Abstract:
The present invention provides a device for in-situ monitoring of material, process and dynamic properties of a MEMS device. The monitoring device includes a pair of comb drives, a cantilever suspension comprising a translating shuttle operatively connected with the pair of comb drives, structures for applying an electrical potential to the comb drives to displace the shuttle, structures for measuring an electrical potential from the pair of comb drives; measuring combs configured to measure the displacement of the shuttle, and structures for measuring an electrical capacitance of the measuring combs. Each of the comb drives may have differently sized comb finger gaps and a different number of comb finger gaps. The shuttle may be formed on two cantilevers perpendicularly disposed with the shuttle, whereby the cantilevers act as springs to return the shuttle to its initial position after each displacement.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority from U.S. provisional patent application No. 60/793,424, filed Apr. 19, 2006, the entirety of which is incorporated herein by reference. 
    
    
     STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT 
     This invention was made with Government support under NSF Grant Nos. EIA-0122599 and EEC-0318642. The Government has certain rights to this invention. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to a Micro-Electromechanical System (MEMS) metrology device and techniques for using the device to measure geometric, dynamic, and material properties of MEMS devices. 
     MEMS devices are known. MEMS devices typically include a two-and-one-half dimension (2.5D) structure that has movable and anchored portions. A common MEMS device having such a structure is a MEMS accelerometer such as the NASA Electronic Part and Package (NEEP) 2001. Such a device includes a generally planar structure having one or more beams, one or more springs, stationary polysilicon fingers and capacitive sense plates. Such devices are made from poly or single crystal silicon as well as from SiGe and SiC and other silicon-based materials. It has been determined that the performance of such devices changes in time due in part to changes in their material properties, which result in part from changes in thermal and load or shock cycles. There is therefore a need for monitoring the changes in material and geometric properties of MEMS devices. 
     In the CMOS world E-test devices are available that allow for various in-situ measurements. For example, E-test resistivity bridges can be used to measure resistivities. Also, E-test devices are available for the in-situ measurement of device line widths. So, while in the CMOS world E-test devices are available that can be incorporated in the masks used to manufacture the CMOS devices, no such equivalent in-situ device exists in the MEMS world. This lack of in-situ measurement capability for MEMS devices is further complicated due to the fact that MEMS devices are movable and that an aspect of their related measurement is directed to measuring the mechanical properties of the MEMS devices. 
     Various approaches are currently available for the measurement of parameters related to MEMS devices. Some of these approaches involve the use of Scanning Electron Microscopy (SEM), optical microscopy, interferometry, surface profileometry and nanoindentation. These approaches tend to be very expensive to implement requiring the use of expensive equipment and highly qualified operators. Another known test is the M-test, which involves using a test chip having arrays of cantilevered beams that are electrostatically actuated. Such a test is sensitive to process variations and cannot measure geometric properties, such as the line width of the device. Another test uses the out-of-plane bending of a cantilevered beam. This test is however restricted to the measurement of out-of-plane characteristics of a beam. 
