Patent Publication Number: US-2021184654-A1

Title: Acoustic wave device, radio-frequency front end circuit, and communication device

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority to Japanese Patent Application No. 2018-168266 filed on Sep. 7, 2018 and is a Continuation Application of PCT Application No. PCT/JP2019/035184 filed on Sep. 6, 2019. The entire contents of each application are hereby incorporated herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to an acoustic wave device including two or more acoustic wave filters, a radio-frequency front end circuit including the acoustic wave device, and a communication device. 
     2. Description of the Related Art 
     Heretofore, a plurality of acoustic wave filters have been widely used in radio-frequency front end circuits of mobile phones and smartphones. For example, in a splitter disclosed in Japanese Unexamined Patent Application Publication No. 2014-68123, first ends of two or more band pass filters of different frequencies are connected to each other forming a common connection. Each band pass filter is formed of a surface acoustic wave filter chip. Each surface acoustic wave filter chip includes a plurality of surface acoustic wave resonators. 
     Japanese Unexamined Patent Application Publication No. 2010-187373 discloses an acoustic wave device in which an acoustic wave resonator is formed by stacking an insulating film including silicon dioxide and a piezoelectric substrate including lithium tantalate on a silicon support substrate. Heat resistance is improved by bonding the layers at the (111) plane of silicon. 
     In the acoustic wave device disclosed in Japanese Unexamined Patent Application Publication No. 2014-68123, a plurality of acoustic wave filters having different frequencies are connected to each other forming a common connection on the side near an antenna terminal. 
     The inventors of preferred embodiments of the present invention discovered that a plurality of spurious responses appear on the high-frequency side of the utilized main mode in an acoustic wave resonator in which a lithium tantalate film is directly or indirectly stacked on a silicon support substrate. When such an acoustic wave resonator is used in an acoustic wave filter having a lower pass band in an acoustic wave device, there is a danger that a spurious response that appears in this acoustic wave filter will appear in the pass band of another acoustic wave filter having a higher pass band. Therefore, there is a risk of the filter characteristics of the other acoustic wave filter being degraded. 
     SUMMARY OF THE INVENTION 
     Preferred embodiments of the present invention provide acoustic wave devices in each of which a ripple due to such a spurious response is unlikely to occur inside the pass band of another band pass filter, radio-frequency front end circuits including such acoustic wave devices, and communication devices. 
     An acoustic wave device according to a first preferred embodiment of the present invention includes N (N is an integer greater than or equal to 2) band pass filters including first ends connected to each other so as to define a common connection and having different pass bands from each other. When the N band pass filters are referred to as a band pass filter ( 1 ), a band pass filter ( 2 ), . . . and a band pass filter (N) in ascending order of pass band frequency, at least one band pass filter (n) (1≤n&lt;N), other than the band pass filter having the highest pass band frequency, among the N band pass filters is an acoustic wave filter that includes one or more acoustic wave resonators. At least one acoustic wave resonator (t) among the one or more acoustic wave resonators includes a silicon support substrate having Euler angles (φ Si , θ Si , ψ Si ), a silicon oxide film stacked above the silicon support substrate, a lithium tantalate film stacked above the silicon oxide film and having Euler angles (φ LT =0°±5°, θ LT , ψ LT =0°±15°), an IDT electrode provided above the lithium tantalate film and including electrode fingers, and a protective film that covers at least a portion of the IDT electrode. In the acoustic wave resonator (t), when λ is a wavelength determined by an electrode finger pitch of the IDT electrode and a thickness normalized using the wavelength λ is referred to as a wavelength-normalized thickness, and when T LT  is a wavelength-normalized thickness of the lithium tantalate film, θ LT  is a Euler angle of the lithium tantalate film, T S  is a wavelength-normalized thickness of the silicon oxide film, T E  is a wavelength-normalized thickness of the IDT electrode converted into a thickness of aluminum obtained as the product of a wavelength-normalized thickness of the IDT electrode and a value obtained by dividing a density of the IDT electrode by a density of aluminum, T P  is a wavelength-normalized thickness of the protective film obtained as the product of a value obtained by dividing a density of the protective film by a density of silicon oxide and a wavelength-normalized thickness obtained by normalizing a thickness of the protective film using the wavelength λ, ψ Si  is a propagation direction inside the silicon support substrate, and T Si  is a wavelength-normalized thickness of the silicon support substrate, a first frequency f h1_t   (n) , where s=1 among first, second, and third frequencies f hs_t   (n)  determined by Formula (1) and Formula (2) below determined using T LT , θ LT , T S , T E , T P , ψ Si , and T Si  and all band pass filters (m) (n&lt;m≤N) having pass bands located at a higher frequency than the pass band of the band pass filter (n) satisfy Formula (3) or Formula (4) below: 
     
       
         
           
             
               
                 
                   
                     
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     where s=1 in Formulas (2) to (4). 
     λ t   (n)  is a wavelength determined by an electrode finger pitch of the IDT electrode of the acoustic wave resonator (t) included in the band pass filter (n), f u   (m)  is a frequency of a high-frequency end of the pass band of the band pass filter (m), f l   (m)  is a frequency of a low-frequency end of the pass band of the band pass filter (m), and coefficients in Formula (1) have values as listed below in Table 1 for the various crystal orientations of the silicon support substrate: 
     
       
         
           
               
               
             
               
                   
                 TABLE 1 
               
             
            
               
                   
                   
               
               
                   
                 s = 1 
               
            
           
           
               
               
               
               
            
               
                   
                 Si(100) 
                 Si(110) 
                 Si(111) 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                 a TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 a TLT   (2)   
                 0 
                 0 
                 0 
               
               
                 a TLT   (1)   
                 −436.3811104 
                 0 
                 0 
               
               
                 b TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 b TLT   (2)   
                 0 
                 0 
                 0 
               
               
                 c TLT   
                 0.251271186 
                 0 
                 0 
               
               
                 a TS   (2)   
                 0 
                 0 
                 0 
               
               
                 a TS   (1)   
                 −370.8189665 
                 0 
                 −174.7116877 
               
               
                 b TS   (2)   
                 0 
                 0 
                 0 
               
               
                 c TS   
                 0.250529661 
                 0 
                 0.24371308 
               
               
                 a TE   (1)   
                 0 
                 0 
                 0 
               
               
                 c TE   
                 0 
                 0 
                 0 
               
               
                 a ψSi   (5)   
                 0 
                 −6.73542E−06 
                 0 
               
               
                 a ψSi   (4)   
                 0 
                 −4.84328E−05 
                 0.000703463 
               
               
                 a ψSi   (3)   
                 0.044803063 
                 0.020121569 
                 −4.77016E−05 
               
               
                 a ψSi   (2)   
                 −0.11149637 
                 0.237494527 
                 −0.961938987 
               
               
                 a ψSi   (1)   
                 −43.37704861 
                 −1.22341255 
                 0.091605753 
               
               
                 b ψSi   (5)   
                 0 
                 −78830.27657 
                 0 
               
               
                 b ψSi   (4)   
                 0 
                 1406271.562 
                 340944.6167 
               
               
                 b ψSi   (3)   
                 51.32996847 
                 −17.77613547 
                 −41.61537323 
               
               
                 b ψSi   (2)   
                 280.2660593 
                 895.2921635 
                 446.7591732 
               
               
                 c ψSi   
                 22.32521186 
                 45.02689779 
                 30.11392405 
               
               
                 a θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 a θLT   (1)   
                 0 
                 0 
                 0 
               
               
                 b θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 c θLT   
                 −90 
                 −90 
                 −90 
               
               
                 d TLTTS   
                 0 
                 0 
                 0 
               
               
                 d TLTTE   
                 0 
                 0 
                 0 
               
               
                 d TLT   ψSi   
                 0 
                 0 
                 0 
               
               
                 e 
                 5141.869703 
                 5073.066348 
                 4781.489451 
               
               
                 a TP   
                 0 
                 0 
                 0 
               
               
                 b TP   
                 0.02 
                 0.02 
                 0.02 
               
               
                 c TP   
                 −0.0102 
                 −0.0102 
                 −0.0102 
               
               
                 d TP   
                 1.0002 
                 1.0002 
                 1.0002. 
               
               
                   
               
            
           
         
       
     
     An acoustic wave device according to a second preferred embodiment of the present invention includes N (N is an integer greater than or equal to 2) band pass filters including first ends connected to each other so as to define a common connection and having different pass bands from each other. When the N band pass filters are referred to as a band pass filter ( 1 ), a band pass filter ( 2 ), . . . and a band pass filter (N) in ascending order of pass band frequency, at least one band pass filter (n) (1≤n&lt;N), other than the band pass filter having the highest pass band frequency, among the N band pass filters is an acoustic wave filter that includes one or more acoustic wave resonators. At least one acoustic wave resonator (t) among the one or more acoustic wave resonators includes a silicon support substrate having Euler angles (T Si , θ Si , ψ Si ), a silicon oxide film stacked above the silicon support substrate, a lithium tantalate film stacked above the silicon oxide film and having Euler angles (φ LT =0°±5°, θ LT , ψ LT =0°±15°), an IDT electrode provided above the lithium tantalate film and including electrode fingers, and a protective film that covers at least a portion of the IDT electrode. In the acoustic wave resonator (t), when λ is a wavelength determined by an electrode finger pitch of the IDT electrode and a thickness normalized using the wavelength λ is referred to as a wavelength-normalized thickness, and when T LT  is a wavelength-normalized thickness of the lithium tantalate film, θ LT  is a Euler angle of the lithium tantalate film, T S  is a wavelength-normalized thickness of the silicon oxide film, T E  is a wavelength-normalized thickness of the IDT electrode converted into a thickness of aluminum obtained as a product of a wavelength-normalized thickness of the IDT electrode and a value obtained by dividing a density of the IDT electrode by a density of aluminum, T P  is a wavelength-normalized thickness of the protective film obtained as a product of a value obtained by dividing a density of the protective film by a density of silicon oxide and a wavelength-normalized thickness obtained by normalizing a thickness of the protective film using the wavelength λ, ψ Si  is a propagation direction inside the silicon support substrate, and T Si  is a wavelength-normalized thickness of the silicon support substrate, a second frequency f h2_t   (n) , where s=2 among first, second, and third frequencies f hs_t   (n)  determined by Formula (1) and Formula (2) below determined using T LT , θ LT , T S , T E , T P , ψ Si , and T Si  and all band pass filters (m) (n&lt;m≤N) having pass bands located at a higher frequency than the pass band of the band pass filter (n) satisfy Formula (3) or Formula (4): 
     
       
         
           
             
               
                 
                   
                     
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                                           S 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   
                                     T 
                                     S 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 T 
                                 S 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   S 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     S 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 T 
                                 E 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   E 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     E 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 5 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   5 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     5 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 4 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   4 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     4 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 3 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   3 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     3 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 si 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   ψ 
                                   Si 
                                 
                                 - 
                                 
                                   c 
                                   
                                     ψ 
                                     Si 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 θ 
                                 LT 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         LT 
                                       
                                       - 
                                       
                                         c 
                                         
                                           θ 
                                           LT 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   
                                     θ 
                                     LT 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 θ 
                                 LT 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   θ 
                                   LT 
                                 
