Abstract:
Silicon-based capacitive accelerometers are relatively simple to fabricate and offer low cost, small size, low power, low noise and provide high sensitivity, good DC response, low drift, and low temperature sensitivity. However, tri-axial accelerometers, as opposed to using multiple discrete accelerometers, require very low cross-axis sensitivity and close sensitivities across the three directions. It would be beneficial to provide a design methodology for such tri-axial accelerometers which is compatible with commercial MEMS manufacturing processes in order to remove requirements for device specific processing, non-standard processing, etc. Accordingly, tri-axial accelerometers with low cross axis sensitivity have been established exploiting decoupled frames in conjunction with axis specific spring designs. Further, exploitation of differential capacitive transduction using an asymmetric configuration for in-plane measurements along X- and Y-axis and an absolute measurement along Z-axis allows the manufacturing upon a commercial MEMS foundry process.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of priority from U.S. Provisional Patent Application Ser. No. 62/182,703 filed Jun. 22, 2015 entitled “Capacitive Accelerometer Devices and Wafer Level Vacuum Encapsulation Methods” the entire contents of which are included herein by cross-reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    This invention relates to MEMS based capacitive accelerometers and more particularly to multiple axis accelerometers with low cross-axis sensitivity and compatibility with commercial wafer level encapsulation and MEMS fabrication processes. 
       BACKGROUND OF THE INVENTION 
       [0003]    The market potential for silicon-based low cost, miniaturized and low power multi-axis Micro-Electro-Mechanical System (MEMS) accelerometers is growing rapidly and these sensors are found in a variety applications including smartphones, gaming devices, digital cameras, automobiles, wearable devices, structural health monitoring, energy exploration and industrial manufacturing. Many leading and emerging semiconductor companies are currently marketing silicon based tri-axial accelerometers and the development of silicon based tri-axial accelerometers has been extensively studied by several research groups using various custom or proprietary microfabrication processes with surface micromachining, bulk micromachining, combined surface and bulk micromachining, and Complementary Metal-Oxide-Semiconductor (CMOS) MEMS processes. 
         [0004]    Silicon-based capacitive accelerometers are relatively simple to fabricate and offer low cost, small size, low power, low noise and provide high sensitivity, good DC response, low drift, and low temperature sensitivity. Tri-axial accelerometers, as opposed to the use of multiple discrete accelerometers, require very low cross-axis sensitivity and close sensitivities across the three directions. Accordingly, it would be beneficial to provide such tri-axial accelerometers using a commercial MEMS manufacturing process in order to remove requirements for device specific processing, non-standard processing, etc. 
         [0005]    It would be further beneficial to provide such tri-axial accelerometers using a commercial MEMS manufacturing process that supports wafer level vacuum encapsulation of the MEMS devices. Wafer level vacuum encapsulation of MEMS devices plays a key role in improving the sensor performance and long term reliability. Further, wafer level vacuum encapsulation can provide extraordinary benefits in comparison to the existing state-of-the-art microfabrication processes for the development of MEMS sensors in reducing the overall product cost, simplifying packaging constraints, and easing supply-chain logistics. The encapsulation of vibrational inertial MEMS sensors such as accelerometers with a pressure below atmospheric pressure influences quality factor, response time, stiction, damping and humidity exposure 
         [0006]    Accordingly, the inventors have established a strategic new design methodology to provide this beneficial low cross axis sensitivity via decoupled frames. Further, by exploiting a differential capacitive transduction using asymmetric configuration for in-plane measurement along X- and Y-axis and an absolute measurement along Z-axis the inventors beneficially provide such as tri-axial accelerometer upon a commercial MEMS foundry process. 
         [0007]    Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying figures. 
       SUMMARY OF THE INVENTION 
       [0008]    It is an object of the present invention to mitigate limitations in the prior art relating to MEMS based capacitive accelerometers and more particularly to multiple axis accelerometers with low cross-axis sensitivity and compatibility with commercial wafer level encapsulation and MEMS fabrication processes. 
         [0009]    In accordance with an embodiment of the invention there is provided a device comprising:
       a substrate;   a microelectromechanical system (MEMS) comprising a plurality of integer N frames disposed within each other with the inner N−1 frames being suspended with respect to the substrate, the outermost frame attached to the substrate, and the innermost frame providing a proof mass;   a plurality of N−1 sets of springs, each set of springs being of a predetermined design and attached from a first predetermined frame of the plurality of N frames to a second predetermined frame of the plurality of N frames;   a plurality of M comb drive pairs, each comb drive pair comprising first and second comb drives attached at opposite sides of a third predetermined frame of the plurality of N frames to a fourth predetermined frame of the plurality of N frames.       
 
         [0014]    In accordance with an embodiment of the invention there is provided a device comprising:
       a fixed frame;   a first frame disposed within an opening within the fixed frame connected to the fixed frame by a plurality of first springs;   a second frame disposed within an opening within the first frame connected to the first frame by a plurality of second springs;   a proof mass disposed within an opening within the second frame connected to the second frame by a plurality of third springs; wherein   each of the first and second frames have a first axis of the respective frame longer than a second axis of the respective frame and the first axes of the first and second frames are orthogonal to each other.       
 
