Abstract:
A Capacitive Micromachined Ultrasonic Transducer (CMUT) having a membrane operatively connected to a top electrode and having a bottom electrode having a concave void. When a DC bias voltage is applied, the membrane is deflected towards the bottom electrode such that a peripheral edge region of the membrane is brought into close proximity with the bottom electrode and an electrostatic force proximal to the peripheral edge region of the membrane is increased.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The invention relates to an improved a Capacitive Micromachined Ultrasonic Transducer (CMUT) and method for manufacturing the CMUT. 
         [0003]    2. Description of the Background Art 
         [0004]      FIGS. 1 to 3  illustrate a conventional working principle of a Capacitive Micromachined Ultrasonic Transducer (CMUT)  100  with a flat bottom electrode  140 . 
         [0005]    Referring to  FIG. 1 , a CMUT  100  is similar to a parallel plate capacitor having a top electrode  110  on a dielectric membrane  120  that is isolated by a vacuum or air cavity  130  to a bottom electrode  140 . The bottom electrode  140  is usually formed on a flat conductive substrate. The top electrode  110  and the bottom electrode  140  may be made from a conductive material such as a conductive silicon substrate. The membrane  120  is made from conductive material or is coated with a conductive material. When actuated by electrostatic force with an AC voltage, the membrane  120  can vibrate to generate ultrasound like a drum diaphragm. Therefore, the CMUT  100  can be used as an ultrasound emitter and receiver. Only 25% of the area near the center of the membrane  120  is patterned with a top electrode  110  since the remaining 75% area has much less capacitance change, which is considered as parasitic capacitance to be removed. In other words, only 25% of the central area of the membrane  120  is patterned with a top electrode  110  to conduct effective capacitance. 
         [0006]    In  FIG. 2 , when a DC bias voltage is applied, the electrostatic force pushes the membrane  120  toward the bottom electrode  140 . The effective capacitance is inversely proportional to the gap distance of the air cavity  130  between the top electrode  110  and the bottom electrode  140 . In other words, effective capacitance can be achieved only when the gap distance is small. Only the middle section of membrane  120  can produce effective capacitance even if the entire membrane  120  is patterned with top electrode  110  because the bottom electrode  140  has a flat bottom. For instance, the capacitance produced in area  150  is considered parasitic capacitance. 
         [0007]    To increase the sensitivity, the DC bias voltage is applied to load up the capacitor with charges, which can also pull the membrane  120  closer to the bottom electrode  140  to get a higher capacitance. The maximum sensitivity can be achieved when the membrane  120  is closest to the bottom electrode  140  without collapsing to the bottom electrode  140 . 
         [0008]    As the DC bias voltage increases, deflection of the membrane  120  also increases. However, when the DC bias voltage is increased above a certain voltage, electrostatic forces pressure the membrane  120  to collapse on the bottom electrode  140 . 
         [0009]      FIG. 3  illustrates a situation where the DC bias voltage is used to collapse the membrane  120 . As a result, the contribution of the affected areas  160  to the effective capacitance is significantly reduced. When the DC bias voltage is large enough to bring the membrane  120  to be deflected to more than ⅓ of the gap distance of the air gap  130 , the membrane  120  will collapse and make contact with the bottom electrode  140 . 
         [0010]      FIG. 4  illustrates the conventional CMUT arrays. The top electrodes  310  can only cover part of the membrane. 
         [0011]    Referring to  FIG. 5 , the capacitance is simply a series combination of two parallel plate capacitors, capacitance C 1  is the capacitance of dielectric membrane, and C 2  is the capacitance of the air cavity, where d 1  is the thickness of the membrane, d 2  is the depth of the air cavity, b is the radius of the top electrode, ∈ 1  and ∈ 2  are the relative dielectric constants, ∈ 0  is the vacuum permittivity. 
         [0000]    
       
         
           
             C 
             = 
             
               
                 1 
                 
                   
                     1 
                     
                       C 
                       1 
                     
                   
                   + 
                   
                     1 
                     
                       C 
                       2 
                     
                   
                 