     There is therefore a need for a tool for monitoring the material and geometric properties of MEMS devices that does not suffer from the above shortcomings. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention provides a device for in-situ monitoring of material, process and dynamic properties of a MEMS device. The monitoring device includes a pair of comb drives; a cantilever suspension comprising a translating shuttle operatively connected with the pair of comb drives; structures for applying an electrical potential to the comb drives to displace the shuttle; structures for measuring an electrical potential from the pair of comb drives; measuring combs configured to measure the displacement of the shuttle; and structures for measuring an electrical capacitance of the measuring combs. Each of the comb drives may have differently sized comb finger gaps and a different number of comb finger gaps. The shuttle may be formed on two cantilevers perpendicularly disposed with the shuttle, whereby the cantilevers act as springs to return the shuttle to its initial position after each displacement. 
     In another aspect, the present invention provides a method for monitoring of material, process and dynamic properties of a MEMS device. The methods includes forming the above described monitoring device adjacent to the MEMS device. A first one of the comb drives is actuated to displace the shuttle and an electrical capacitance of the measuring combs is used to determine when the displacement of the shuttle has reached a set distance. A first electrical potential placed on the first comb drive to accomplish the displacement of the shuttle by the set distance is measured. A second one of the comb drives is actuated to displace the shuttle and the electrical capacitance of the measuring combs is used to determine when the displacement of the shuttle has reached the set distance. A second electrical potential placed on the second comb drive to accomplish the displacement of the shuttle by the set distance is measured. Moreover, an electrical capacitance of the first comb drive may be measured while the second comb drive is displacing the shuttle by the set distance. 
     The embodiments of the present invention enable the in-situ monitoring of material, process and dynamic properties of MEMS devices. The MEMS metrology devices of the present invention enable a nondestructive metrology scheme, are inexpensive and thus are deployable in arrays. They are also compact in size and so are ideal for on-chip integration and thus can be constructed in close proximity to the MEMS devices being monitored. In addition, the MEMS metrology devices in accordance with the embodiments of the present invention provide for accurate measurements and monitoring of material, process and dynamic properties of MEMS devices. 
     For a further understanding of the nature and advantages of the invention, reference should be made to the following description taken in conjunction with the accompanying figures. It is to be expressly understood, however, that each of the figures is provided for the purpose of illustration and description only and is not intended as a definition of the limits of the embodiments of the present invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a device layout of a MEMS metrology device in accordance with the embodiments of the present invention. 
         FIG. 2  is a detail view of the complementary measuring combs of the device of  FIG. 1 .  FIG. 2  also shows the complementary comb-drive that is used for increasing measurement sensitivity. 
         FIGS. 2A-F  are simplified drawings showing the relative movement of the measuring combs  102  and the varying capacitance vs. displacement for measuring combs.  FIG. 2G  is a simplified drawing showing how the differential capacitance of the complementary comb drive is measured. 
         FIG. 3  is an exemplary schematic diagram of an embodiment of the device of  FIG. 1 . 
         FIG. 4  is an exemplary schematic diagram of one embodiment of the measuring combs of  FIG. 3 . 
         FIG. 5  is an exemplary schematic diagram of a second embodiment of the measuring combs of  FIG. 3 . 
         FIG. 6  is an exemplary schematic diagram of an embodiment of the gap closing sense array of  FIG. 3 . 
         FIG. 7  is an exemplary schematic diagram of the drive/sense comb-drive of  FIG. 3 . 
         FIG. 8  is an exemplary circuit diagram of the differential sense portion of the device of  FIG. 3 , and which does not include capacitive bypassing of the power supply. 
         FIG. 9  is an exemplary overall circuit diagram for the device of  FIG. 3 . 