         [0020]    In accordance with an embodiment of the invention there is provided a method comprising:
       linking a proof mass to a first frame by a plurality of first springs that support motion of the proof mass out of the plane within which the proof mass and the plurality of first springs are manufactured;   linking the first frame to an outer frame by a plurality of second springs that support motion of the first frame in the plane within which the first frame and the plurality of second springs are manufactured; and   providing a pair of drive combs attached to the outer frame.       
 
         [0024]    Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying figures. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0025]    Embodiments of the present invention will now be described, by way of example only, with reference to the attached Figures, wherein: 
           [0026]      FIG. 1  depicts schematically the structural design of a tri-axial accelerometer sensor with decoupled frames according to an embodiment of the invention; 
           [0027]      FIGS. 2A and 2B  respectively depict 3D models of the complete accelerometer sensor and partial sensor showing only the fixed and decoupled frame structures respectively according to embodiments of the invention; 
           [0028]      FIGS. 2C and 2D  depict schematically the structures of the spring elements employed within an accelerometer sensor according to an embodiment of the invention; 
           [0029]      FIGS. 2E and 2F  depict electrical transduction mechanisms of differential capacitance measurement with asymmetric comb-finger configuration along X- and Y-axes and absolute capacitance measurement along Z-axis; 
           [0030]      FIGS. 3A  depicts schematically the structural design of a tri-axial accelerometer sensor with decoupled frames according to an embodiment of the invention; 
           [0031]      FIG. 3B  depicts schematically the springs within the tri-axial accelerometer of  FIG. 3A  for each of the X-axis, Y-axis and Z-axis respectively; 
           [0032]      FIG. 4A  depicts a low cross-axis sensitivity technique with extended top electrode along X- and Y-axes according to an embodiment of the invention; 
           [0033]      FIG. 4B  depicts a low cross-axis sensitivity technique with recessed fingers with 4 μm height difference according to an embodiment of the invention; 
           [0034]      FIG. 4C  depicts a low cross-axis sensitivity technique capacitive inter-digitated compensation electrodes according to an embodiment of the invention; 
           [0035]      FIGS. 5A, 5B and 5C  respectively depict Finite Element Method analysis of in-plane X- and Y-axes modes and out-of-plane Z-axis mode for a tri-axial accelerometer sensor with decoupled frames according to an embodiment of the invention; 
           [0036]      FIGS. 6A and 6B  respectively depict frequency-domain analysis via lumped element modelling for a tri-axial accelerometer sensor with decoupled frames according to an embodiment of the invention; 
           [0037]      FIGS. 7A, 7B and 7C  respectively depict simulated calibration curves for differential measurements along the X- and Y-axes and absolute measurement in the Z-axis for a tri-axial accelerometer sensor with decoupled frames according to an embodiment of the invention; 
           [0038]      FIGS. 8A, 8B, 8C and 8D  respectively depict damping force and coefficient for the central proof mass arising from the squeezed effect and slide air-film effect for a tri-axial accelerometer sensor with decoupled frames according to an embodiment of the invention; 
           [0039]      FIG. 9A  depicts cross-section views of the layers employed in the tri-axial accelerometer sensor manufacture according to an embodiment of the invention; 
           [0040]      FIG. 9B  depicts a cross-sectional optical micrograph of a fabricated inertial tri-axial accelerometer sensor; 
           [0041]      FIGS. 10A, 10B, 10C and 10D  depict the experimental setup for testing tri-axial accelerometer sensors according to embodiments of the invention; 
           [0042]      FIGS. 11A, 11B and 11C  respectively depict experimental results for a tri-axial accelerometer sensor manufacture according to an embodiment of the invention in the X-, Y- and Z-axes; 
           [0043]      FIGS. 12A, 12B, 12C, 12D, 12E and 12F  depict experimental results for evaluating the cross-axis sensitivity of a tri-axial accelerometer sensor manufacture according to an embodiment of the invention by testing at 45° to different pairs of the X-, Y- and Z-axes; and 
           [0044]      FIG. 13  depicts a total equivalent noise (TENA) measurement of an accelerometer according to an embodiment of the invention performed under 1 g input acceleration. 
       
    
    