                                 - 
                                 
                                   c 
                                   
                                     θ 
                                     LT 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 d 
                                 
                                   
                                     T 
                                     LT 
                                   
                                    
                                   
                                     T 
                                     S 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     LT 
                                   
                                   - 
                                   
                                     c 
                                     
                                       T 
                                       LT 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   S 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     S 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 d 
                                 
                                   
                                     T 
                                     LT 
                                   
                                    
                                   
                                     T 
                                     E 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     LT 
                                   
                                   - 
                                   
                                     c 
                                     
                                       T 
                                       LT 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   E 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     E 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 d 
                                 
                                   
                                     T 
                                     LT 
                                   
                                    
                                   
                                     ψ 
                                     Si 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     LT 
                                   
                                   - 
                                   
                                     c 
                                     
                                       T 
                                       LT 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   ψ 
                                   Si 
                                 
                                 - 
                                 
                                   c 
                                   
                                     ψ 
                                     Si 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           e 
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               a 
                               
                                 T 
                                 P 
                               
                             
                              
                             
                               T 
                               P 
                               3 
                             
                           
                           + 
                           
                             
                               b 
                               
                                 T 
                                 P 
                               
                             
                              
                             
                               T 
                               P 
                               2 
                             
                           
                           + 
                           
                             
                               c 
                               
                                 T 
                                 P 
                               
                             
                              
                             
                               T 
                               P 
                               2 
                             
                           
                           + 
                           
                             d 
                             
                               T 
                               P 
                             
                           
                         
                         ) 
                       
                     
                   
                   ; 
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         f 
                         
                           
                             h 
                             s 
                           
                            
                           _ 
                            
                           
                               
                           
                            
                           t 
                         
                         
                           ( 
                           n 
                           ) 
                         
                       
                       = 
                       
                         
                           V 
                           
                             
                               h 
                               s 
                             
                              
                             _ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         
                           λ 
                           t 
                           
                             ( 
                             n 
                             ) 
                           
                         
                       
                     
                     ; 
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         f 
                         
                           hs 
                            
                           
                               
                           
                            
                           _ 
                            
                           
                               
                           
                            
                           t 
                         
                         
                           ( 
                           n 
                           ) 
                         
                       
                       &gt; 
                       
                         f 
                         u 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                     ; 
                     and 
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         f 
                         
                           hs 
                            
                           
                               
                           
                            
                           _ 
                            
                           
                               
                           
                            
                           t 
                         
                         
                           ( 
                           n 
                           ) 
                         
                       
                       &lt; 
                       
                         f 
                         l 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
     where s=2 in Formulas (2) to (4). 
     λ t   (n)  is a wavelength determined by an electrode finger pitch of the IDT electrode of the acoustic wave resonator (t) included in the band pass filter (n), f u   (m)  is a frequency of a high-frequency end of the pass band of the band pass filter (m), f l   (m)  is a frequency of a low-frequency end of the pass band of the band pass filter (m), and coefficients in Formula (1) have values as listed in Table 2 below for the various crystal orientations of the silicon support substrate: 
     
       
         
           
               
               
             
               
                   
                 TABLE 2 
               
             
            
               
                   
                   
               
               
                   
                 s = 2 
               
            
           
           
               
               
               
               
            
               
                   
                 Si(100) 
                 Si(110) 
                 Si(111) 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                 a TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 a TLT   (2)   
                 −3168.980655 
                 −3017.48047 
                 −6490.588929 
               
               
                 a TLT   (1)   
                 −1070.770975 
                 −1740.800945 
                 −1736.124534 
               
               
                 b TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 b TLT   (2)   
                 0.012520877 
                 0.011673567 
                 0.012932291 
               
               
                 c TLT   
                 0.25 
                 0.26454918 
                 0.244042365 
               
               
                 a TS   (2)   
                 0 
                 0 
                 −5447.157686 
               
               
                 a TS   (1)   
                 −1175.713239 
                 −1443.794269 
                 −1260.230106 
               
               
                 b TS   (2)   
                 0 
                 0 
                 0.012411464 
               
               
                 c TS   
                 0.250104384 
                 0.264754098 
                 0.243336275 
               
               
                 a TE   (1)   
                 −440.9839549 
                 −631.5739347 
                 −613.3480905 
               
               
                 c TE   
                 0.249686848 
                 0.254030055 
                 0.249646955 
               
               
                 a ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 a ψSi   (4)   
                 0 
                 −8.39122E−05 
                 0 
               
               
                 a ψSi   (3)   
                 −0.01385161 
                 −0.000219614 
                 0 
               
               
                 a ψSi   (2)   
                 0.040113138 
                 0.288564318 
                 0.113517332 
               
               
                 a ψSi   (1)   
                 10.06827612 
                 −1.021332326 
                 0.01604077 
               
               
                 b ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 b ψSi   (4)   
                 0 
                 1489555.068 
                 0 
               
               
                 b ψSi   (3)   
                 −14.09179662 
                 −1671.679914 
                 0 
               
               
                 b ψSi   (2)   
                 281.2490194 
                 936.1444126 
                 446.7377612 
               
               
                 c ψSi   
                 22.53131524 
                 46.09631148 
                 29.70873786 
               
               
                 a θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 a θLT   (1)   
                 −2.926997767 
                 0 
                 0 
               
               
                 b θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 c θLT   
                 −50.02087683 
                 −90 
                 −90 
               
               
                 d TLTTS   
                 0 
                 0 
                 0 
               
               
                 d TLTTE   
                 −2151.133017 
                 −2785.629029 
                 −2639.620065 
               
               
                 d TLT   ψSi   
                 17.13925013 
                 0 
                 0 
               
               
                 e 
                 5188.573706 
                 5052.799929 
                 5187.888817 
               
               
                 a TP   
                 0 
                 0 
                 0 
               
               
                 b TP   
                 −0.1336 
                 −0.1336 
                 −0.1336 
               
               
                 c TP   
                 −0.0552 
                 −0.0552 
                 −0.0552 
               
               
                 d TP   
                 1.001 
                 1.001 
                 1.001. 
               
               
                   
               
            
           
         
       
     
     An acoustic wave device according to a third preferred embodiment of the present invention includes N (N is an integer greater than or equal to 2) band pass filters including first ends connected to each other so as to define a common connection and having different pass bands from each other. When the N band pass filters are referred to as a band pass filter ( 1 ), a band pass filter ( 2 ), . . . and a band pass filter (N) in ascending order of pass band frequency, at least one band pass filter (n) (1≤n&lt;N), other than the band pass filter having the highest pass band frequency, among the N band pass filters is an acoustic wave filter that includes one or more acoustic wave resonators. At least one acoustic wave resonator (t) among the one or more acoustic wave resonators includes a silicon support substrate having Euler angles (φ Si , θ Si , ψ Si ), a silicon oxide film stacked above the silicon support substrate, a lithium tantalate film stacked above the silicon oxide film and having Euler angles (φ LT =0°±5°, θ LT , ψ LT =0°±15°), an IDT electrode provided above the lithium tantalate film and including electrode fingers, and a protective film that covers at least a portion of the IDT electrode. In the acoustic wave resonator (t), when λ is a wavelength determined by an electrode finger pitch of the IDT electrode and a thickness normalized using the wavelength λ is referred to as a wavelength-normalized thickness, and when T LT  is a wavelength-normalized thickness of the lithium tantalate film, θ LT  is a Euler angle of the lithium tantalate film, T S  is a wavelength-normalized thickness of the silicon oxide film, T E  is a wavelength-normalized thickness of the IDT electrode converted into a thickness of aluminum obtained as a product of a wavelength-normalized thickness of the IDT electrode and a value obtained by dividing a density of the IDT electrode by a density of aluminum, T P  is a wavelength-normalized thickness of the protective film obtained as a product of a value obtained by dividing a density of the protective film by a density of silicon oxide and a wavelength-normalized thickness obtained by normalizing a thickness of the protective film using the wavelength λ, ψ Si  is a propagation direction inside the silicon support substrate, and T Si  is a wavelength-normalized thickness of the silicon support substrate, a third frequency f h3_t   (n) , where s=3 among first, second, and third frequencies f hs_t   (n)  determined by Formula (1) and Formula (2) below determined using T LT , θ LT , T S , T E , T P , ψ Si , and T Si  and all band pass filters (m) (n&lt;m≤N) having pass bands located at a higher frequency than the pass band of the band pass filter (n) satisfy Formula (3) or Formula (4) below: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       h 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               a 
                               
                                 T 
                                 LT 
                               
                               
                                 ( 
                                 3 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         T 
                                         LT 
                                       
                                       - 
                                       
                                         c 
                                         
                                           T 
                                           LT 
                                         
                                       
                                     
                                     ) 
                                   
                                   3 
                                 
                                 - 
                                 
                                   b 
                                   
                                     T 
                                     LT 
                                   
                                   
                                     ( 
                                     3 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 T 
                                 LT 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         T 
                                         LT 
                                       
                                       - 
                                       
                                         c 
                                         
                                           T 
                                           LT 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   
                                     T 
                                     LT 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 T 
                                 LT 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   LT 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     LT 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 T 
                                 S 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         T 
                                         S 
                                       
                                       - 
                                       
                                         c 
                                         
                                           T 
                                           S 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   
                                     T 
                                     S 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 T 
                                 S 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   S 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     S 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 T 
                                 E 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   E 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     E 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 5 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   5 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     5 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 4 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   4 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     4 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 3 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   3 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     3 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 Si 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         Si 
                                       
                                       - 
                                       
                                         c 
                                         
                                           ψ 
                                           Si 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   
                                     ψ 
                                     Si 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 ψ 
                                 si 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   ψ 
                                   Si 
                                 
                                 - 
                                 
                                   c 
                                   
                                     ψ 
                                     Si 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 θ 
                                 LT 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         LT 
                                       
                                       - 
                                       
                                         c 
                                         
                                           θ 
                                           LT 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 - 
                                 
                                   b 
                                   
                                     θ 
                                     LT 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               a 
                               
                                 θ 
                                 LT 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   θ 
                                   LT 
                                 
                                 - 
                                 
                                   c 
                                   
                                     θ 
                                     LT 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 d 
                                 
                                   
                                     T 
                                     LT 
                                   
                                    
                                   
                                     T 
                                     S 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     LT 
                                   
                                   - 
                                   
                                     c 
                                     
                                       T 
                                       LT 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   S 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     S 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 d 
                                 
                                   
                                     T 
                                     LT 
                                   
                                    
                                   
                                     T 
                                     E 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     LT 
                                   
                                   - 
                                   
                                     c 
                                     
                                       T 
                                       LT 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   E 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     E 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 d 
                                 
                                   
                                     T 
                                     LT 
                                   
                                    
                                   
                                     ψ 
                                     Si 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     LT 
                                   
                                   - 
                                   
                                     c 
                                     
                                       T 
                                       LT 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   ψ 
                                   Si 
                                 
                                 - 
                                 
                                   c 
                                   
                                     ψ 
                                     Si 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           e 
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               a 
                               
                                 T 
                                 P 
                               
                             
                              