         FIGS. 10A-E  are exemplary device stack schematic drawings showing an embodiment of the fabrication of the MEMS metrology device in accordance with the embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1  shows a device layout of a MEMS metrology device in accordance with an embodiment of the present invention.  FIG. 2  is a detail view of the complementary measuring combs  102  of the device of  FIG. 1 .  FIG. 2  also shows the complementary comb-drive  104  that is used for increasing measurement sensitivity. In the device of  FIG. 1 , capacitance changes across the measuring combs  102  correspond with the lateral translation of the shuttle  106  as it is actuated by the comb-drives (e.g., drives  108  and  110 ). The device of  FIG. 1  can be a compact device, fitting inside a 1 mm by 1 mm square or smaller, that can accurately measure in-plane over- or under-cut, effective Young&#39;s Modulus, and the comb-drive force for the material and process in which it is made. This device is useful for nanoscale calipers, manipulators, and force gauges, as well as for the scientific exploration of physical forces, developing fabrication processes, calibrating simulations of devices, and automatically recalibrating MEMS devices for environmental changes. Details of the complementary measuring combs  102  are shown in  FIG. 2 . As used herein over- or under-cut refer to a deviation of a fabricated gap from its designed value. 
     In one embodiment, the procedure for measuring over- or under- cut, Young&#39;s Modulus, and the comb-drive force involves displacing the central cantilevered suspension  112  with the comb-drives  108  and  110  and measuring the deflection amount (x) with the complementary measuring combs  102 . For example, in connection with  FIG. 1 , two of the comb-drives  108  and  110  are actuated, which causes the cantilevers  112  to deflect and translate in the lateral direction  114  by deflection amount (x). Cantilevers  112  may act as springs that bias the shuttle  106  back to its original position after each displacement driven by the comb-drives. Using the measuring combs  102  to measure the deflection amount (x), the over- or under-cut can be determined. In order to increase the sensitivity of the measuring combs  102 , they can be driven closer together with the secondary measuring comb drive  104 . Then, by actuating one of the comb-drives (e.g., drive  108 ) and measuring the capacitance change with the other comb-drive (e.g., drive  110 ), the fringing field correction factor can be obtained for accurately determining the comb-drive force. 
     And, by assuming a uniform over- or undercut, the effective Young&#39;s Modulus of the system can be obtained. When over- or under-cut varies as a function of beam width and beam gap, then the same measurements may be repeated for several different comb-drives each with varying beam width and gaps. 
       FIGS. 2A-F  show simplified drawings of the relative movement of the complementary measuring combs  102  and the varying capacitance vs. displacement for the combs  102 . As is shown in  FIGS. 2A-F , as the central suspension  102   a  is displaced (corresponding to the displacement of shuttle  106 ), the capacitance value of the combs  102  varies from a maximum value ( FIG. 2A ) to a minimum value ( FIG. 2C ) and back to a maximum value ( FIG. 2F ). As is shown in  FIG. 2F , the spacing between comb fingers  202  can be measured as the displacement corresponding to the peak capacitance values.  FIG. 2G  is a simplified drawing that extends the concept of  FIGS. 2A-F  to demonstrate that differential capacitance of the complementary comb drive can be measured as the difference in capacitance between two sets of measuring combs  102 - 1  and  102 - 2 . 
     A more detailed description of the method described above follows. First, the measuring combs  102  are used to find what applied voltages, V 1  and V 2 , on two separate comb drives  108  and  110 , will allow the cantilever  112  to displace by a fixed amount, (x). The comb drives  108  and  110  can have different gap sizes between comb fingers, g 1  and g 2 , and possibly different number of drive finger gaps, N 1  and N 2 . 
     The over- or under-cut, (e), may be calculated as:
 