     DETAILED DESCRIPTION 
       [0045]    The present invention is directed to MEMS based capacitive accelerometers and more particularly to multiple axis accelerometers with low cross-axis sensitivity and compatibility with commercial wafer level encapsulation and MEMS fabrication processes. 
         [0046]    The ensuing description provides exemplary embodiment(s) only, and is not intended to limit the scope, applicability or configuration of the disclosure. Rather, the ensuing description of the exemplary embodiment(s) will provide those skilled in the art with an enabling description for implementing an exemplary embodiment. It being understood that various changes may be made in the function and arrangement of elements without departing from the spirit and scope as set forth in the appended claims. 
         [0047]    0. Background 
         [0048]    Within the prior art accelerometers have been implemented not only on research semiconductor processing lines but also upon unmodified commercial Micro-Electro-Mechanical System (MEMS) processes such as MEMSCAP&#39;s Multi-User MEMS Processes (MUMPS), Sandia National Laboratories&#39; SUMMiT V, IMEC&#39;s SiGe, and STMicroelectronics&#39; ThELMA. In contrast, the Teledyne DALSA MEMS Integrated Design for Inertial Sensors (MIDIS) process is not only a commercial MEMS process but it also includes ultra-clean wafer level vacuum encapsulation of the MEMS devices. Wafer level vacuum encapsulation of MEMS devices is currently extensively studied by various research groups as it plays a key role in improving both performance and long term reliability of the sensor. The availability of wafer level vacuum encapsulation processing provides benefits in reducing the overall product cost, simplifying packaging constraints, and easing supply-chain logistics. The encapsulation of vibrational inertial MEMS sensors such as accelerometers with a pressure below atmospheric pressure also influences quality factor, response time, stiction, damping and humidity exposure. Whilst wafer level vacuum packaging of MEMS accelerometers has been demonstrated within the prior art using custom microfabrication processes. In contrast, the inventors employ wafer level vacuum packaging of MEMS accelerometers in a high volume commercial MEMS process. The commercial MIDIS process is based on high aspect ratio bulk micromachining of single-crystal silicon wafer (referred to as the device layer within this specification) that can either be vacuum encapsulated at 10 mTorr or at sub-atmospheric pressure of 150 Torr between two other silicon wafers (referred to as top interconnect and bottom handling wafers within this specification). The achievable total leak rate equivalent in the MIDIS process is 45 molecules/s (7.5×10 −13  atm·cm 3 /s). The top silicon wafer includes Through Silicon Vias (TSVs) with sealed anchors for compact flip-chip integration and interconnection with external microelectronic signal processing circuitry. Within the following description the inventors present a tri-axial accelerometer sensor permitting simultaneous acceleration detection along the 3 principal axes (X, Y and Z). 
         [0049]    In order to achieve this the inventors have established a novel design for the decoupled frames as well as established out-of-plane measurements upon a MIDIS process specifically optimized for in-plane inertial sensors. In order to decouple the frames, the inventors established different kinds of spring structures that are made selectively more sensitive across one specific axis of input acceleration. Further, as described and depicted below the novel methodology exploits recessed comb-fingers that are used to enable sensing along the Z-direction. Accordingly, tri-axial accelerometers according to embodiments of the invention employ differential capacitive transduction using asymmetric configurations for the measurements along the X- and Y-axes which are in-plane and absolute measurement along Z-axis, i.e. out of plane. Accordingly, tri-axial accelerometers according to embodiments of the invention may be interfaced to capacitance to digital converter circuits such as those implemented in CMOS allowing direct CMOS-MEMS integration methodologies to be supported. 
         [0050]    1. Accelerometer Sensor Design 
         [0051]    Referring to  FIG. 1  there is depicted schematically the design of a proposed tri-axial accelerometer according to an embodiment of the invention. The design consists of a central Proof Mass  170  supported by multiple frames, Fixed Frame  110 , Frame-1  150  and Frame-2  130 , that are connected to each other by different kinds of spring structures. The accelerometer uses a single central Proof Mass  170  structure moving along the main principle axes which reduces the total dimensional area of the inertial sensor. The central proof mass, the Frame-1  150  and Frame-2  130  are sensitive to Z-, X- and Y-axis directions, respectively. Thus, each axis can be modeled by a spring-mass system with its associated damping coefficients, b x ;b y ;b z  along X-, Y- and Z-axis, respectively, as given by Equation (1). Each mass component, m x ;m y ;m z  experiences a stimulus force due to the applied accelerations, a x ;a y ;a z , along X-, Y- and Z-axis, respectively, generating a small displacement that can be converted into electrical measurements. Within the embodiments of the invention described below two mechanisms are employed for converting displacement to an electrical parameter variation that can be measured. These are differential capacitance measurements along the X- and Y-axis and absolute capacitance measurements along the Z-axis. The transmissibility function, T(jω), between the acceleration and the displacement is expressed by Equation (2), where, ω 0  is the natural frequency and ξ is the damping ratio. 
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         [0052]    1.1. Electro-Mechanical Design 
         [0053]    Now referring to  FIG. 2A  in first image  200 A a three-dimensional (3D) model of the device structure is depicted together with first and second close-up views of the first and second spring structures  210  and  220  in the top-right and bottom-right corners that are used to support the three different masses. These first and second structures  210  and  220  respectively are depicted in isolation in  FIGS. 2C and 2D  respectively as third and fourth images  200 C and  200 D respectively. The central proof mass has a square shape to allow uniform out-of-plane motion sensing. The fixed and decoupled frames have a rectangular shape enabling more sensitive in-plane motion sensing along each specific axis. The device thickness is fixed by the 
         [0054]    MEMS fabrication process at 30 μm. The large size of the proof mass helps to reduce the noise and increase the sensitivity of the accelerometer. At low operational frequencies (f&lt;&lt;f 0 ), the mechanical sensitivity, S MECH , is inversely proportional to the natural frequency, f 0 , for each axis and is given by the Equation (3), where, Δx is the displacement for a specific variation of the input acceleration, Δa. 