                             
                               T 
                               P 
                               3 
                             
                           
                           + 
                           
                             
                               b 
                               
                                 T 
                                 P 
                               
                             
                              
                             
                               T 
                               P 
                               2 
                             
                           
                           + 
                           
                             
                               c 
                               
                                 T 
                                 P 
                               
                             
                              
                             
                               T 
                               P 
                               2 
                             
                           
                           + 
                           
                             d 
                             
                               T 
                               P 
                             
                           
                         
                         ) 
                       
                     
                   
                   ; 
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         f 
                         
                           
                             h 
                             s 
                           
                            
                           _ 
                            
                           
                               
                           
                            
                           t 
                         
                         
                           ( 
                           n 
                           ) 
                         
                       
                       = 
                       
                         
                           V 
                           
                             
                               h 
                               s 
                             
                              
                             _ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         
                           λ 
                           t 
                           
                             ( 
                             n 
                             ) 
                           
                         
                       
                     
                     ; 
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         f 
                         
                           hs 
                            
                           
                               
                           
                            
                           _ 
                            
                           
                               
                           
                            
                           t 
                         
                         
                           ( 
                           n 
                           ) 
                         
                       
                       &gt; 
                       
                         f 
                         u 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                     ; 
                     and 
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         f 
                         
                           hs 
                            
                           
                               
                           
                            
                           _ 
                            
                           
                               
                           
                            
                           t 
                         
                         
                           ( 
                           n 
                           ) 
                         
                       
                       &lt; 
                       
                         f 
                         l 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
     where s=3 in Formulas (2) to (4). 
     λ t   (n)  is a wavelength determined by an electrode finger pitch of the IDT electrode of the acoustic wave resonator (t) included in the band pass filter (n), f u   (m)  is a frequency of a high-frequency end of the pass band of the band pass filter (m), f l   (m)  is a frequency of a low-frequency end of the pass band of the band pass filter (m), and coefficients in Formula (1) have values as listed in below Table 3 for the various crystal orientations of the silicon support substrate; 
     
       
         
           
               
               
             
               
                   
                 TABLE 3 
               
             
            
               
                   
                   
               
               
                   
                 s = 3 
               
            
           
           
               
               
               
               
            
               
                   
                 Si(100) 
                 Si(110) 
                 Si(111) 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                 a TLT   (3)   
                 24973.69604 
                 0 
                 0 
               
               
                 a TLT   (2)   
                 1430.312613 
                 0 
                 −6156.310187 
               
               
                 a TLT   (1)   
                 −1027.290569 
                 −737.4915071 
                 −1476.525631 
               
               
                 b TLT   (3)   
                 −0.000364885 
                 0 
                 0 
               
               
                 b TLT   (2)   
                 0.011041041 
                 0 
                 0.009760415 
               
               
                 c TLT   
                 0.277376171 
                 0.283128295 
                 0.290986602 
               
               
                 a TS   (2)   
                 0 
                 0 
                 0 
               
               
                 a TS   (1)   
                 −617.0601012 
                 0 
                 0 
               
               
                 b TS   (2)   
                 0 
                 0 
                 0 
               
               
                 c TS   
                 0.273226238 
                 0 
                 0 
               
               
                 a TE   (1)   
                 −254.6645301 
                 −400.406917 
                 −524.8090015 
               
               
                 c TE   
                 0.262449799 
                 0.265026362 
                 0.25499391 
               
               
                 a ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 a ψSi   (4)   
                 0 
                 −0.000208283 
                 −0.000637441 
               
               
                 a ψSi   (3)   
                 0 
                 −0.004709453 
                 −0.001817349 
               
               
                 a ψSi   (2)   
                 −0.126294383 
                 0.574442977 
                 0.749991624 
               
               
                 a ψSi   (1)   
                 2.746835794 
                 7.891650217 
                 −0.116425099 
               
               
                 b ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 b ψSi   (4)   
                 0 
                 1654327.754 
                 303345.3736 
               
               
                 b ψSi   (3)   
                 0 
                 −7229.860818 
                 −201.8986483 
               
               
                 b ψSi   (2)   
                 266.9400494 
                 985.3381236 
                 411.1815157 
               
               
                 c ψSi   
                 22.10843373 
                 47.8602812 
                 30.42021924 
               
               
                 a θLT   (2)   
                 0 
                 0 
                 0.593925697 
               
               
                 a θLT   (1)   
                 0 
                 0 
                 −2.75182308 
               
               
                 b θLT   (2)   
                 0 
                 0 
                 67.60597649 
               
               
                 c θLT   
                 −90 
                 −90 
                 −49.6589525 
               
               
                 d TLTTS   
                 −2391.060714 
                 0 
                 0 
               
               
                 d TLTTE   
                 0 
                 0 
                 −2773.934251 
               
               
                 d TLT   ψSi   
                 −12.5134053 
                 15.88855602 
                 0 
               
               
                 e 
                 5661.675305 
                 5417.174527 
                 5440.57579 
               
               
                 a TP   
                 10.443 
                 10.443 
                 10.443 
               
               
                 b TP   
                 −0.0316 
                 −0.0316 
                 −0.0316 
               
               
                 c TP   
                 −0.0653 
                 −00653 
                 −0.0653 
               
               
                 d TP   
                 1.0011 
                 1.0011 
                 1.0011. 
               
               
                   
               
            
           
         
       
     