 e =( N   1   V   1   2   g   2   −N   2   V   2   2   g   1 )/( N   1   V   1   2   −N   2   V   2   2 ).
 
     Alternatively, (e) can be determined by finding the scale factor, (S), applied to V 2 , to match the measuring comb&#39;s capacitive profile for the second comb drive (e.g., drive  110 ) with that of the first comb drive (e.g., drive  108 ). Then,
 
 e =( S   2   N   1   g   2   −N   2   g   1 )/( S   2   N   1   −N   2 ).
 
     The fringing field factor (a) can then be determined by driving the second comb drive (e.g., drive  110 ) to (x) and measuring the change of capacitance, (C), on the first comb drive (e.g., drive  108 ):
 
 a=C ( g   1   −e )/( xN   1 ε 0 ε air   h ), where (h) is the thickness of the silicon.
 
Next for a uniform over- or under-cut, the effective Young&#39;s Modulus, (E), of the silicon is found by:
 
 E=CV   1   2 ( L−e ) 3 /(4 x   2   h ( w+e ) 3 ),
 
     where (w) is the width of the cantilever beams  112  and (L) is the length of the cantilever beams  112 . 
     Multiple comb drives (e.g., drives  108 ,  110 ) may be used to verify the over- or under-cut for beams and gaps of different sizes. 
     To improve the accuracy of the pair of complementary combs  102  for measuring capacitance changes, one set of the complementary combs  102  (shown in  FIG. 2 ) can be offset (as shown in  FIG. 2G ). With the one offset pair of complementary combs  102 - 2 , the capacitance of one pair of combs can reach a maximum when the other pair (e.g., combs  102 - 1 ) reaches a minimum as the shuttle  106  is actuated. This allows a differential sense circuit to be used to measure the capacitance difference between the pair of complementary combs  102 - 1  and  102 - 2 . 
     A design layout, circuits, and measurement steps used for characterizing over- or under-cut, effective Young&#39;s Modulus, and comb-drive forces using the MEMS metrology device in accordance with the embodiments of the present invention is described below. The components of the characterization device are labeled in Design Layout section. The measurement circuit and set-up are shown in the Circuit Layout section. The measurement procedures are shown in Measurements section. 
     Design Layout 
     Overall Design 
       FIG. 3  shows an exemplary schematic diagram of an embodiment of the device of  FIG. 1 . In  FIGS. 3-9 , electrical terminals are shown as cross-hatched boxes (e.g., terminals A 1 , S 1 , and S 2  of  FIG. 3 ). As can be seen, the design includes the following components or subsystems:
         A. an anchored guided suspension  112  with cantilever width, cw, and cantilever length, cl,   B. a shuttle  106  that joins the suspension  112  and the driving and sensing apparatus,   C. a set of driving and sensing comb-drives  108 / 110 ,   D. measuring combs  102  to sense displacement of the shuttle  106 , and   E. an optional gap-closing sense array  302  for further assessing cut-error.       

     The following guidelines can be used in the overall design of the device. The width of the guided suspension  112 , cw, can be chosen as small as possible to keep the required length of the guided suspension, cl, as low as possible. When the beam-anchor compliance is being studied, the second structure should contain another guided suspension of a different width, a good value may be cw*1.25 (this doubles the stiffness) but the optimal value may depend on the process. The length of the guided suspension  112 , cl, should be chosen small enough that processing steps will not break it, stiction will not immobilize it, and it will fit in the desired amount of space. cl should be chosen large enough that the shuttle  106  may move far enough (without breaking the suspension) that the measuring combs  102  senses a capacitive peak. Also, cl should be chosen large enough that the required voltage to achieve the desired displacement (x) of the shuttle  106  is kept small enough for the equipment to handle. The shuttle  106  should be kept as stiff as possible while still being releasable. For improved sensitivity and simplicity, driving and sensing comb-drives (e.g., drives  108 / 110 ) should be matched on both sides of the shuttle  106 . When measuring cut-errors for many different geometries, it may be more economical to use differing comb-drive dimensions for comb-drives located on opposite sides. 
     Complementary Measuring Comb Design 
       FIG. 4  shows an exemplary schematic diagram of one embodiment of the measuring combs  102  of  FIG. 3 . Vertical members  102   a  in  FIG. 4  correspond with the vertical members  102   a  of  FIG. 3 . Electrical terminals C 1  and C 2  are located adjacent anchors  410  and  420 , respectively. Terminal C 3  is located within comb drive  104 . As shown in  FIG. 4 , the design of the measuring combs  102  includes the following:
         F. one set of matched teeth  102 - 1 ,   G. one set of mismatched teeth  102 - 2 ,   H. suspensions  402  that allow the teeth sets to be moved towards each other by moving combs  102   b , and   I. comb-drive  104  to actuate the suspensions  402 .       