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         [0055]    The springs supporting the Proof Mass  170  and moveable (decoupled) frame structures Frame-1  150  and Frame-2  130  each consist of four flexible springs supporting the inner element from its outer element. However, the inventors have established a novel configuration of these springs such that they are made selectively more sensitive across one specific axis of input acceleration and reduce cross-coupling to the other axes of input acceleration. 
         [0056]    First Spring  210 : This comprises first and second Mounts  210 A and  210 B disposed at one side of the first Spring  210 . Between these first and second Mounts  210 A and  210 B is a first Serpentine  210 C comprised of a number of first U-Springs  210 D disposed sequentially. These are connected in series at one common side, such that first U-Springs  210 D are disposed to one common side of a line joining the first and second mounts  210 A and  210 B. 
         [0057]    Second Spring  220 : This comprises third and fourth Mounts  220 A and  220 B which are centrally disposed with respect to second Spring  220 . Between these third and fourth Mounts  220 A and  220 B is a second Serpentine  220 C comprised of a number of second U-Springs  220 D. These are disposed sequentially in series and alternate, such that the second U-Springs  220 D are disposed alternately either side of a line joining the third and fourth Mounts  220 A or  220 B. Within the embodiments of the invention presented with respect to  FIGS. 1 to 13  the number of second U-Springs  220 D is two. 
         [0058]    Spring Set 1: Frame-1  150  has four third Springs  160  of spring constant K z  attached to the central Proof Mass  170 . Each of these springs is a second Spring  220  disposed at the mid-point of each edge of the square proof mass. 
         [0059]    Spring Set 2: Frame-2  130  has four second Springs  140  of spring constant K x  attached to Frame-1  150 . Each of these springs is a first Spring  210  and these are disposed in pairs along two edges of Frame-1  150  to Frame-2  130  starting from a corner of Frame-1  150  towards the middle of the edge they are disposed upon. With Frame-1  130  being rectangular with larger dimension along the X-axis these four Springs  140  are disposed on the short edge of Frame-1  150  and along the long edge of Frame-2  130 . 
         [0060]    Spring Set 3: Fixed Frame  110  has four first Springs  120  of spring constant K Y  attached to Frame-2  130 . Each of these springs is a first Spring  210  and these are disposed in pairs along two edges of Fixed Frame  110  to Frame-2  130  starting from a corner of Frame-2  130  towards the middle of the edge they are disposed upon. With Frame-2  110  being rectangular with larger dimension along the Y-axis these four Springs  120  are disposed on the short edge of Frame-2  130  to the Fixed Frame  110 . The Fixed Frame  110  is also patterned such that a portion of it runs along the long edges of Frame-2  130  near the four first Springs  120  to form Limiters  230  which based upon their dimension may limit the X-axis motion of the Frame 2  130 . 
         [0061]    In each case of first to third Springs  120 ,  140  and  160  their corners within the first Serpentine  210 C are filleted to reduce the stresses due to elevated processing temperatures, especially, during the bonding process. The Proof Mass  170  as depicted is square in order to improve uniform out-of-plane sensing (i.e. vertical motion in Z-axis). In contrast Frame-1  150  and Frame-2  130  are rectangular in order to enhance their sensitivity in their respective directions. Similarly, the performance variations in the spring designs selected between each pair of sequential frames aid motion within the respective axis whilst three decoupled masses, Proof Mass  170 , Fame-1  150 , and Frame-2  130  aids isolation of motion in all three planes as nothing is referenced to a fixed set of electrodes. This is particularly true for the X-axis where the sensing electrode and “fixed” electrodes are both free to move in the Y-direction. 
         [0062]    Additionally, the inventors exploit comb drives on both sides of the Proof Mass  170  such that the overlapping area remains constant, independent of the out of plane motion. Further, exploitation of recessed fingers with a height difference means that the constant overlapping area is maintained independent of the vertical motion. 
         [0063]    Based on the lumped modeling, the spring constants, K X  and K Y  along the X- and Y-axes, respectively and K Z  along Z axis can be expressed by Equations (4A) and (4B), where E,I,h,w,L represent Young&#39;s modulus, the inertial moment, the height, the width and the length for each beam, respectively. The aspect ratio of thickness to width is appropriately designed to insure high stiffness allowing only one movement along the intended specific axis. 
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         [0064]    Referring to  FIGS. 2E and 2F  respectively, the two electrical transduction mechanisms exploited by the inventors within the novel tri-axial accelerometer are depicted. First image  200 E in FIG.  2 E_depicts a differential capacitance measurement with asymmetric configuration through inter-digitated fingers of MEMS combs. This differential capacitance measurement is employed with two different gaps, d 1 ; d 2  along the X- and Y-axis respectively. In contrast, the electrical transduction is based on absolute measurement along the Z-axis with a gap, d, between two electrodes as depicted in second image  200 F in  FIG. 2F . For small displacements, the electrical sensitivities along the X- and Y-axes are expressed by Equation (5A) and along the Z-axis by Equation (5B). Here, the small gap d 1  between the inter-digitized fingers is fixed at d 1 =2 μm and the optimal ratio between d 1  and d 2  is determined to be 6, i.e. d 2 =6*d 1 . The gap, d, between the top electrode and the central proof mass was similarly set at d=2 μm. The overall sensitivity of the sensor along the X- and Y-axes are expressed by Equation (6A) and along the Z-axis by Equation (6B). 
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         [0065]    1.2. Cross-Axis Sensitivity 
         [0066]    The output signal of the tri-axial accelerometer that uses a single proof-mass structure and performs simultaneous measurement across the three principle axes, can generally be expressed by Equation (7). Ideally, to achieve 0% (zero) cross-axis sensitivity (CrossSens), the matrix shown in Equation (7) should only have terms S XX ;S YY ;S ZZ  with remainder being equal to zero. Thus, the cross-axis sensitivity which is defined as the output induced on a sense axis from the application of acceleration on a perpendicular axis, expressed as a percentage of the sensitivity is given by Equations (8A) to (8C) respectively for X-, Y-, and Z-axes respectively. 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       X 
                     