     With each of the acoustic wave devices according to preferred embodiments of the present invention, a spurious response generated by at least one acoustic wave resonator of a band pass filter having a lower pass band is unlikely to occur inside the pass band of another band pass filter having a higher pass band. Thus, degradation of the filter characteristics of the other band pass filter is unlikely to occur. Therefore, radio-frequency front end circuits and communication devices that each include such an acoustic wave device provide excellent filter characteristics. 
     The above and other elements, features, steps, characteristics and advantages of the present invention will become more apparent from the following detailed description of the preferred embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a circuit diagram of an acoustic wave device according to a first preferred embodiment of the present invention. 
         FIG. 2  is a circuit diagram illustrating a first acoustic wave filter that is used in the acoustic wave device of the first preferred embodiment of the present invention. 
         FIG. 3A  is a schematic front cross-sectional view of an acoustic wave resonator used in the acoustic wave device of the first preferred embodiment of the present invention and  FIG. 3B  is a schematic plan view illustrating the electrode structure of the acoustic wave resonator. 
         FIG. 4  is a schematic diagram illustrating the pass bands of first to fourth acoustic wave filters in the first preferred embodiment of the present invention. 
         FIG. 5  is a diagram illustrating an admittance characteristic of the acoustic wave resonator. 
         FIG. 6  is a diagram illustrating the relationship between a propagation direction ψ Si  of a silicon support substrate and the acoustic velocities of waves of a main mode and a spurious response A. 
         FIG. 7  is a diagram illustrating the relationship between the wavelength-normalized thickness of a lithium tantalate film and the acoustic velocities of waves of the main mode and the spurious response A. 
         FIG. 8  is a diagram illustrating the relationship between the cut angle (about 90°−θ LT ) of the lithium tantalate film and the acoustic velocities of waves of the main mode and the spurious response A. 
         FIG. 9  is a diagram illustrating the relationship between the wavelength-normalized thickness of a silicon oxide film and the acoustic velocities of waves of the main mode and the spurious response A. 
         FIG. 10  is a diagram illustrating the relationship between the wavelength-normalized thickness of an IDT electrode and the acoustic velocities of waves of the main mode and the spurious response A. 
         FIG. 11  is a diagram illustrating the relationship between the wavelength-normalized thickness of a protective film including a silicon oxide film and the acoustic velocity of waves of the spurious response A. 
         FIG. 12A  is a diagram illustrating the filter characteristics of an acoustic wave device of a comparative example and  FIG. 12B  is a diagram illustrating the filter characteristics of the acoustic wave device of the first preferred embodiment of the present invention. 
         FIG. 13  is a diagram illustrating the relationship between the wavelength-normalized thickness of the silicon support substrate and the maximum values of the phases of spurious response A, spurious response B, and spurious response C. 
         FIG. 14  is a diagram illustrating the relationship between a propagation direction ψ Si  of the silicon support substrate and the acoustic velocities of waves of the main mode and the spurious response B. 
         FIG. 15  is a diagram illustrating the relationship between the wavelength-normalized thickness of a lithium tantalate film and the acoustic velocities of waves of the main mode and the spurious response B. 
         FIG. 16  is a diagram illustrating the relationship between the cut angle (about 90°−θ LT ) of the lithium tantalate film and the acoustic velocities of waves of the main mode and the spurious response B. 
         FIG. 17  is a diagram illustrating the relationship between the wavelength-normalized thickness of a silicon oxide film and the acoustic velocities of waves of the main mode and the spurious response B. 
         FIG. 18  is a diagram illustrating the relationship between the wavelength-normalized thickness of the IDT electrode and the acoustic velocities of waves of the main mode and the spurious response B. 
         FIG. 19  is a diagram illustrating the relationship between the wavelength-normalized thickness of a protective film including a silicon oxide film and the acoustic velocity of waves of the spurious response B. 
         FIG. 20  is a diagram illustrating the relationship between the propagation direction ψ Si  of the silicon support substrate and the acoustic velocities of waves of the main mode and the spurious response C. 
         FIG. 21  is a diagram illustrating the relationship between the wavelength-normalized thickness of the lithium tantalate film and the acoustic velocities of waves of the main mode and the spurious response C. 
         FIG. 22  is a diagram illustrating the relationship between the cut angle (about 90°−θ LT ) of the lithium tantalate film and the acoustic velocities of waves of the main mode and the spurious response C. 
         FIG. 23  is a diagram illustrating the relationship between the wavelength-normalized thickness of the silicon oxide film and the acoustic velocities of waves of the main mode and spurious C. 
         FIG. 24  is a diagram illustrating the relationship between the wavelength-normalized thickness of the IDT electrode and the acoustic velocities of waves of the main mode and the spurious response C. 
         FIG. 25  is a diagram illustrating the relationship between the wavelength-normalized thickness of the protective film including a silicon oxide film and the acoustic velocity of waves of the spurious response C. 
         FIG. 26  is a diagram illustrating the relationship between the film thickness of a LiTaO 3  film and a Q value in an acoustic wave device. 
         FIG. 27  is a diagram illustrating the relationship between the film thickness of a LiTaO 3  film and a temperature coefficient of frequency TCF in an acoustic wave device. 
         FIG. 28  is a diagram illustrating the relationship between the film thickness of a LiTaO 3  film and acoustic velocity in an acoustic wave device. 
         FIG. 29  is a diagram illustrating the relationship between the thickness of a piezoelectric film including LiTaO 3  and a relative band width. 
         FIG. 30  is a diagram illustrating the relationship between the film thickness of a SiO 2  film, the material of a high-acoustic-velocity film, and acoustic velocity. 
         FIG. 31  is a diagram illustrating the relationships between the film thickness of a SiO 2  film, the material of a high-acoustic-velocity film, and the electromechanical coupling coefficient. 
         FIG. 32  is a partial enlarged front cross-sectional view for describing a modification in which the thickness of the protective film is different in different portions of the protective film. 
         FIG. 33  is a partial enlarged front cross-sectional view for describing another modification in which the thickness of the protective film is different in different portions of the protective film. 
         FIG. 34  is a partial enlarged front cross-sectional view for describing yet another modification in which the thickness of the protective film is different in different portions of the protective film. 
         FIG. 35  is a front cross-sectional view illustrating a modification of an acoustic wave resonator used in an acoustic wave device according to a preferred embodiment of the present invention. 
         FIG. 36  is a front cross-sectional view illustrating yet another modification of an acoustic wave resonator used in an acoustic wave device according to a preferred embodiment of the present invention. 
         FIG. 37  is a partial enlarged front cross-sectional view for describing a modification in which the protective film is a multilayer film. 
         FIG. 38  is a schematic diagram for explaining a crystal orientation Si(100). 
         FIG. 39  is a schematic diagram for explaining a crystal orientation Si(110). 
         FIG. 40  is a schematic diagram for explaining a crystal orientation Si(111). 
         FIG. 41  is a schematic configuration diagram of a communication device including a radio-frequency front end circuit according to a preferred embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Hereafter, the present invention will be made clearer by describing specific preferred embodiments of the present invention with reference to the drawings. 
     The preferred embodiments of the present invention described in the present specification are illustrative examples and it should be noted that portions of the configurations illustrated in different preferred embodiments can be substituted for one another or combined with one another. 
     First Preferred Embodiment 
       FIG. 1  is a circuit diagram of an acoustic wave device according to a first preferred embodiment of the present invention. An acoustic wave device  1  includes an antenna terminal  2 . The antenna terminal  2  is connected to the antenna of a smartphone, for example. 
     In the acoustic wave device  1 , first to fourth acoustic wave filters  3  to  6  are connected to the antenna terminal  2  via a common connection. The first to fourth acoustic wave filters  3  to  6  preferably are band pass filters. Note that although the first to fourth acoustic wave filters  3  to  6  are used as a plurality of band pass filters, it is not necessary for all of the plurality of band pass filters to be acoustic wave filters in the present invention. In other words, in an acoustic wave device that includes N band pass filters including first ends that are connected to each other so as to define a common connection and that have different pass bands from each other, it is sufficient that at least one band pass filter, other than the band pass filter having the highest pass band frequency, among the N band pass filters is an acoustic wave filter that includes the acoustic wave resonator described below. Therefore, the commonly connected band pass filters may be, for example, LC filters or another type of filter other than acoustic wave filters. 
       FIG. 4  is a schematic diagram illustrating the relationship between the pass bands of the first to fourth acoustic wave filters. As illustrated in  FIG. 4 , the first to fourth acoustic wave filters have different pass bands from each other. The pass bands of the first to fourth acoustic wave filters are respectively referred to as first to fourth pass bands. 
     The first pass band is nearest the low frequency side and the frequencies of the second pass band, the third pass band, and the fourth pass band increase in this order. In other words, first pass band&lt;second pass band&lt;third pass band&lt;fourth pass band. In the second to fourth pass bands, a low-frequency end is denoted as f l   (m)  and a high-frequency end is denoted as f u   (m) . The low-frequency end is the low-frequency end of the pass band. In addition, the high-frequency end is the high-frequency end of the pass band. For example, the ends of the frequency bands of the bands standardized in 3GPP or the like may preferably be used as the low-frequency ends and high-frequency ends of the pass bands. 
     Here, (m) is 2, 3, or 4 for the second to fourth pass bands. In other words, m represents the numbers of the acoustic wave filters other than the aforementioned acoustic wave filter. 
     The first to fourth acoustic wave filters  3  to  6  each include a plurality of acoustic wave resonators.  FIG. 2  is a circuit diagram of the first acoustic wave filter  3 . The first acoustic wave filter  3  includes series arm resonators S 1  to S 3  and parallel arm resonators P 1  and P 2 , which are each defined by an acoustic wave resonator. In other words, the first acoustic wave filter  3  is a ladder filter. However, the number of series arm resonators and the number of parallel arm resonators in the ladder filter are not limited to the numbers illustrated in this example. 
     Furthermore, the second to fourth acoustic wave filters  4  to  6  are similarly each defined by a ladder filter and include a plurality of series arm resonators and a plurality of parallel arm resonators. 
     Provided that the first to fourth acoustic wave filters  3  to  6  include a plurality of acoustic wave resonators, the first to fourth acoustic wave filters  3  to  6  may have a circuit configuration other than a ladder filter. For example, the first to fourth acoustic wave filters  3  to  6  may each be an acoustic wave filter in which an acoustic wave resonator is serially connected to a longitudinally coupled resonator acoustic wave filter. In addition, the first to fourth acoustic wave filters  3  to  6  may each be an acoustic wave filter in which a ladder filter is connected to a longitudinally coupled resonator acoustic wave filter. 
       FIG. 3A  is a schematic front cross-sectional view of an acoustic wave resonator used for the series arm resonators S 1  to S 3  or the parallel arm resonators P 1  and P 2  of the first acoustic wave filter  3  and  FIG. 3B  is a schematic plan view illustrating the electrode structure thereof. 
     An acoustic wave resonator  11  includes a silicon support substrate  12 , a silicon oxide film  13  stacked above the silicon support substrate  12 , and a lithium tantalate film  14  stacked above the silicon oxide film  13 . 
     The silicon support substrate  12  is preferably made of silicon, for example. The silicon support substrate  12  preferably includes single crystal silicon, for example, but it is sufficient that the silicon support substrate  12  have a crystal orientation and does not need to be a perfect single crystal. The silicon oxide film  13  is preferably a SiO 2  film, for example. The silicon oxide film  13  may include SiO 2  doped with fluorine or the like, for example, as long as the silicon oxide film  13  includes silicon oxide. The silicon oxide film  13  may have a multilayer structure including a plurality of layers made of silicon oxide. Intermediate layers made of, for example, titanium or nickel may be included between the plurality of layers. The thickness of the silicon oxide film  13  in this case refers to the total thickness of the multilayer structure. The lithium tantalate film  14  is preferably a single crystal lithium tantalate film, but it is sufficient that the lithium tantalate film  14  have a crystal orientation and does not have to be a perfect single crystal. 
     The thickness of the silicon oxide film  13  may be 0. In other words, the silicon oxide film  13  does not necessarily have to be provided. 
     An interdigital transducer (IDT) electrode  15  is provided above the upper surface of the lithium tantalate film  14 . More specifically, reflectors  16  and  17  are provided on both sides of the IDT electrode  15  in the acoustic wave propagation direction, thus defining a one-port surface acoustic wave resonator. 
     A protective film  18  covers the IDT electrode  15  and the reflectors  16  and  17 . The protective film  18  is preferably made of, for example, a silicon oxide film in the present preferred embodiment. However, the protective film  18  may be made of any of various dielectric films, such as a silicon oxynitride film or a silicon nitride film, for example. In addition, in the present preferred embodiment, the protective film  18  covers not only the electrode fingers of the IDT electrode  15  but also the upper surface of the lithium tantalate film  14  and both side surfaces of the electrode fingers. However, the protective film  18  is not limited to this configuration. 
     The inventors of preferred embodiments of the present invention discovered that in an acoustic wave filter device in which the lithium tantalate film  14  is directly or indirectly stacked above the silicon support substrate  12 , when the IDT electrode  15  is excited, in addition to the response of a main mode that is to be used, a plurality of spurious responses appear on the high-frequency side of the main mode. This plurality of spurious responses will be described with reference to  FIG. 5 . 
       FIG. 5  is a diagram illustrating the admittance characteristic of an example of an acoustic wave resonator in which a silicon oxide film and a lithium tantalate film are stacked above a silicon support substrate. As is clear from  FIG. 5 , spurious response A, spurious response B, and spurious response C appear at higher frequency positions than the main mode response that appears around 3.9 GHz. Spurious response A appears around at around 4.7 GHz. Spurious response B appears at a higher point at around 5.2 GHz. Spurious response C appears around 5.7 GHz. When the frequency of spurious response A is f 1 , the frequency of spurious response B is f 2 , and the frequency of spurious response C is f 3 , f 1 &lt;f 2 &lt;f 3 . The frequencies of spurious response A, spurious response B, and spurious response C are the peak positions of the impedance phase characteristic of the waves of spurious response A, spurious response B, and spurious response C. 
     As described above, in an acoustic wave device in which a plurality of acoustic wave filters having different pass bands are connected to each other so as to define a common connection at the side near the antenna terminal, when a spurious response generated by an acoustic wave filter having a pass band at a lower frequency appears in the pass band of another acoustic wave filter having a pass band at a higher frequency, a ripple is generated. Therefore, it is preferable that at least one of spurious response A, spurious response B, and spurious response C does not appear in the pass bands of the second to fourth acoustic wave filters  4  to  6 . It is preferable that two spurious responses out of spurious response A, spurious response B, and spurious response C do not appear in the pass bands of the second to fourth acoustic wave filters  4  to  6 . For example, it is preferable that spurious response A and spurious response B, spurious response A and spurious response C, or spurious response B and spurious response C do not appear in the pass bands of the second to fourth acoustic wave filters  4  to  6 . In addition, it is preferable that none of spurious response A, spurious response B, and spurious response C appear in the pass bands of the second to fourth acoustic wave filters  4  to  6 . 
     In the acoustic wave device  1  of the present preferred embodiment it is preferable that spurious response A of at least one acoustic wave resonator included in the first acoustic wave filter  3  does not appear in the second to fourth pass bands illustrated in  FIG. 4 . Therefore, degradation of the filter characteristics of the second to fourth acoustic wave filters  4  to  6  is unlikely to occur. 
     This feature of the present preferred embodiment is detailed in i) below. 
     i) The values of a wavelength-normalized thickness T LT  of the lithium tantalate film  14 , an Euler angle θ LT  of the lithium tantalate film  14 , a wavelength-normalized thickness T S  of the silicon oxide film  13 , a wavelength-normalized thickness T E  of the IDT electrode  15  converted into a thickness of aluminum obtained as the product of the wavelength-normalized thickness of the IDT electrode  15  and a value obtained by dividing the density of the IDT electrode  15  by the density of aluminum, a wavelength-normalized thickness T P  of the protective film  18  obtained as the product of a value obtained by dividing the density of the protective film  18  by the density of silicon oxide and a wavelength-normalized thickness obtained by normalizing the thickness of the protective film  18  using a wavelength λ, the propagation direction ψ Si  in the silicon support substrate  12 , and a wavelength-normalized thickness T Si  of the silicon support substrate  12  are set so that a frequency f h1_t   (n)  of spurious response A determined by Formula (1) and Formula (2) below satisfies Formula (3) or Formula (4) below for all values of m where m&gt;n. 
     As a result, spurious response A is positioned outside the pass bands of the second to fourth acoustic wave filters  4  to  6 . Therefore, degradation of the filter characteristics of the second to fourth acoustic wave filters  4  to  6  caused by spurious response A is unlikely to occur. Positioning of the frequency of spurious response A outside the second to fourth pass bands by satisfying the above conditions will be described in more detail below. 
     The density of the IDT electrode  15  is not a measured value, but rather a value obtained from the density of the metal material of the IDT electrode  15 . In addition, the density of aluminum is 2698.9 kg/m 3 . This value is provided in Handbook of Chemistry, Pure Chemistry II, Revised Fourth Edition, edited by The Chemical Society of Japan, published by Maruzen (1993), p. 26. 
     The density of the protective film  18  is not a measured value, but rather a value obtained from the density of the material of the protective film  18 . In addition, the density of silicon oxide is 2200 kg/m 3 . This value is provided in Handbook of Chemistry, Applied Chemistry II, Materials, Revised Fourth Edition, edited by The Chemical Society of Japan, published by Maruzen (1993), p. 922. 
     In addition, “the thickness of the protective film  18 ” in this specification refers to the thickness of the protective film in portions located above the electrode fingers of the IDT electrode. 
     