               
               = 
               
                 
                   1 
                   
                     
                       1 
                       
                         
                           ɛ 
                           0 
                         
                          
                         
                           ɛ 
                           1 
                         
                          
                         
                           A 
                           
                             d 
                             1 
                           
                         
                       
                     
                     + 
                     
                       1 
                       
                         
                           ɛ 
                           0 
                         
                          
                         
                           ɛ 
                           2 
                         
                          
                         
                           A 
                           
                             d 
                             2 
                           
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       πɛ 
                       0 
                     
                      
                     
                       b 
                       2 
                     
                   
                   
                     
                       
                         d 
                         1 
                       
                       
                         ɛ 
                         1 
                       
                     
                     + 
                     
                       
                         d 
                         2 
                       
                       
                         ɛ 
                         2 
                       
                     
                   
                 
               
             
           
         
       
     
         [0012]    Referring to  FIG. 6 , for a flat bottom electrode with deflected membrane of a conventional CMUT, the deflected circular membrane is assumed to be a spherical shell partially covered by the top electrode, where R a  is the inner shell radius, R b  is the outer shell radius, and h is the height of the inner shell. C 1  from the deflected dielectric membrane is calculated by the equation of parallel plate capacitor with the area of the partial spherical shell. 
         [0000]    
       
         
           
             
               C 
               1 
             
             = 
             
               4 
                
               
                   
               
                
               
                 πɛ 
                 0 
               
                
               
                 ɛ 
                 1 
               
                
               h 
                
               
                   
               
                
               
                 
                   
                     R 
                     a 
                   
                    
                   
                     R 
                     b 
                   
                 
                 
                   d 
                   1 
                 
               
             
           
         
       
     
         [0013]    The capacitance in the air cavity between the bottom of the deflected membrane with radius R b  and the flat bottom electrode is calculated as follows. 
         [0014]    C 2 =(The parallel plate capacitance between the flat bottom electrode and the virtual flat plate (dashed line))−(The capacitance between the spherical shell with radius R b  and the virtual flat plate). 
         [0000]    
       
         
           
             
               C 
               2 
             
             = 
             
               1 
               
                 
                   1 
                   
                     
                       ɛ 
                       0 
                     
                      
                     
                       ɛ 
                       2 
                     
                      
                     
                       
                         π 
                          
                         
                             
                         
                          
                         
                           b 
                           2 
                         
                       
                       
                         d 
                         2 
                       
                     
                   
                 
                 - 
                 
                   1 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       πɛ 
                       0 
                     
                      
                     
                       ɛ 
                       2 
                     
                      
                     
                       R 
                       b 
                     
                      
                     ln 
                      
                     
                         
                     
                      
                     
                       H 
                       
                         
                           
                             
                               R 
                               b 
                               2 
                             
                             - 
                             
                               b 
                               2 
                             
                           
                         
                         - 
                         
                           
                             
                               R 
                               b 
                               2 
                             
                             - 
                             
                               a 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   C 
                   = 
                   
                     1 
                     
                       
                         1 
                         
                           C 
                           1 
                         
                       
                       + 
                       
                         1 
                         
                           C 
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       πɛ 
                       0 
                     
                     
                       
                         
                           d 
                           1 
                         
                         
                           4 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             1 
                           
                            
                           
                             hR 
                             a 
                           
                            
                           
                             R 
                             b 
                           
                         
                       
                       + 
                       
                         
                           d 
                           2 
                         
                         
                           
                             ɛ 
                             2 
                           
                            
                           
                             b 
                             2 
                           
                         
                       
                       - 
                       
                         1 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             2 
                           
                            
                           
                             R 
                             b 
                           
                            
                           ln 
                            
                           
                               
                           
                            
                           
                             H 
                             
                               
                                 
                                   
                                     R 
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     b 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     R 
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 R 
                 a 
               
               = 
               
                 
                   1 
                   
                     2 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                   ( 
                   
                     
                       H 
                       2 
                     
                     + 
                     
                       a 
                       2 
                     
                   
                   ) 
                 
               
             