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         [0067]    The capability to displace the central proof mass along Z-axis is achieved by the four Second Springs  220  where the stiffness is predominant over the First Springs  210 . This allows dominant and unidirectional displacement of the bottom electrode. The area in the top wafer with Through Silicon Vias (TSVs) exceeds the central proof mass dimensions by 15 μm as shown in  FIG. 4A , which is considered sufficient margin to enable the bottom electrode to move along X- or Y-axis without causing any DC offset in the electrical measurement that could lead to cross-axis sensitivity. Further, there is no effect of the vertical displacement from the inter-digitized fingers as the difference in the height between the fixed and the moving fingers is 4 μm, leading to null DC offset and thus leading to low cross-axis sensitivity as shown in  FIG. 4B . The cross-axis sensitivity issue along X- and Y-axis is addressed through the use of two moving frames which permit the decoupling of the motion along the X- and Y-axis. 
         [0068]    As shown in second image  200 B in  FIG. 2B , the Frame-1  150  is supported by Frame-2  130 , and thus, there is negligible effect of the X-axis acceleration on the Frame-2  130 . However, any movement in the Y-axis direction has direct effect on the Frame-1  150 . Accordingly, this issue was mitigated by the inventors by employing another inter-digitized compensation capacitor, as shown in  FIG. 4C , which keeps the same initial capacitance value regardless of the motion of the Frame-1  150  along the Y-axis. Thus, there is no mutual effect from the two frames on the output capacitance measurement which ultimately leads to low cross-axis sensitivity between a X  and a Y . 
         [0069]    Now referring to  FIG. 3A  there is depicted a schematic  300 A of a tri-axial accelerometer  300  according to an embodiment of the invention exploiting a similar nested sequence of Fixed Frame  340 , first Free Frame  350 , second Free Frame  360  and Proof Mass  370  wherein the first Free Frame  350  is connected to the Fixed Frame  340  via four Y-axis Springs  310 . The second Free Frame  360  is disposed within the first Free Frame  350  and supported from it by four X-axis Springs  320 . Similarly, the Proof Mass  370  is disposed within the second Free Frame  360  and supported from it by four Z-axis Springs  330 . As evident in  FIG. 3A  the X-axis Springs  310  and Y-axis Spring  320  are located at the corner of the inner frame they support whereas the Z-axis Springs  330  are disposed at the center of the Proof Mass  370 . In contrast to  FIG. 1  the Proof Mass  370  is circular, the second free Frame  360  square, first Free Frame  350  slightly rectangular but could be square, and Fixed Frame  340  is similarly slightly rectangular but may also be square. Now referring to  FIG. 3B  enlarged images of X-axis Springs  310 , Y-axis Springs  320 , and Z-axis springs  330  with respect to the tri-axial accelerometer  300 . 
         [0070]    2. Simulation Results and Discussion 
         [0071]    The simulations performed by the inventors consisted of modal and damping analysis using Finite Element Method (FEM) analysis with Coventorware software. The Architect module was used to perform lumped modeling where one could perform co-integration analysis combining signal conditioning circuitry and the sensor device. 
         [0072]    2.1. Electromechanical Results 
         [0073]    Modal analysis was used to show the dynamic characteristics of the sensor. The shapes for the first, second and third modes are illustrated in  FIGS. 5A to 5C  respectively with first to third images  500 A to  500 C respectively for X-, Y- and Z-axes respectively. The resonant frequencies along sensitive X-, Y- and Z-axes were designed to be          =4.37 kHz,          =4.16 kHz and          =8.37 kHz. Lumped modeling uses frequency analysis as illustrated in  FIGS. 6A and 6B  respectively depicting the amplitude  600 A and the phase  600 B respectively of the sensitive axis with resonant frequencies established around          =4 kHz,          =4.3 kHz, and          =7.3 kHz along the X-, Y- and Z-axes, respectively. The inventors note that the FEM method takes into account the rounded shapes of the springs as well as the complete structure shape. However, lumped modeling uses a perfect model for beams and structures which introduces a small difference between the two results. 