       
         
           
             
               
                 
                   
                     V 
                     h 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             a 
                             
                               T 
                               LT 
                             
                             
                               ( 
                               3 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       T 
                                       LT 
                                     
                                     - 
                                     
                                       c 
                                       
                                         T 
                                         LT 
                                       
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                               - 
                               
                                 b 
                                 
                                   T 
                                   LT 
                                 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               T 
                               LT 
                             
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       T 
                                       LT 
                                     
                                     - 
                                     
                                       c 
                                       
                                         T 
                                         LT 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 b 
                                 
                                   T 
                                   LT 
                                 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               T 
                               LT 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 T 
                                 LT 
                               
                               - 
                               
                                 c 
                                 
                                   T 
                                   LT 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               T 
                               S 
                             
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       T 
                                       S 
                                     
                                     - 
                                     
                                       c 
                                       
                                         T 
                                         S 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 b 
                                 
                                   T 
                                   S 
                                 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               T 
                               S 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 T 
                                 S 
                               
                               - 
                               
                                 c 
                                 
                                   T 
                                   S 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               T 
                               E 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 T 
                                 E 
                               
                               - 
                               
                                 c 
                                 
                                   T 
                                   E 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               ψ 
                               Si 
                             
                             
                               ( 
                               5 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       ψ 
                                       Si 
                                     
                                     - 
                                     
                                       c 
                                       
                                         ψ 
                                         Si 
                                       
                                     
                                   
                                   ) 
                                 
                                 5 
                               
                               - 
                               
                                 b 
                                 
                                   ψ 
                                   Si 
                                 
                                 
                                   ( 
                                   5 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               ψ 
                               Si 
                             
                             
                               ( 
                               4 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       ψ 
                                       Si 
                                     
                                     - 
                                     
                                       c 
                                       
                                         ψ 
                                         Si 
                                       
                                     
                                   
                                   ) 
                                 
                                 4 
                               
                               - 
                               
                                 b 
                                 
                                   ψ 
                                   Si 
                                 
                                 
                                   ( 
                                   4 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               ψ 
                               Si 
                             
                             
                               ( 
                               3 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       ψ 
                                       Si 
                                     
                                     - 
                                     
                                       c 
                                       
                                         ψ 
                                         Si 
                                       
                                     
                                   
                                   ) 
                                 
                                 3 
                               
                               - 
                               
                                 b 
                                 
                                   ψ 
                                   Si 
                                 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               ψ 
                               Si 
                             
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       ψ 
                                       Si 
                                     
                                     - 
                                     
                                       c 
                                       
                                         ψ 
                                         Si 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 b 
                                 
                                   ψ 
                                   Si 
                                 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               ψ 
                               si 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 ψ 
                                 Si 
                               
                               - 
                               
                                 c 
                                 
                                   ψ 
                                   Si 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               θ 
                               LT 
                             
                             
                               ( 
                               2 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       LT 
                                     
                                     - 
                                     
                                       c 
                                       
                                         θ 
                                         LT 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 b 
                                 
                                   θ 
                                   LT 
                                 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             a 
                             
                               θ 
                               LT 
                             
                             
                               ( 
                               1 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 θ 
                                 LT 
                               
                               - 
                               
                                 c 
                                 
                                   θ 
                                   LT 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             
                               d 
                               
                                 
                                   T 
                                   LT 
                                 
                                  
                                 
                                   T 
                                   S 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   LT 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     LT 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 T 
                                 S 
                               
                               - 
                               
                                 c 
                                 
                                   T 
                                   S 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             
                               d 
                               
                                 
                                   T 
                                   LT 
                                 
                                  
                                 
                                   T 
                                   E 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   LT 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     LT 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 T 
                                 E 
                               
                               - 
                               
                                 c 
                                 
                                   T 
                                   E 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             
                               d 
                               
                                 
                                   T 
                                   LT 
                                 
                                  
                                 
                                   ψ 
                                   Si 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   T 
                                   LT 
                                 
                                 - 
                                 
                                   c 
                                   
                                     T 
                                     LT 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 ψ 
                                 Si 
                               
                               - 
                               
                                 c 
                                 
                                   ψ 
                                   Si 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         e 
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             a 
                             
                               T 
                               P 
                             
                           
                            
                           
                             T 
                             P 
                             3 
                           
                         
                         + 
                         
                           
                             b 
                             
                               T 
                               P 
                             
                           
                            
                           
                             T 
                             P 
                             2 
                           
                         
                         + 
                         
                           
                             c 
                             
                               T 
                               P 
                             
                           
                            
                           
                             T 
                             P 
                             2 
                           
                         
                         + 
                         
                           d 
                           
                             T 
                             P 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       f 
                       
                         
                           h 
                           s 
                         
                          
                         _ 
                          
                         
                             
                         
                          
                         t 
                       
                       
                         ( 
                         n 
                         ) 
                       
                     
                     = 
                     
                       
                         V 
                         
                           
                             h 
                             s 
                           
                            
                           _ 
                            
                           
                               
                           
                            
                           t 
                         
                       
                       
                         λ 
                         t 
                         
                           ( 
                           n 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       f 
                       
                         hs 
                          
                         
                             
                         
                          
                         _ 
                          
                         
                             
                         
                          
                         t 
                       
                       
                         ( 
                         n 
                         ) 
                       
                     
                     &gt; 
                     
                       f 
                       u 
                       
                         ( 
                         m 
                         ) 
                       
                     
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       f 
                       
                         hs 
                          
                         
                             
                         
                          
                         _ 
                          
                         
                             
                         
                          
                         t 
                       
                       
                         ( 
                         n 
                         ) 
                       
                     
                     &lt; 
                     
                       f 
                       l 
                       
                         ( 
                         m 
                         ) 
                       
                     
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
     s=1 in Formulas (2) to (4). 
     In Formulas (1) to (4), h represents spurious a response located at a higher frequency than the main mode, n represents the n-th filter, t represents the t-th element (resonator) of the nth filter, and m represents the m-th (m&gt;n) filter. Furthermore, in this specification, “wavelength-normalized thickness” refers to a thickness obtained by normalizing a thickness using the wavelength of the IDT electrode. Here, “wavelength” refers to a wavelength λ determined by the electrode finger pitch of the IDT electrode. Therefore, a “wavelength-normalized thickness” is a thickness obtained by normalizing the actual thickness by treating λ as 1 and is a value obtained by dividing the actual thickness by λ. The wavelength λ determined by the electrode finger pitch of the IDT electrode may be determined as the average value of the electrode finger pitch. 
     The inventors of preferred embodiments of the present invention discovered that the frequency position of spurious response A is affected by the various parameters described above. 
     As illustrated in  FIG. 6 , the acoustic velocity of the main mode negligibly changes, whereas the acoustic velocity of the waves of spurious response A significantly changes when the propagation direction ψ Si  of the silicon support substrate is changed. As illustrated in  FIG. 7 , the acoustic velocity of the waves of spurious response A changes with the wavelength-normalized thickness T LT  of the lithium tantalate film. As illustrated in  FIG. 8 , the acoustic velocity of the waves of spurious response A also changes with the cut angle, i.e., (about 90°−θ LT ), of the lithium tantalate film. As illustrated in  FIG. 9 , the acoustic velocity of the waves of spurious response A also slightly changes with the wavelength-normalized thickness T S  of the silicon oxide film. As illustrated in  FIG. 11 , the acoustic velocity of the waves of spurious response A also changes with the wavelength-normalized thickness T P  of the protective film, which is a silicon oxide film. In addition, as illustrated in  FIG. 10 , the acoustic velocity of the waves of spurious response A also slightly changes with the wavelength-normalized thickness T E  of the IDT electrode. The inventors of preferred embodiments of the present invention obtained the acoustic velocity of the waves of spurious response A while freely changing these parameters. As a result, it was discovered that the acoustic velocity of waves of spurious response A is expressed by Formula (1). It was confirmed that the coefficients in Formula (1) preferably have the values listed in below Table 4 for the various crystal orientations of the silicon support substrate. 
     
       
         
           
               
               
             
               
                   
                 TABLE 4 
               
             
            
               
                   
                   
               
               
                   
                 s = 1 
               
            
           
           
               
               
               
               
            
               
                   
                 Si(100) 
                 Si(110) 
                 Si(111) 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                 a TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 a TLT   (2)   
                 0 
                 0 
                 0 
               
               
                 a TLT   (1)   
                 −436.3811104 
                 0 
                 0 
               
               
                 b TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 b TLT   (2)   
                 0 
                 0 
                 0 
               
               
                 c TLT   
                 0.251271186 
                 0 
                 0 
               
               
                 a TS   (2)   
                 0 
                 0 
                 0 
               
               
                 a TS   (1)   
                 −370.8189665 
                 0 
                 −174.7116877 
               
               
                 b TS   (2)   
                 0 
                 0 
                 0 
               
               
                 c TS   
                 0.250529661 
                 0 
                 0.24371308 
               
               
                 a TE   (1)   
                 0 
                 0 
                 0 
               
               
                 c TE   
                 0 
                 0 
                 0 
               
               
                 a ψSi   (5)   
                 0 
                 −6.73542E−06 
                 0 
               
               
                 a ψSi   (4)   
                 0 
                 −4.84328E−05 
                 0.000703463 
               
               
                 a ψSi   (3)   
                 0.044803063 
                 0.020121569 
                 −4.77016E−05 
               
               
                 a ψSi   (2)   
                 −0.11149637 
                 0.237494527 
                 −0.961938987 
               
               
                 a ψSi   (1)   
                 −43.37701861 
                 −1.22341255 
                 0.091605753 
               
               
                 b ψSi   (5)   
                 0 
                 −78830.27657 
                 0 
               
               
                 b ψSi   (4)   
                 0 
                 1406271.562 
                 340944.6167 
               
               
                 b ψSi   (3)   
                 51.32996847 
                 −17.77613547 
                 −41.61537323 
               
               
                 b ψSi   (2)   
                 280.2660593 
                 895.2921635 
                 446.7591732 
               
               
                 c ψSi   
                 22.32521186 
                 45.02689779 
                 30.11392405 
               
               
                 a θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 a θLT   (1)   
                 0 
                 0 
                 0 
               
               
                 b θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 c θLT   
                 −90 
                 −90 
                 90 
               