             , 
             
               
                 R 
                 b 
               
               = 
               
                 
                   R 
                   a 
                 
                 + 
                 
                   d 
                   1 
                 
               
             
             , 
             
               h 
               = 
               
                 
                   R 
                   a 
                 
                 - 
                 
                   
                     
                       R 
                       a 
                       2 
                     
                     - 
                     
                       b 
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0015]      FIG. 7  is a graph of effective capacitance with respect to membrane deflection of a conventional CMUT with flat bottom electrodes. The diameter of the silicon nitride membrane is 100 μm, the thickness of the membrane is 0.2 μm, and the depth of the air cavity is 1 μm. These values are applied into the derived equations above. The relative dielectric constant of silicon nitride film is 7.5. For CMUTs with flat bottom electrodes, the capacitance change reaches its maximum of 22% when the diameter of the top electrode is 84 μm. The capacitance change drops to 9% when the top electrode fills the membrane. The maximum capacitance at the collapsed mode can only reach 0.075 pF. 
       SUMMARY OF THE INVENTION 
       [0016]    It is therefore an object of the present invention to provide a Capacitive Micromachined Ultrasonic Transducer (CMUT), that includes a membrane operatively connected to a top electrode; and a bottom electrode having a concave void. Whereby, when a DC bias voltage is applied, the membrane is deflected towards the bottom electrode such that a peripheral edge region of the membrane is brought into close proximity with the bottom electrode and an electrostatic force proximal to the peripheral edge region of the membrane is increased. 
         [0017]    When the DC bias voltage is applied, the distance between the peripheral edge region of the membrane and the bottom electrode may be less than the distance between a central region of the membrane and the bottom electrode. 
         [0018]    When the DC bias voltage applied is above a predetermined amount to collapse the membrane to the bottom electrode, contact between the membrane and the bottom electrode may be minimised to a central region of the membrane. 
         [0019]    About 25% of the membrane is in contact with the bottom electrode when the membrane is collapsed to the bottom membrane. 
         [0020]    The top electrode may have the same diameter as the void of the bottom electrode. 
         [0021]    The membrane may be flat or deflected. 
         [0022]    The size of the membrane may be from about 500 μm to 5 μm with a frequency range from 100 kHz up to 100 MHz in air. 
         [0023]    The thickness of the membrane may be from about 0.1 μm to 10 μm. 
         [0024]    The CMUT may have an array of membranes where each top electrode fills the entire area of each membrane leaving only small voids for anchoring each membrane. 
         [0025]    In another embodiment, a method for manufacturing a Capacitive Micromachined Ultrasonic Transducer (CMUT) is provided, whereby the method includes the features of sputtering a layer of Cr/Au as a seed layer on a silicon substrate that includes a layer of silicon nitride to form a CMUT membrane; coating a patterned photoresist to define the active area of a CMUT cell; melting the patterned photoresist to form a spherical profile by surface tension; and electroplating of nickel with the seed layer to form the bottom electrode by over-plating to cover the patterned photoresist. 
         [0026]    The Young&#39;s modulus of the silicon nitride may be around 200 GPa. 
         [0027]    The method may further include sealing released holes caused by the electroplating using a silicone-based polydimethylsiloxane (PDMS) with air trapped in CMUT cavities. 
         [0028]    The method may further include coating parylene C in a vacuum chamber. 
         [0029]    The method may further include removing the silicon substrate by single-side potassium hydroxide (KOH) etching that stops at the silicon nitride membrane. 
         [0030]    The method may further include patterning the PDMS to define a membrane area and array elements. 
         [0031]    The method may further include wire bonding to front-end electronics. 
         [0032]    Further scope of applicability of the present invention will become apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0033]    The present invention will become more fully understood from the detailed description given hereinbelow and the accompanying drawings which are given by way of illustration only, and thus, are not limitive of the present invention, and wherein: 
           [0034]      FIG. 1  illustrates the conventional working principle of CMUT with a flat bottom electrode; 
           [0035]      FIG. 2  illustrates the conventional working principle of CMUT with a flat bottom electrode when DC bias is applied; 
           [0036]      FIG. 3  illustrates the conventional working principle of CMUT with a flat bottom electrode when DC biasing is applied to collapse the membrane; 
           [0037]      FIG. 4  illustrates the conventional CMUT array arrangement; 
           [0038]      FIG. 5  illustrate the conventional working principle of CMUT with a flat bottom electrode; 
           [0039]      FIG. 6  illustrate the conventional working principle of CMUT with a flat bottom electrode; 
           [0040]      FIG. 7  illustrates a graph of effective capacitance with respect to membrane deflection of a conventional CMUT with a flat bottom electrode; 
           [0041]      FIG. 8  illustrate a CMUT with a concave bottom electrode in accordance with an embodiment of the present invention; 
           [0042]      FIG. 9  illustrate a CMUT with a concave bottom electrode in accordance with an embodiment of the present invention when DC bias is applied; 
           [0043]      FIG. 10  illustrate a CMUT with a concave bottom electrode in accordance with an embodiment of the present invention when DC biasing is applied to collapse the membrane; 
           [0044]      FIG. 11  illustrates an exemplary CMUT array arrangement in accordance with an embodiment of the present invention; 
           [0045]      FIG. 12  illustrate the working principle of CMUT with a concave bottom electrode in accordance with an embodiment of the present invention; 
           [0046]      FIG. 13  illustrate the working principle of CMUT with a concave bottom electrode in accordance with an embodiment of the present invention; 
           [0047]      FIG. 14  illustrates a graph of effective capacitance with respect to membrane deflection of a CMUT with a concave bottom electrode in accordance with the present invention; and 
           [0048]      FIG. 15  illustrates an exemplary fabrication process of CMUT arrays with concave bottom electrodes. 
       