         [0074]    The capacitance measurement is based on differential measurement in order to increase the total capacitance change and consequently to improve the sensor sensitivity along the X- and Y-axis. The initial capacitances values of prototype tri-axial accelerometers according to an embodiment of the invention are C X =2 pF, C Y =2.7 pF, and C Z =0.9 pF along the X-, Y- and Z-axes, respectively. Referring to  FIGS. 7A to 7C  respectively there are depicted the calibration curves  700 A to  700 C along the X-, Y-, and Z-axes respectively for the capacitance output versus the input acceleration for a prototype tri-axial accelerometers according to an embodiment of the invention. The maximum displacement for sensor comb structure is fixed at 25% from the initial gap d 1  in order to ensure an acceptable linearity over the [0-50 g] measurement range. First and second calibration curves  700 A and  700 B in  FIGS. 7A and 7B  contain two curves that represent the variation along each side of the inter-digitized comb fingers. Third calibration curve  700 C in  FIG. 7C  represents only one curve as the measurement is absolute along Z-axis. The sensitivities, given by the slopes of each curve, for the prototype tri-axial accelerometers according to an embodiment of the invention were dC X /dg=8.55          F·g −1 , dC Y /dg=18.03          F·g −1 , and dC Z /dg=2.64          F·g −1  with maximum ranges of 957          F , 2.10 pF , and 269          F along the X-, Y- and Z-axes, respectively. 
         [0075]    2.2. Accelerometer Performance 
         [0076]    An underdamped response has the advantage of increasing the quality factor and thus achieving lower noise performance in an accelerometer. In addition, a higher Q-factor can help to improve the response time of the accelerometer. Both the vacuum maintained inside the sealed cavity and the device geometry control the damping coefficient and consequently the quality factor. The MIDIS commercial foundry process employed by the inventors offers encapsulation of the device wafer at 10 milliTorr vacuum. Further, the damping due to the geometry effect is analyzed by considering two main damping mechanisms, the slide air-film and the squeezed air-film damping, that are confined in the 2 μm gap between the central proof mass and the top wafer.  FIGS. 8A to 8D  respectively depict the results of the damping coefficient over the operating frequency range indicating the dominance of the squeezed airflow mechanism (0.092 μN/μm/μs) over the slide film damping (5.26 e −10  μN/μm/μs). Likewise, the effect of inter-digital fingers was analyzed using Stokes flow regime and the damping coefficients values established as 9.26 e −8  μN/μm/μs and 1.518 e −7  μN/μm/μs along X- and Y-axes, respectively. As depicted first graph  800 A depicts damping force due to the squeezed effect, second graph  800 B the damping coefficient from the squeezed effect, third graph  800 C he damping force due to the size effect, and fourth graph  800 D the damping force coefficient due to the size effect. 
         [0077]    3. Experimental Results and Discussion 
         [0078]    The prototype tri-axial accelerometers according to an embodiment of the invention were using the MEMS Integrated Design for Inertial Sensors (MIDIS) process from Teledyne DALSA Inc. which exploits a 30 μtm device wafer thickness. The cross-sectional view of a typical device fabricated with the commercial MEMS Integrated Design for Inertial Sensors process is depicted in  FIG. 9A  with the different layers involved in the device fabrication, namely silicon  910 , silicon dioxide (SiO 2 )  920 , in-situ doped polysilicon (ISDP)  930 , polymer  940 , and AlCu  950 . Also depicted at comb  960 , first sealed cavity  970 , e.g. at 10 mTorr “vacuum”, and a second seal cavity  980 , e.g. at pressure such as 150 Torr , i.e. 150 Torr 1760 Torr≅0.2 atm. As depicted the prototype tri-axial accelerometers according to an embodiment of the invention are fabricated using a top layer  900 A which is then bonded to a device layer  900 B, which is 30 μm thick and within which the MEMS elements are formed, and handling layer  900 C. Accordingly, bonding these three layers together provides the required sealed cavities around the MEMS elements. Now referring to  FIG. 9B  there is depicted an optical micrograph of the cross-section of the fabricated tri-axial accelerometer device. The TSV cap layer (top layer  900 A) is bonded to the device layer  900 B through thin intermediate conductive layer of 2 μm. Further, the commercial MIDIS process allowed the inventors to manufacture combs with recessed fingers having a height difference of 4 μm between the fixed comb and the moving comb as shown in  FIG. 9A . These two features, namely a gap of 2 μtm between the two electrodes using oxide material as an insulating layer and recessed fingers with a height difference of 4 μm, were exploited to enable detection along Z-axis. Referring to Table 1 below design limits of the MIDIS MEMS process are defined within which prototype tri-axial accelerometers according to embodiments of the invention can be manufactured. It would, however, be evident to one of skill in the art that the prototype tri-axial accelerometers according to embodiments of the invention may be manufactured using other research and commercial MEMS processes although these may offer different design limits, design limitations and/or additional processing complexity to achieve the novel designs provided by prototype tri-axial accelerometers according to embodiments of the invention. 
         [0000]    
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 MIDIS Fabrication Process Features 
               