               
                 d TLTTS   
                 0 
                 0 
                 0 
               
               
                 d TLTTE   
                 0 
                 0 
                 0 
               
               
                 d TLT   ψSi   
                 0 
                 0 
                 0 
               
               
                 e 
                 5141.869703 
                 5073.066348 
                 4781.489451 
               
               
                 a TP   
                 0 
                 0 
                 0 
               
               
                 b TP   
                 0.02 
                 0.02 
                 0.02 
               
               
                 c TP   
                 −0.0102 
                 −0.0102 
                 −0.0102 
               
               
                 d TP   
                 1.0002 
                 1.0002 
                 1.0002 
               
               
                   
               
            
           
         
       
     
     When V h1_t  represent the acoustic velocity of waves of spurious A, the frequency of spurious A is expressed as f h1_t   (n) =V h1_t /λ t   (n)  according to Formula (2). Here, f h1  represents the frequency of spurious response A and t represents the number of an element, such as a resonator of the n-th filter. 
     In the present preferred embodiment, f h1_t   (n)  is higher than f u   (m)  or lower than f l   (m) , as illustrated in Formula (3) or Formula (4). In other words, f h1_t   (n)  is lower than the low-frequency ends or higher than the high-frequency ends of the second pass band, the third pass band, and the fourth pass band illustrated in  FIG. 4 . Therefore, it is clear that the frequency f h1_t   (n)  of spurious response A is not located inside the second to fourth pass bands. 
     In Formula (1), a) in the case where Si(100) (Euler angles (φ Si =0±5°, θ Si =0±5°, ψ Si )) is used, the range of ψ Si  is about 0°≤ψ Si  about 45°. However, due to the symmetry of the crystal structure of Si(100), ψ Si  and ψ Si ±(n×90°) are synonymous (n=1, 2, 3 . . . ). Similarly, ψ Si  and −ψ Si  have the same meaning. 
     (b) In the case where Si(110) (Euler angles (φ Si =−45±5°, θ Si =−90±5°, ψ Si )) is used, the range of ψ Si  is about 0°ψ Si ≤about 90°. Due to the symmetry of the crystal structure of Si(110), ψ Si  and ψ Si ±(n×180°) are synonymous (n=1, 2, 3 . . . ). Similarly, ψ Si  and −ψ Si  have the same meaning. 
     (c) In the case where Si(111) (Euler angles (φ Si =−45±5°, θ Si =−54.73561±5°, ψ Si )) is used, the range of ψ Si  is about 0°&lt;ψ Si  about 60°. However, due to the symmetry of the crystal structure of Si(111), ψ Si  and ψ Si ±(n×120°) are synonymous (n=1, 2, 3 . . . ). 
     In addition, although the range of θ LT  is about −180°&lt;θ LT ≤about 0°, θ LT  and θ LT  180° may be treated as being synonymous. 
     In this specification, in Euler angles (0°±5°, θ, 0°±15°), “0°±5°” means within a range greater than or equal to −5° and less than or equal to +5° and “0°±15°” means within a range greater than or equal to −15° and less than or equal to +15°. In this specification, for example, a range within 0°±5° may be simply referred to as 0°±5°. 
     The wavelength-normalized thickness T E  of the IDT electrode  15  is a thickness converted into a film thickness of an IDT electrode made of aluminum. However, the electrode material is not limited to Al. Various metals such as, for example, Ti, NiCr, Cu, Pt, Au, Mo, and W may be used. In addition, alloys including these metals as main components may be used. Furthermore, a multilayer metal film obtained by stacking a plurality of metal films made of such metals or alloys may be used. 
       FIG. 12A  is a diagram illustrating the filter characteristics of an acoustic wave device of a comparative example in which the acoustic wave resonator does not satisfy Formula (3) and Formula (4), and  FIG. 12B  is a diagram illustrating the filter characteristics of the acoustic wave device of the first preferred embodiment. 
     In  FIGS. 12A and 12B , the filter characteristics of the first acoustic wave filter and the second acoustic wave filter are illustrated. The solid line represents the filter characteristics of the first acoustic wave filter. As illustrated by the broken line in  FIG. 12A , a ripple appears in the pass band in the filter characteristics of the second acoustic wave filter. This is ripple is caused by a spurious response of an acoustic wave resonator inside the first acoustic wave filter. In contrast, as illustrated in  FIG. 12B , in the acoustic wave device of the first preferred embodiment, this ripple does not appear in the pass band of the second acoustic wave filter. In other words, as a result of the acoustic wave resonator being configured to satisfy Formula (3) or Formula (4), the ripple does not appear in the second pass band of the second acoustic wave filter. 
       FIG. 13  is a diagram illustrating the relationship between the wavelength-normalized thickness of the silicon support substrate and the maximum values of the phases of spurious response A, spurious response B, and spurious response C. As is clear from  FIG. 13 , when the wavelength-normalized thickness of the silicon support substrate is greater than about 4, the magnitude of spurious response A is almost constant and it is clear that variations in the wavelength-normalized thickness of the silicon support substrate are easily tolerated. When the wavelength-normalized thickness of the silicon support substrate is greater than about 10, spurious response B and spurious response C are constant and when the wavelength-normalized thickness of the silicon support substrate is greater than about 20, spurious response A, spurious response B, and spurious response C are all constant. Therefore, the wavelength-normalized thickness T Si  of the silicon support substrate is preferably T Si &gt;response 4. The wavelength-normalized thickness T Si  of the silicon support substrate is more preferably T Si &gt;response 10. The wavelength-normalized thickness T Si  of the silicon support substrate is even more preferably T Si &gt;response 20. 
     In the present preferred embodiment, the frequency of spurious response A of at least one acoustic wave resonator among the plurality of acoustic wave resonators of the first acoustic wave filter  3  satisfies Formula (3) or Formula (4). More preferably, it is preferable the frequency of spurious response A of an acoustic wave resonator that is closest to the antenna terminal satisfy Formula (3) or Formula (4). This is because the effect of spurious response A of the acoustic wave resonator that is closest to the antenna terminal tends to appear more significantly in the pass bands of the other second to fourth acoustic wave filters  4  to  6  compared with the other acoustic wave resonators. 
     More preferably, it is preferable that the frequency position of spurious response A satisfy Formula (3) or Formula (4) in all of the acoustic wave resonators. As a result, it is even more unlikely that a ripple caused by spurious response A will be generated in the pass bands of the second to fourth acoustic wave filters  4  to  6 . 
     With the structure of the present preferred embodiment of the present invention, as described above, the waves of spurious response A tend to be confined to the portion where the silicon oxide film  13  and the lithium tantalate film  14  are stacked, but when the wavelength-normalized thickness of the lithium tantalate film  14  is less than or equal to about 3.5×, the portion where the silicon oxide film  13  and the lithium tantalate film  14  are stacked becomes thinner, and therefore the waves of spurious response A are less likely to be confined. 
     More preferably, the wavelength-normalized thickness of the lithium tantalate film  14  is less than or equal to about 2.5×, for example, and in this case, the absolute value of a temperature coefficient of frequency TCF can be made smaller. In addition, the wavelength-normalized thickness of the lithium tantalate film  14  is preferably less than or equal to about 1.5×, for example. In this case, the electromechanical coupling coefficient can be easily adjusted. In addition, more preferably, the wavelength-normalized thickness of the lithium tantalate film  14  is less than or equal to about 0.5×, for example. In this case, the electromechanical coupling coefficient can be easily adjusted over a wide range. 
     Second Preferred Embodiment 
     In a second preferred embodiment of the present invention, a ripple caused by spurious response B rather than spurious response A is not located in the pass bands of the second to fourth acoustic wave filters  4  to  6 . This will be explained with reference to  FIGS. 14 to 19 . 
     As illustrated in  FIG. 14 , the acoustic velocity of the waves of spurious response B changes with the propagation direction ψ Si . Similarly, as illustrated in  FIG. 15 , the acoustic velocity of the waves of spurious response B also changes with the wavelength-normalized thickness T LT  of the lithium tantalate film. As illustrated in  FIG. 16 , the acoustic velocity of the waves of spurious response B also changes with the cut angle (about 90°−θ LT ) of the lithium tantalate film. As illustrated in  FIG. 17 , the acoustic velocity of the waves of spurious response B also changes with the wavelength-normalized thickness T S  of the silicon oxide film. As illustrated in  FIG. 18 , the acoustic velocity of the waves of spurious response B also changes with the wavelength-normalized thickness T E  of the IDT electrode. Furthermore, as illustrated in  FIG. 19 , it is clear that the acoustic velocity of the waves of spurious response B also changes with the wavelength-normalized thickness T P  of the protective film, which is preferably a silicon oxide film, for example. From the results illustrated in  FIGS. 14 to 19 , it was discovered that the acoustic velocity of the waves of spurious response B is also expressed by Formula (1), similarly to the first preferred embodiment. However, the coefficients in Formula (1) preferably have the values listed in Table 5 below for the various crystal orientations of the silicon support substrate in the case of the waves of spurious B. 
     
       
         
           
               
               
             
               
                   
                 TABLE 5 
               
             
            
               
                   
                   
               
               
                   
                 s = 2 
               
            
           
           
               
               
               
               
            
               
                   
                 Si(100) 
                 Si(110) 
                 Si(111) 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                 a TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 a TLT   (2)   
                 −3168.980655 
                 −3017.48047 
                 −6490.588929 
               
               
                 a TLT   (1)   
                 −1070.770975 
                 −1740.800945 
                 −1736.124534 
               
               
                 b TLT   (3)   
                 0 
                 0 
                 0 
               
               
                 b TLT   (2)   
                 0.012520877 
                 0.011673567 
                 0.012932291 
               
               
                 c TLT   
                 0.25 
                 0.26454918 
                 0.244042365 
               
               
                 a TS   (2)   
                 0 
                 0 
                 −5447.157686 
               
               
                 a TS   (1)   
                 −1175.713239 
                 −1443.794269 
                 −1260.230106 
               
               
                 b TS   (2)   
                 0 
                 0 
                 0.012411464 
               
               
                 c TS   
                 0.250104384 
                 0.264754098 
                 0.243336275 
               
               
                 a TE   (1)   
                 −440.9839549 
                 −631.5739347 
                 −613.3480905 
               
               
                 c TE   
                 0.249686848 
                 0.254030055 
                 0.249646955 
               
               
                 a ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 a ψSi   (4)   
                 0 
                 −8.39122E−05 
                 0 
               
               
                 a ψSi   (3)   
                 −0.01385161 
                 −0.000219614 
                 0 
               
               
                 a ψSi   (2)   
                 0.040113138 
                 0.288564318 
                 0.113517332 
               
               
                 a ψSi   (1)   
                 10.06827612 
                 −1.021332326 
                 0.01604077 
               
               
                 b ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 b ψSi   (4)   
                 0 
                 1489555.068 
                 0 
               
               
                 b ψSi   (3)   
                 −14.09179662 
                 −1671.679914 
                 0 
               
               
                 b ψSi   (2)   
                 281.2490194 
                 936.1444126 
                 446.7377612 
               
               
                 c ψSi   
                 22.53131524 
                 46.09631148 
                 29.70873786 
               
               
                 a θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 a θLT   (1)   
                 −2.926997767 
                 0 
                 0 
               
