIG. 7A ) which are configured to provide filtering of the RF signals. 
         [0093]    As shown in  FIGS. 16 and 17 , in certain embodiments, a controller  350  can be provided for controlling various functionalities associated with operations of the filtering module  310  (e.g., the antenna duplexer module  210 ) and/or other operating component(s). In certain embodiments, the at least one processor  370  can be configured to facilitate implementation of various processes for operation of the communication device  300 . The processes performed by the at least one processor  370  may be implemented by computer program instructions. These computer program instructions may be provided to the at least one processor  370 , which can be a general purpose computer, a special purpose computer, or another programmable data processing apparatus to produce a machine, such that the instructions, which execute via the at least one processor of the computer or other programmable data processing apparatus, create a mechanism for operating the communication device  300 . In certain embodiments, these computer program instructions may also be stored in the computer-readable medium  360 . The battery  380  can be any suitable battery for use in the communication device  300 , including, for example, a lithium-ion battery. 
         [0094]    Having described above several aspects of at least one embodiment, it is to be appreciated various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be part of this disclosure and are intended to be within the scope of the invention. Accordingly, the foregoing description and drawings are by way of example only, and the scope of the invention should be determined from proper construction of the appended claims, and their equivalents.