    
    
     DETAILED DESCRIPTION 
       [0049]    Referring to  FIGS. 8 to 10 , a Capacitive Micromachined Ultrasonic Transducer (CMUT)  200  with a concave shaped bottom electrode  240  is depicted. A concave air cavity  230  is defined by the concavity of the bottom electrode  240 . Turning to  FIG. 8 , the top electrode  210  covers 100% of the area  260  of the membrane  220  above the air cavity  230 . Consequently, the effective capacitance for the CMUT  200  can be significantly higher than the conventional CMUT  100  of  FIG. 1  with a top electrode  110  which covers only 25% of the membrane area  120 . 
         [0050]    Turning to  FIG. 9 , when direct current (DC) bias is applied, the entire area  260  of the membrane  220  above the air cavity  230  is considered to produce effective capacitance. The concavity of the bottom electrode  240  substantially conforms to the deflection of the membrane  220  when a DC bias voltage is applied. 
         [0051]    If the bottom electrode  240  is defined with a concave shape or curved profile, and when the membrane  220  is deflected, the membrane  220  can fully comply and conform to the top surface of the bottom electrode  240 , especially around the outer edge  270  of the membrane  220  above the air cavity  230 . This can increase the electrostatic force around the edge  270  of the membrane  220  to pull down the membrane  220  so a smaller DC bias voltage can be used. Using a smaller DC bias voltage is essential when inserting the transducer probe into the human body for an intravascular application. The bandwidth of the CMUT  200  can also be improved since most of the membrane  220  is under the electrostatic force from the DC bias voltage, which can increase the tensile stress on the membrane  220  to reduce the ringing tail. 
         [0052]    Turning to  FIG. 10 , when a DC bias voltage exceeds a certain voltage level, the membrane  220  is collapsed to the bottom electrode  240 . In this situation, only approximately 25% of the membrane  220  is in contact with the bottom electrode  240 . Hence, 75% of the membrane  220  which is the area  270  proximal to the peripheral edge (that is, the area  270  of the membrane  220  that is not in contact with the bottom electrode  240  above the air cavity  230 ) is considered as effective capacitance. 
         [0053]    A CMUT can also operate at the collapsed mode to have an increased sensitivity and bandwidth. The sensitivity is increased from the increased capacitance at the minimum gap distance around the contacting area. The bandwidth can be improved because the movement of the membrane  220  can be damped by the bottom electrode  240  to reduce the ringing tail. When implementing the concave shaped bottom electrode  240  to operate CMUTs  200  at the collapsed mode, the whole membrane  220  is barely touching the bottom electrode  240  to increase the bandwidth and sensitivity. In particular, around the central area of the membrane  220  is damped by the bottom electrode  240 . Thus, the CMUT  200  can increase effective capacitance to improve fill factor, output pressure, bandwidth, and sensitivity of the transducer. 
         [0054]    The resonant frequency of the CMUT depends on the size and thickness of the membrane. The size of the membrane can range from 500 μm to 5 μm with a frequency range from 100 kHz up to 100 MHz in air. The thickness of the membrane can range from 0.1 μm to 10 μm. Since each membrane of the CMUT is very small, it requires an array of membranes for the CMUT to fill the area of a single transducer element. 
         [0055]      FIG. 11  illustrates exemplary CMUT arrays. The top electrode  320  can fill the whole area of the membrane leaving only small voids for anchoring the membrane, which increases the fill factor four times more than conventional CMUTs. The top electrode  320  can also be patterned to make a 1-D CMUT array for 2-D ultrasonic imaging. In addition, the electroplated bottom electrode can also be patterned to isolate 2-D array elements for 3-D ultrasonic imaging. 
         [0056]    The capacitance of a parallel plate capacitor can be determined from the area of the effective capacitance A and the distance between the top and bottom electrodes d, which is expressed as follows: 
         [0000]    
       