               
                 b θLT   (2)   
                 0 
                 0 
                 0 
               
               
                 c θLT   
                 −50.02087683 
                 −90 
                 −90 
               
               
                 d TLTTS   
                 0 
                 0 
                 0 
               
               
                 d TLTTE   
                 −2151.133017 
                 −2785.629029 
                 −2639.620065 
               
               
                 d TLT   ψSi   
                 17.13925013 
                 0 
                 0 
               
               
                 e 
                 5188.573706 
                 5052.799929 
                 5187.888817 
               
               
                 a TP   
                 0 
                 0 
                 0 
               
               
                 b TP   
                 −0.1336 
                 −0.1336 
                 −0.1336 
               
               
                 c TP   
                 −0.0552 
                 −0.0552 
                 −0.0552 
               
               
                 d TP   
                 1.001 
                 1.001 
                 1.001 
               
               
                   
               
            
           
         
       
     
     The frequency position of spurious response B is obtained from the frequency position of spurious B f h2_t   (n) =V h2_t /λ t   (n)  using Formula (2) from the obtained acoustic velocity V h2_t  of the waves of spurious response B, as described above. In addition, in the second preferred embodiment, the frequency position f h2_t   (n)  of spurious response B is set so that the frequency position of spurious response B satisfies Formula (3A) or Formula (4A) below. Therefore, in the second preferred embodiment, spurious response B is located outside the second to fourth pass bands of the second to fourth acoustic wave filters  4  to  6 . Therefore, ripples are unlikely to be generated in the filter characteristics of the second to fourth acoustic wave filters  4  to  6  due to spurious response B. 
         f   h2_t   (n)   &gt;f   u   (m)   Formula (3A)
 
         f   h2_t   (n)   &lt;f   l   (m)   Formula (4A)
 
     More preferably, the frequency position of spurious response B satisfy Formula (3A) or Formula (4A) in all of the acoustic wave resonators. As a result, it is even more unlikely that a ripple caused by spurious response B will be generated in the pass bands of the second to fourth acoustic wave filters  4  to  6 . However, it is sufficient that the frequency position of spurious response B satisfy Formula (3A) or Formula (4A) in at least one acoustic wave resonator of the first acoustic wave filter  3 . 
     Third Preferred Embodiment 
     In a third preferred embodiment of the present invention, a ripple caused by spurious response C rather than spurious response A is not located in the pass bands of the second to fourth acoustic wave filters  4  to  6 . This will be explained with reference to  FIGS. 20 to 25 . 
     As illustrated in  FIG. 20 , the acoustic velocity of the waves of spurious response C changes with the propagation direction ψ Si . Similarly, as illustrated in  FIG. 21 , the acoustic velocity of the waves of spurious response C also changes with the wavelength-normalized thickness T LT  of the lithium tantalate film. As illustrated in  FIG. 22 , the acoustic velocity of the waves of spurious response C also changes with the cut angle (about 90°−θ LT ) of the lithium tantalate film. As illustrated in  FIG. 23 , the acoustic velocity of the waves of spurious response C also changes with the wavelength-normalized thickness T S  of the silicon oxide film. As illustrated in  FIG. 24 , the acoustic velocity of the waves of spurious response C also changes with the wavelength-normalized thickness T E  of the IDT electrode. In addition, as illustrated in  FIG. 25 , the acoustic velocity of the waves of spurious response C also changes with the wavelength-normalized thickness T P  of the protective film, which is preferably a silicon oxide film, for example. From the results illustrated in  FIGS. 20 to 25 , it was discovered that the acoustic velocity of the waves of spurious response C is also expressed by Formula (1), similarly to as in the case of the first preferred embodiment. However, the coefficients in Formula (1) preferably have the values listed in below Table 6 for the various crystal orientations of the silicon support substrate in the case of the waves of spurious response C. 
     
       
         
           
               
               
             
               
                   
                 TABLE 6 
               
             
            
               
                   
                   
               
               
                   
                 s = 3 
               
            
           
           
               
               
               
               
            
               
                   
                 Si(100) 
                 Si(110) 
                 Si(111) 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                 a TLT   (3)   
                 24973.69604 
                 0 
                 0 
               
               
                 a TLT   (2)   
                 1430.312613 
                 0 
                 −6156.310187 
               
               
                 a TLT   (1)   
                 −1027.290569 
                 −737.4915071 
                 −1476.525631 
               
               
                 b TLT   (3)   
                 −0.000364885 
                 0 
                 0 
               
               
                 b TLT   (2)   
                 0.011041041 
                 0 
                 0.009760415 
               
               
                 c TLT   
                 0.277376171 
                 0.283128295 
                 0.290986602 
               
               
                 a TS   (2)   
                 0 
                 0 
                 0 
               
               
                 a TS   (1)   
                 −617.9601012 
                 0 
                 0 
               
               
                 b TS   (2)   
                 0 
                 0 
                 0 
               
               
                 c TS   
                 0.273226238 
                 0 
                 0 
               
               
                 a TE   (1)   
                 −254.6645301 
                 −400.406917 
                 −524.8090015 
               
               
                 c TE   
                 0.262449799 
                 0.265026362 
                 0.25499391 
               
               
                 a ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 a ψSi   (4)   
                 0 
                 −0.000208283 
                 −0.000637441 
               
               
                 a ψSi   (3)   
                 0 
                 −0.004709453 
                 −0.001817349 
               
               
                 a ψSi   (2)   
                 −0.126294383 
                 0.574442977 
                 0.749991624 
               
               
                 a ψSi   (1)   
                 2.746835794 
                 7.891650217 
                 −0.116425099 
               
               
                 b ψSi   (5)   
                 0 
                 0 
                 0 
               
               
                 b ψSi   (4)   
                 0 
                 1654327.754 
                 303345.3736 
               
               
                 b ψSi   (3)   
                 0 
                 −7229.860818 
                 −201.8986483 
               
               
                 b ψSi   (2)   
                 266.9400494 
                 985.3381236 
                 411.1815157 
               
               
                 c ψSi   
                 22.10843373 
                 47.8602812 
                 30.42021924 
               
               
                 a θLT   (2)   
                 0 
                 0 
                 0.593925697 
               
               
                 a θLT   (1)   
                 0 
                 0 
                 −2.75182308 
               
               
                 b θLT   (2)   
                 0 
                 0 
                 67.60597649 
               
               
                 c θLT   
                 −90 
                 −90 
                 −49.6589525 
               
               
                 d TLTTS   
                 −2391.060714 
                 0 
                 0 
               
               
                 d TLTTE   
                 0 
                 0 
                 −2773.934251 
               
               
                 d TLT   ψSi   
                 −12.5134053 
                 15.88055602 
                 0 
               
               
                 e 
                 5661.675305 
                 5417.174527 
                 5440.57579 
               
               
                 a TP   
                 10.443 
                 10.443 
                 10.443 
               
               
                 b TP   
                 −0.0316 
                 −0.0316 
                 −0.0316 
               
               
                 c TP   
                 −0.0653 
                 −0.0653 
                 −0.0653 
               
               
                 d TP   
                 1.0011 
                 1.0011 
                 1.0011 
               
               
                   
               
            
           
         
       
     
     The frequency position of spurious response C is obtained from the frequency position of spurious response C f h3_t   (n) =V h3_t /λt (n)  using Formula (2) from the obtained acoustic velocity V h3_t  of the waves of spurious response C, as described above. In the third preferred embodiment, the frequency position of spurious response C is set so that the frequency position of spurious response C satisfies Formula (3B) or Formula (4B) below. Therefore, in the third preferred embodiment, spurious response C is located outside the second to fourth pass bands of the second to fourth acoustic wave filters  4  to  6 . Therefore, ripples caused by spurious response C are unlikely to be generated in the filter characteristics of the second to fourth acoustic wave filters  4  to  6 . 
         f   h3_t   (n)   &gt;f   u   (m)   Formula (3B)
 
         f   h3_t   (n)   &lt;f   l   (m)   Formula (4B)
 