             
          
           
               
                   
                   
                 Minimum 
                 Minimum 
                 Maximum 
               
               
                   
                 Thickness 
                 Feature 
                 Spacing 
                 Spacing 
               
               
                 Feature 
                 (μm) 
                 (μm) 
                 (μm) 
                 (μm) 
               
               
                   
               
             
          
           
               
                 Device Wafer 
                 3 
                 1.5 
                 1.5 
                 — 
               
               
                 Top Interconnect Wafer 
                 180 
                 10 
                 50 
                 — 
               
               
                 Bonding Plane 
                 2 
                 50 
                 10 
                 700 
               
               
                 Handling Wafer 
                 380 
                 20 
                 50 
                 — 
               
               
                   
               
             
          
         
       
     
         [0079]    Referring to  FIGS. 10A and 10B  respectively there are depicted an experimental setup in first image  1000 A of  FIG. 10A  for testing the sensor performance using a horizontal shaker ARMS-200 rotary motion simulator where sensor testing is performed by placing the chip in different positions along the principle axes (X, Y and Z) as shown in second image  1000 B in  FIG. 10B . In initial experiments, the measurements determined the sensor sensitivity and resolution along the various axes, while relying on the measured Total Noise Equivalent Acceleration (TENA). In subsequent experiments, the inventors performed experiments to measure cross-axis sensitivity performance by placing the accelerometer at 45° for each pair of axes (X,Y), (X,Z), and (Y,Z) as shown in third image  1000 C in  FIG. 10C , through which the acceleration components along two axes can be measured. Here, the objective is to compare the measured acceleration components in the second experiment to the value obtained from the first experiment. The sensor readout circuit employed was a 24-bit 2-channel Σ/Δ capacitance-to-digital convertor (AD7746 by Analog Devices Inc.) as evident from fourth image  1000 D in  FIG. 10D . The AD7746 includes an oversampling 24-bit Σ-Δ modulator and signal processing circuitry for noise shaping and filtering which helps to significantly minimize the noise and therefore enhances the measurement accuracy. The shaker allows acceleration only along one axis in range of 0-6 g. The AD7446 readout circuit was interfaced to a microcontroller via an I2C interface to transmit the acquired measurements to a computer. 
         [0080]    Referring to  FIGS. 11A to 11C  respectively there are illustrated the different calibration curves, that is, capacitance variation versus the input acceleration, with its linear regression curves through which the sensitivity was deduced from the slope. First to third images  1100 A to  1100 C in  FIGS. 11A to 11C  respectively depict the X-, Y-, and Z-axes respectively The experimental results in  FIG. 11  are very close to the results found in the simulation studies. Sensitivity values of 10.5          F·g −1 , 16.4          F·g −1  and 3          F·g −1  were obtained for the X-, Y- and Z-axes, respectively. 
         [0081]    Now referring to  FIGS. 12A to 12F  respectively there are depicted experimental results for the calibration curves of each axis pair, namely (X,Y), (X,Z), and (Y,Z). The slope of the curve represents the sensor sensitivity and is extremely close to the values found in the case of single acceleration component where the sensor is directed along the radial axis. Here, the small differences in the experimental results can be explained by the noise generated due to the electric motor in the ARMS-200 rotary motion simulator. Table 2 summarizes the important specifications that describe the novel tri-axial accelerometer sensor implemented according to an embodiment of the invention. As depicted in  FIGS. 12A to 12F  the images_are:
       First image  1200 A in  FIG. 12A  depicts measured X-axis component from coupled X-Y axis excitation;   Second image  1200 B in  FIG. 12B  depicts measured X-axis component from coupled X-Z axis excitation;   Third image  1200 C in  FIG. 12C  depicts measured Y-axis component from coupled X-Y axis excitation;   Fourth image  1200 D in  FIG. 12D  depicts measured Y-axis component from coupled Y-Z axis excitation;   Fifth image  1200 E in  FIG. 12E  depicts measured Z-axis component from coupled X-Y axis excitation; and   Sixth image  1200 F in  FIG. 12F  depicts measured Z-axis component from coupled Z-Y axis excitation.       
 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
             
               
               
             
               
               
               
               
             
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Performance of Prototype Tri-Axial Accelerometer Sensor 
               
             
          
           
               
                   
                 In-Plane 
                 Out-of-Plane 
               
               
                   
                 Measurement 
                 Measurement 
               
             
          
           
               
                   
                 X-Axis 
                 Y-Axis 
                 Z-Axis 
               
               
                   
                   
               
             
          
           
               
                 Measurement Range (g) 
                 ±50 
               
             
          
           
               
                 Sensor Sensitivity (fF/g) 
                 10.5 
                 16.4 
                 3.0 
               
               
                 Resolution (mg) 
                 2.8 
                 1.8 
                 10.0 
               
               
                 Spring Stiffness (μN/μm) 
                 23.4 
                 38.7 
                 51.8 
               
               
                 Resonant Frequency (kHz) 
                 4.3 
                 4.1 
                 8.3 
               