         
           
             
               C 
               = 
               
                 
                   Q 
                   
                     V 
                     ab 
                   
                 
                 = 
                 
                   
                     ɛ 
                     0 
                   
                    
                   
                     ɛ 
                     r 
                   
                    
                   
                     A 
                     d 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    for series capacitor 
         [0000]    
       
         
           
             
               1 
               C 
             
             = 
             
               
                 1 
                 
                   C 
                   1 
                 
               
               + 
               
                 1 
                 
                   C 
                   2 
                 
               
             
           
         
       
     
         [0057]    Based on the geometry of the CMUT, the capacitance of the CMUT can be calculated as follows, where the electrode diameter is much greater than the cavity depth (2c&gt;2b&gt;&gt;d 2 ) and the capacitance C 2  is assumed to be a parallel-plate capacitor. 
         [0058]    Referring to  FIG. 12 , for a concave bottom electrode with a flat membrane, the capacitance from dielectric membrane C 1  is calculated by the parallel-plate capacitor equation. The capacitance C 2  between the bottom surface of the flat membrane to the concave bottom electrode with spherical surface is calculated also using the spherical shell to flat plate capacitance equation: 
         [0000]    
       
         
           
             
               C 
               1 
             
             = 
             
               
                 ɛ 
                 0 
               
                
               
                 ɛ 
                 1 
               
                
               
                 
                   π 
                    
                   
                       
                   
                    
                   
                     c 
                     2 
                   
                 
                 
                   d 
                   1 
                 
               
             
           
         
       
       
         
           
             
               C 
               2 
             
             = 
             
               2 
                
               
                   
               
                
               
                 πɛ 
                 0 
               
                
               
                 ɛ 
                 2 
               
                
               R 
                
               
                   
               
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               ln 
                
               
                   
               
                
               
                 
                   d 
                   2 
                 
                 
                   d 
                   0 
                 
               
             
           
         
       
       
         
           
             C 
             = 
             
               
                 1 
                 
                   
                     1 
                     
                       C 
                       1 
                     
                   
                   + 
                   
                     1 
                     
                       C 
                       2 
                     
                   
                 
               
               = 
               
                 
                   πɛ 
                   0 
                 
                 
                   
                     
                       d 
                       1 
                     
                     
                       
                         ɛ 
                         1 
                       
                        
                       
                         c 
                         2 
                       
                     
                   
                   + 
                   
                     1 
                     
                       2 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         2 
                       
                        
                       R 
                        
                       
                           
                       
                        
                       ln 
                        
                       
                           