     More preferably, the frequency position of spurious response C satisfy Formula (3B) or Formula (4B) in all of the acoustic wave resonators. As a result, it is even more unlikely that a ripple caused by spurious response C will be generated in the pass bands of the second to fourth acoustic wave filters  4  to  6 . However, it is sufficient that the frequency position of spurious response C satisfy Formula (3B) or Formula (4B) in at least one acoustic wave resonator of the first acoustic wave filter  3 . 
     Fourth Preferred Embodiment 
     A fourth preferred embodiment of the present invention satisfies all of content of the first preferred embodiment, the second preferred embodiment, and the third preferred embodiment. The specific structure of an acoustic wave device of the fourth preferred embodiment is the same or substantially the same as that of the first to third preferred embodiments. 
     In the fourth preferred embodiment, the frequency positions of spurious response A, spurious response B, and spurious response C represented by Formula (2) are expressed as f hs_t   (n) =V hs_t /λ t   (n) , where the acoustic velocities of the waves of spurious response A, spurious response B, and spurious response C are represented by V h1_t , V h2_t , and V h3_t , respectively. Here, s is 1, 2, or 3. In the fourth preferred embodiment, a first frequency f h1_t   (n) , which is the frequency of spurious response A, a second frequency f h2_t   (n) , which is the frequency of spurious response B, and a third frequency f h3_t   (n) , which is the frequency of spurious response C, are all higher than f u   (m)  or lower than f l   (m) . Therefore, spurious response A, spurious response B, and spurious response C are located outside the second to fourth pass bands of the second to fourth acoustic wave filters  4  to  6 . Therefore, degradation of the filter characteristics of the second to fourth acoustic wave filters  4  to  6  is even less likely to occur. 
     Therefore, summarizing the conditions of the fourth preferred embodiment, f hs_t   (n)  (s is 1, 2, or 3) satisfies f hs_t   (n) &gt;f u   (m)  or f hs_t   (n) &lt;f l   (m)  when s is 1, 2, or 3. In the fourth preferred embodiment, it is preferable that T Si &gt;about 20 in order that the magnitudes of spurious response A, spurious response B, and spurious response C can be maintained constant or substantially constant. 
     In the fourth preferred embodiment, spurious response A, spurious response B, and spurious response C are preferably not located in the pass bands of the second to fourth acoustic wave filters, which are the other acoustic wave filters, but two of spurious response A, spurious response B, and spurious response C, such as spurious response A and spurious response B, spurious response A and spurious response C, or spurious response B and spurious response C may be located outside the pass bands of the second to fourth acoustic wave filters. In this case, the effects of the spurious responses can be further reduced compared with the first to third preferred embodiments. 
       FIG. 26  is a diagram illustrating the relationship between the film thickness of a LiTaO 3  film and the Q value in an acoustic wave device in which a low-acoustic-velocity film including a SiO 2  film having a thickness of about 0.35λ and a LiTaO 3  film having Euler angles (0°, 140.0°, 0°) are stacked above a high-acoustic-velocity silicon support substrate. The vertical axis in  FIG. 26  represents the product of a Q characteristic and a relative bandwidth (Δf) of a resonator. In addition,  FIG. 27  is a diagram illustrating the relationship between the film thickness of a LiTaO 3  film and the temperature coefficient of frequency TCF.  FIG. 28  is a diagram illustrating the relationship between the film thickness of a LiTaO 3  film and acoustic velocity. From  FIG. 26 , the film thickness of the LiTaO 3  film is preferably less than or equal to about 3.5λ, for example. In this case, the Q value is higher than in the case where the film thickness is greater than about 3.5λ. More preferably, the film thickness of the LiTaO 3  film is less than or equal to about 2.5λ, for example, in order to further increase the Q value. 
     Furthermore, from  FIG. 27 , in the case where the film thickness of the LiTaO 3  film is less than or equal to about 2.5λ, the absolute value of the temperature coefficient of frequency TCF can be reduced compared with the case where the film thickness is greater than about 2.5λ. More preferably, the film thickness of the LiTaO 3  film is less than or equal to about 2λ, for example, and in this case, the absolute value of the temperature coefficient of frequency TCF is less than or equal to about 10 ppm/° C. It is further preferable that the film thickness of the LiTaO 3  film is less than or equal to about 1.5λ, for example, in order to reduce the absolute value of the temperature coefficient of frequency TCF. 
     From  FIG. 28 , if the film thickness of the LiTaO 3  film is greater than about 1.5λ, the change in acoustic velocity is very small. 
     As illustrated in  FIG. 29 , the relative bandwidth varies by a large amount in a film thickness range from about 0.05λ to about 0.5λ for the LiTaO 3  film. Therefore, the electromechanical coupling coefficient can be adjusted over a wider range. Therefore, it is preferable that the film thickness of the LiTaO 3  film is in a range from about 0.05λ to about 0.5λ, for example, in order to widen the range over which the electromechanical coupling coefficient and the relative bandwidth can be adjusted. 
       FIGS. 30 and 31  are diagrams respectively illustrating the relationships between the film thickness (λ) of the SiO 2  film and the acoustic velocity and the electromechanical coupling coefficient. Here, a silicon nitride film, an aluminum oxide film, and diamond were used as the high-acoustic-velocity film underneath the low-acoustic-velocity film including SiO 2 . The film thickness of the high-acoustic-velocity film was about 1.5λ. The acoustic velocity of a bulk wave in silicon nitride is about 6000 m/s, the acoustic velocity of a bulk wave in aluminum oxide is about 6000 m/s, and the acoustic velocity of a bulk wave in diamond is about 12800 m/s. As illustrated in  FIGS. 30 and 31 , the electromechanical coupling coefficient and the acoustic velocity do not change substantially even when the material of the high-acoustic-velocity film and the film thickness of the SiO 2  film are changed. In particular, the electromechanical coupling coefficient does not substantially change regardless of the material of the high-acoustic-velocity film when the film thickness of the SiO 2  film lies in the range from about 0.1λ to about 0.5λ. Furthermore, it is clear from  FIG. 30  that the acoustic velocity does not change regardless of the material of the high-acoustic-velocity film when the film thickness of the SiO 2  film lies in a range from about 0.3λ to about 2λ. Therefore, the film thickness of the low-acoustic-velocity film including silicon oxide is preferably less than or equal to about 2λ and more preferably less than or equal to about 0.5λ, for example. 
       FIGS. 32 to 34  are partial enlarged front cross-sectional views for describing modifications in which the thickness of the protective film is different in different portions of the protective film in an acoustic wave resonator used in a preferred embodiment of the present invention. In the modifications illustrated in  FIGS. 32 to 34 , the protective film  18  covers the lithium tantalate film  14  and the upper surfaces and side surfaces of electrode fingers  15   a  of the IDT electrode  15 . In the modification illustrated in  FIG. 32 , the thickness of the portions of the protective film  18  covering the side surfaces of the electrode fingers  15   a  is smaller than the thickness of the portions of the protective film  18  covering the upper surfaces of the electrode fingers  15   a . In this case, the Q value can be increased and the electromechanical coupling coefficient can be increased. In more detail, since the Qm of the protective film  18  is small, the Q value of the acoustic wave resonator can be increased by making the portions of the protective film  18  on the side surfaces of the electrode fingers  15   a  thinner. Therefore, the loss of the acoustic wave filter can be reduced. In addition, when the protective film  18  is provided on the lithium tantalate film  14 , the difference in acoustic impedance between the portions where the electrode fingers  15   a  are provided and the gaps between the electrode fingers  15   a  becomes smaller. Therefore, the electromechanical coupling coefficient is reduced. However, the electromechanical coupling coefficient can be increased by reducing the thickness of the portions of the protective film  18  on the side surfaces of the electrode fingers  15   a.    
     On the other hand, in the modification illustrated in  FIG. 33 , the thickness of the portions of the protective film  18  on the lithium tantalate film  14  is smaller than the thickness of the portions of the protective film  18  that cover the upper surfaces of the electrode fingers  15   a . In this case as well, the electromechanical coupling coefficient can be increased. In other words, the electromechanical coupling coefficient can be increased by reducing the thickness of the portions of the protective film  18  that cover the lithium tantalate film  14 . 
     In the modification illustrated in  FIG. 34 , the thickness of the portions of the protective film  18  on the lithium tantalate film  14  is larger than the thickness of the portions of the protective film  18  that cover the upper surfaces of the electrode fingers  15   a . In this case, the electromechanical coupling coefficient can be reduced and a narrower bandwidth can be achieved. 
       FIG. 35  is a front sectional view for describing a modification of an acoustic wave resonator used in a preferred embodiment of the present invention. The acoustic wave resonator of this modification is configured in the same or substantially the same way as the acoustic wave resonator  11  illustrated in  FIG. 3A , except that the silicon oxide film  13  is not provided. Thus, an acoustic wave resonator used in a preferred embodiment of the present invention may have a structure in which the lithium tantalate film  14  is directly stacked above the silicon support substrate  12 . In this case, the thickness of the silicon oxide film  13  is 0. 
       FIG. 36  is a front cross-sectional view illustrating yet another modification of an acoustic wave resonator used in an acoustic wave device according to a preferred embodiment of the present invention. In an acoustic wave resonator  11 A, the protective film  18  is stacked on the upper surfaces of the electrode fingers of the IDT electrode  15 . The protective film  18  does not extend onto the side surfaces of the electrode fingers of the IDT electrode  15 . Thus, the protective film  18  may be stacked on only the upper surfaces of the electrode fingers. 
       FIG. 37  is a partial front cross-sectional view for describing a structure in a case where the protective film  18  is multilayer film. The protective film  18  has a structure in which a first protective film layer  18   a , a second protective film layer  18   b , and a third protective film layer  18   c  are stacked on top of one another. Thus, the protective film  18  may be a multilayer film including a plurality of protective film layers. In this case, the wavelength-normalized thickness T P  of the protective film  18  is obtained as the sum of the products of values obtained by dividing the density of each protective film layer by the density of silicon oxide and the wavelength normalized thicknesses of these protective film layers. For example, the wavelength-normalized thickness T P  of the protective film  18  is provided by T P =(d1/d0)t1+(d2/d0)t2+(d3/d0)t3, where d0 is the density of silicon oxide, d1 is the density and t1 is the wavelength-normalized thickness of the first protective film layer  18   a , d2 is the density and t2 is the wavelength-normalized thickness of the second protective film layer  18   b , and d3 is the density and t3 is the wavelength-normalized thickness of the third protective film layer  18   c.    
     As illustrated in  FIG. 38 , Si(100) refers to a substrate that has been cut along the (100) plane perpendicular or substantially perpendicular to the crystal axis represented by the Miller index [100] in a silicon crystal structure having a diamond structure. Crystallographically equivalent planes such as Si(010) are also included. 
     As illustrated in  FIG. 39 , Si(110) refers to a substrate that has been cut along the (110) plane perpendicular or substantially perpendicular to the crystal axis represented by the Miller index [110] in a silicon crystal structure having a diamond structure. Other crystallographically equivalent planes are also included. 
     As illustrated in  FIG. 40 , Si(111) refers to a substrate that has been cut along the (111) plane perpendicular or substantially perpendicular to the crystal axis represented by the Miller index [111] in a silicon crystal structure having a diamond structure. Other crystallographically equivalent planes are also included. 
     The acoustic wave devices of the above-described preferred embodiments can be used as a component, such as a duplexer of a radio-frequency front end circuit, for example. An example of such a radio-frequency front end circuit will be described below. 
       FIG. 41  is a schematic configuration diagram of a communication device that includes a radio-frequency front end circuit. A communication device  240  includes an antenna  202 , a radio-frequency front end circuit  230 , and an RF signal processing circuit  203 . The radio-frequency front end circuit  230  is a circuit portion that is connected to the antenna  202 . The radio-frequency front end circuit  230  includes an acoustic wave device  210  and amplifiers  221  to  224 . The acoustic wave device  210  includes first to fourth filters  211  to  214 . The acoustic wave device of a preferred embodiment of the present invention described above can be used as the acoustic wave device  210 . The acoustic wave device  210  includes an antenna common terminal  225  that is connected to the antenna  202 . First terminals of the first to third filters  211  to  213 , which are reception filters, and a first terminal of the fourth filter  214 , which is a transmission filter, are all commonly connected to the antenna common terminal  225 . Output terminals of the first to third filters  211  to  213  are respectively connected to the amplifiers  221  to  223 . In addition, the amplifier  224  is connected to an input terminal of the fourth filter  214 . 
     The output terminals of the amplifiers  221  to  223  are connected to the RF signal processing circuit  203 . The input terminal of the amplifier  224  is connected to the RF signal processing circuit  203 . 
     An acoustic wave device according to a preferred embodiment of the present invention can be suitably used as the acoustic wave device  210  in the communication device  240 . 
     An acoustic wave device according to a preferred embodiment of the present invention can be used for various communication bands, but for example, it is suitable to use the pass bands of communication bands defined in the 3GPP standards as the pass band of the acoustic wave filters. 
     In addition, one or more acoustic wave resonators of an acoustic wave device according to a preferred embodiment of the present invention are preferably all acoustic wave resonators that satisfy Formula (3) or Formula (4). An acoustic wave device according to a preferred embodiment of the present invention can be used in various communication applications, but is preferably used as a carrier aggregation composite filter, for example. In other words, a carrier aggregation composite filter further includes an antenna terminal to which first ends of a plurality of acoustic wave filters are commonly connected, and three or more acoustic wave filters are commonly connected to the antenna terminal side. The plurality of acoustic wave filters simultaneously transmit and receive signals of a plurality of communication bands. 
     An acoustic wave device according to a preferred embodiment of the present invention may include only a plurality of transmission filters or may include a plurality of reception filters. The acoustic wave device includes n band pass filters, where n is greater than or equal to two. Therefore, a duplexer is also an acoustic wave device according to a preferred embodiment of the present invention. 
     Preferred embodiments of the present invention are widely applicable to communication devices such as mobile phones in the form of filters, acoustic wave devices applicable to multi-band systems, front end circuits, and communication devices. 
     While preferred embodiments of the present invention have been described above, it is to be understood that variations and modifications will be apparent to those skilled in the art without departing from the scope and spirit of the present invention. The scope of the present invention, therefore, is to be determined solely by the following claims.