               
                 Mechanical Noise (μg/{square root over (Hz)}) 
                 1.04 × 10 −2   
                 0.1415 
                 19.2 
               
               
                 Cross-Axis Sensitivity (%) 
                 1.3 
                 0.86 
                 1.05 
               
             
          
           
               
                 Dimensions (μm 3 ) 
                 1000 × 1000 × 30 
               
               
                   
               
             
          
         
       
     
         [0088]    As noted supra low noise operation is an important parameter in the specification of an accelerometer. The noise floor is mainly established by two factors, the signal conditioning circuit and the mechanical noise within the accelerometer. The former is generally considered external to the accelerometer even where the MEMS accelerometer is integrated with a CMOS circuit as the electrical circuit design drives its electrical noise. The latter arises from the air viscosity inside the accelerometer package. The noise spectral density referred to the input acceleration is given by Equation (9) where k B , T, and Q are the Boltzmann constant, the working temperature and the quality factor, respectively. 
         [0000]    
       
         
           
             
               
                 
                   
                     a 
                     n 
                   
                   = 
                   
                     
                       
                         4 
                          
                         
                           k 
                           B 
                         
                          
                         T 
                          
                         
                             
                         
                          
                         
                           ω 
                           n 
                         
                       
                       Q 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0089]    Amongst the features of the commercial MIDIS process to facilitating low noise performance are that it includes a pair of unique features that are not currently available through any other commercial MEMS foundry. The first of these is that a thick structural device wafer of 30 μm adds significant mass, and the second is the ultra-clean wafer level vacuum encapsulation at 10 mTorr which leads to a high Q factor through reduction in the damping effect. Experimentally, a total equivalent noise (TENA) measurement of an accelerometer is performed under 1 g input acceleration and is deduced from the 6σ uncertainty of the electrical output as depicted in  FIG. 13 . Accordingly, for the Y-axis of a novel tri-axial accelerometer according to an embodiment of the invention this measurement yielded a result of approximately 0.33          F which leads to an acceleration resolution of 1.8 mg for a given bandwidth of 32 Hz. Referring to  FIG. 13  it is evident that the data near the start and end shows a large signal variation due to the tilting of test chip to collect suitable data. As predicted through the damping mechanism, in the current design the dominant mechanical noise is generated along Z-axis and it is estimated to be 19.2 μg/√{square root over (Hz)}. As evident from Table 2 the mechanical noise on the X- and Y-axes was significantly lower. By comparison a commercial 3-axis accelerometer, a Dytran 7503A5 High Precision tri-axial MEMS accelerometer which achieves 75 μg/√{square root over (Hz)} such that the inventive accelerometer of the inventors has only 25% of the mechanical noise on its worst axis, the Z-axis, and is orders of magnitude lower in mechanical noise on the X- and Y-axes. Equally cross-axis sensitivity is at worst 45% that of the commercial MEMS device. 
         [0090]    Accordingly, the inventors have presented the design, fabrication and testing of a wafer level vacuum encapsulated tri-axial capacitive accelerometer with low cross-axis sensitivity. The novel accelerometer is fabricated using a commercial MEMS foundry process provides a promising option allowing highly efficient and reproducible manufacturing at large volumes, lower cost, and high yields. The wafer level vacuum encapsulation of the novel accelerometer provides benefits in reducing the overall product cost, simplifying packaging constraints, and easing supply-chain logistics. The novel accelerometer includes several novel features including:
       integrated structure using decoupled frames supported by strategically designed springs; and   capacitive compensators for the purpose of achieving low cross-axis sensitivity.       
 
         [0093]    It would be evident that whilst accelerometers have been described performing measurements in 3 axes it would be evident that the design principles embodied in the invention may be applied to accelerometers performing measurements in 1 axis or 2 axes. 
         [0094]    Specific details are given in the above description to provide a thorough understanding of the embodiments. However, it is understood that the embodiments may be practiced without these specific details. For example, circuits may be shown in block diagrams in order not to obscure the embodiments in unnecessary detail. In other instances, well-known circuits, processes, algorithms, structures, and techniques may be shown without unnecessary detail in order to avoid obscuring the embodiments. 
         [0095]    The foregoing disclosure of the exemplary embodiments of the present invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many variations and modifications of the embodiments described herein will be apparent to one of ordinary skill in the art in light of the above disclosure. The scope of the invention is to be defined only by the claims appended hereto, and by their equivalents. 
         [0096]    Further, in describing representative embodiments of the present invention, the specification may have presented the method and/or process of the present invention as a particular sequence of steps. However, to the extent that the method or process does not rely on the particular order of steps set forth herein, the method or process should not be limited to the particular sequence of steps described. As one of ordinary skill in the art would appreciate, other sequences of steps may be possible. Therefore, the particular order of the steps set forth in the specification should not be construed as limitations on the claims. In addition, the claims directed to the method and/or process of the present invention should not be limited to the performance of their steps in the order written, and one skilled in the art can readily appreciate that the sequences may be varied and still remain within the spirit and scope of the present invention.