                       
                        
                       
                         
                           d 
                           2 
                         
                         
                           d 
                           0 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               R 
               = 
               
                 
                   1 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       d 
                       2 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       d 
                       2 
                       2 
                     
                     + 
                     
                       a 
                       2 
                     
                   
                   ) 
                 
               
             
             , 
             
               
                 d 
                 0 
               
               = 
               
                 
                   
                     
                       R 
                       2 
                     
                     - 
                     
                       c 
                       2 
                     
                   
                 
                 - 
                 R 
                 + 
                 
                   d 
                   2 
                 
               
             
           
         
       
     
         [0059]    Referring to  FIG. 13 , for a concave bottom electrode with a deflected membrane, the membrane is assumed to deform into a spherical shape similar to the case of  FIG. 6 . The concave bottom electrode is also assumed to have a spherical surface. For membrane capacitance C 1 , it is calculated similar to  FIG. 6  as described above. As to C 2 , the cavity capacitance is first calculated relative to the virtual flat plate. Then, the cavity capacitance is subtracted with the capacitance between two spherical surfaces to obtain C 2 . The equations for obtaining C 1  and C 2  are as follows: 
         [0000]    
       
         
           
             
               C 
               1 
             
             = 
             
               4 
                
               
                   
               
                
               
                 πɛ 
                 0 
               
                
               
                 ɛ 
                 1 
               
                
               
                 h 
                 ′ 
               
                
               
                 
                   
                     R 
                     a 
                   
                    
                   
                     R 
                     b 
                   
                 
                 
                   d 
                   1 
                 
               
             
           
         
       
     
         [0060]    C 2 =(The capacitance between the bottom electrode and the virtual flat plate (dashed line))−(The capacitance between the spherical shell with radius R b  and the virtual flat plate). 
         [0000]    
       
         
           
             
               C 
               2 
             
             = 
             
               1 
               
                 
                   1 
                   
                     2 
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                      
                     
                       πɛ 
                       0 
                     
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                     R 
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                      
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                      
                     
                       
                         d 
                         2 
                       
                       
                         d 
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                       πɛ 
                       0 
                     
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                      
                     
                       R 
                       b 
                     
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                      
                     
                         
                     
                      
                     
                       H 
                       
                         
                           
                             
                               R 
                               b 
                               2 
                             
                             - 
                             
                               c 
                               2 
                             
                           
                         
                         - 
                         
                           
                             
                               R 
                               b 
                               2 
                             
                             - 
                             
                               a 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   C 
                   = 
                   
                     1 
                     
                       
                         1 
                         
                           C 
                           1 
                         
                       
                       + 
                       
                         1 
                         
                           C 
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       πɛ 
                       0 
                     
                     
                       
                         
                           d 
                           1 
                         
                         
                           4 
                            
                           
                               
                           
                            
                           
                             ɛ 
                             1 
                           
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                             h 
                             ′ 
                           
                            
                           
                             R 
                             a 
                           
                            
                           
                             R 
                             b 
                           
                         
                       
                       + 
                       
                         1 
                         
                           2 
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                            
                           
                             ɛ 
                             2 
                           
                            
                           R 
                            
                           
                               
                           
                            
                           ln 
                            
                           
                               
                           
                            
                           
                             
                               d 
                               2 
                             
                             
                               d 
                               0 
                             
                           
                         
                       
                       - 
                       
                         1 
                         
                           2 
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                            
                           
                             ɛ 
                             2 
                           
                            
                           
                             R 
                             b 
                           
                            
                           ln 
                            
                           
                               
                           
                            
                           
                             H 
                             
                               
                                 
                                   
                                     R 
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     c 
                                     2 
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     R 
                                     b 
                                     2 
                                   
                                   - 
                                   
                                     a 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 R 
                 a 
               
               = 
               
                 
                   1 
                   
                     2 
                      
                     
                         
                     
                      
                     H 
                   
                 
                  
                 
                   ( 
                   
                     
                       H 
                       2 
                     
                     + 
                     
                       a 
                       2 
                     
                   
                   ) 
                 
               
             
             , 
             
               
                 R 
                 b 
               
               = 
               
                 
                   R 
                   a 
                 
                 + 
                 
                   d 
                   2 
                 
               
             
             , 
             
               
                 h 
                 ′ 
               
               = 
               
                 
                   R 
                   a 
                 
                 - 
                 
                   
                     
                       R 
                       a 
                       2 
                     
                     - 
                     
                       c 
                       2 
                     
                   
                 
               
             
           
         
       
     
         [0061]      FIG. 14  illustrates a graph of effective capacitance with respect to membrane deflection of a CMUT with concave bottom electrodes. CMUTs using concave bottom electrodes the capacitance change can increase up to 79% when enlarging the diameter of the top electrode up to 99.8 μm. The maximum capacitance at the collapsed mode can reach up to 0.7 pF, which is almost ten times more compared to the CMUT using flat bottom electrodes. From Coulomb&#39;s Law, the electrostatic force of a parallel capacitor is expressed as follows, where Q is the electrical charge, E is the electrical field, and V is the voltage. 
         [0000]    
       
         
           
             
               F 
               = 
               
                 
                   Q 
                   · 
                   E 
                 
                 = 
                 
                   
                     Q 
                      
                     
                         
                     
                      
                     
                       Q 
                       
                         2 
                          
                         
                             
                         
                          
                         A 
                          
                         
                             
                         
                          
                         
                           ɛ 
                           0 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ɛ 
                         0 
                       
                        
                       
                         AV 
                         2 
                       
                     
                     
                       2 
                        
                       
                           
                       
                        
                       
                         d 
                         2 
                       
                     
                   
                 
               
             
             , 
             
               
                 where 
                  
                 
                     
                 
                  
                 Q 
               
               = 
               
                 A 
                  
                 
                     
                 
                  
                 
                   ɛ 
                   0 
                 
                  
                 
                   V 
                   d 
                 
               
             
           
         
       
     
         [0062]    For output pressure, since the electrostatic force is inversely proportional to the square of the cavity depth, which means shorten the cavity depth can estimate to have two orders of magnitude increase on the output pressure when the capacitance increase is one order of magnitude. 
         [0063]      FIG. 15  illustrates a method of fabricating CMUT arrays with concave bottom electrodes. The fabrication starts with step S 601  using a silicon wafer that includes a layer of silicon nitride to serve as the CMUT membrane. The Young&#39;s modulus of the silicon nitride is around 200 GPa. In step S 602 , a layer of Cr/Au is sputtered to serve as the seed layer for electroplating. Photoresist is coated by a spin coater to get a thickness of around 1 μm and patterned to define the active area of the CMUT cell. Next, in step S 603 , a thermal reflow process is carried out at 150° C. for 30 minutes to melt the patterned photoresist to form a spherical profile by surface tension. In step S 604 , electroplating of nickel is performed with the Cr/Au seed layer to form the bottom electrode by over-plating to cover the photoresist sacrificial layer. Step S 605  illustrates that the electroplating leaves a small hole for removal of photoresist and Cr/Au in the cavity. 
         [0064]    Finally, in step S 606 , the released holes are first sealed by silicone-based polydimethylsiloxane (PDMS) with air trapped in CMUT cavities. This is followed by a coating of parylene C in a vacuum chamber. The vacuum chamber sucks the trapped air out through the gas permeable PDMS for vacuum sealed cavities since parylene is not gas permeable. The silicon substrate is then removed by single-side potassium hydroxide (KOH) etching that stops at the silicon nitride membrane. This eliminates the membrane stiction problem because the cavity remains dry during the wet etching with the protection of the PDMS and parylene coating. The PDMS now serves as the flexible substrate with silicon nitride membrane ready to be deposited with the metal for the top electrode. The PDMS is then patterned to define the membrane area and array elements. After wire bonding to front-end electronics, the CMUT array is ready to be used. 
         [0065]    It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the scope or spirit of the invention as broadly described. The present embodiments are, therefore, to be considered in all respects illustrative and not restrictive.