     The following guidelines can be used in the overall design of the device of  FIG. 4 . The set of matched teeth  102 - 1  and mismatched teeth  102 - 2  should have the same spacing between comb teeth. In addition, the suspension drive  104  may be joined or separated depending on whether more simplicity or flexibility is needed. 
       FIG. 5  shows an exemplary schematic diagram of a second embodiment of a measuring combs  102  of  FIG. 3 . In the alternative configuration of  FIG. 5 , four teeth sets,  502 - 1 , - 2 , - 3 , and - 4 , may be used having four different alignments, the alignment of each set shifted by a quarter of the distance between adjacent comb teeth. The sinusoidal-like capacitance function is then shifted by 90 degrees, and a Hariharan type algorithm may be applied to find the displacement (x) (described in the measurements section below). 
     Gap-Closing Sense Array Design 
       FIG. 6  shows an exemplary schematic diagram of an embodiment of the gap closing sense array  302  of  FIG. 3 . An optional gap-closing sense array  302  may be used to further refine the measurements of cut-error and also measure layer thickness. Each gap-closing sensor has two fixed gaps  601  and  602 , each defined between a beam of the gap sensor and a beam carried on the shuttle  106 . For gap  602 , beam  604  of the gap sensor overlaps with beam  606  of the shuttle by a set distance. The second set of beams  603  and  605  overlap by an additional distance, dw, compared to the overlap distance between beams  604  and  606 . The differential capacitance between the two sets of beams  604 / 606  and  603 / 605  can be used to determine the value of gaps  601  and  602 , and therefore cut-error, the method of which is discussed in further detail below. With reference to  FIG. 6 , G 1  and G 2  can be tied to A 1  if gap-closing sensing is not used (to avoid charging effects). In general, A 1 &#39;s DC offset (if used) should also be applied to G 1  and G 2 . 
     Drive/Sense Comb-Drive Design 
       FIG. 7  shown an exemplary schematic diagram of the drive/sense comb drives (e.g., such as drives  108 / 110 ) of  FIG. 3 . In a balanced configuration, such as that shown in  FIG. 7 , comb-drives  702 / 703 / 704 / 705  are disposed on both sides of shuttle  106 , one of which may be driven while the differential capacitance is sensed between them by combs  102  (not shown). Each comb-drive (e.g.,  702  and  703 ) may have different finger widths, w, and gap spacings, g, between fingers. The finger overlap can be kept small to reduce levitation effects. Comb-drives  704  and  705  on the opposite side of the shuttle  106  may have fewer comb-fingers but more overlap to reduce tilt, or may be identical to the opposing comb-drives  702  and  703 , respectively. 
     Circuit Layout 
       FIG. 8  shows an exemplary circuit diagram of the differential sense portion of the device of  FIG. 3 , and which does not include capacitive bypassing of the power supply. In one characterization procedure, sensitive differential capacitance measurements (with a resolution around or below 10 aF) are made while a voltage sweep is supplied to the driving comb-set. There are two variations for measuring the differential capacitance—one using a charge-integrator scheme and another using a differentiation scheme, both of which can be used by changing capacitors and resistors. In  FIG. 8 , resistor values are denoted with an “R” and capacitor values are denoted with a “C”. In connection with  FIG. 8 , the signal from the DC biasing (Vb 1  and Vb 2 ) is low-pass filtered through resistor, R 1 , and capacitor, C 1  (large). When an integrator is desired, C 2  is set to a small capacitive value and R 2  is set to a large resistive value. The gain (V/|Vac|) is roughly 2*d/C 2 . When a differentiator is desired, C 2  is not used, and R 2  is set to around a 100 Kohm level. The gain (V/|Vac|) is roughly 2*d*R 2 *omega (where omega is the frequency of Vac). The second stage amplifier can further be used to amplify the signal. The gain is roughly R 4 /R 3 . Additionally, the alternating signal can be converted to a DC signal through an RMS to DC converter. 
       FIG. 9  shows an exemplary overall circuit diagram for the device of  FIG. 3 . Using the terminal notation from the previous figures, terminals not shown are grounded. Vac is an alternating signal (&gt;200 kHz) applied to the suspension and Vdc is a bias applied to the suspension (and other components) to correct for the levitation effect. 
     Measurements 
     Using the systems and circuits of  FIGS. 1-9  described above, the MEMS metrology in accordance with the embodiments of the present invention are described below. A first measurement methodology is described first, followed by alternative methodologies that are used to counteract the levitation effect, refine the measuring comb measurements, and refine the measurement of cut error. The description below shows how the comb-drive force (F), cut error (e), and then effective Young&#39;s Modulus (E) are measured using capacitance and voltage measurements based on an known layer thickness (h). This methodology includes the following steps: 
     Step A: The differential sense circuit shown in  FIG. 8  is calibrated so that the capacitance, d, is known from a measurement of V (note that there is a factor of 2). An LCR meter such as an HP4824A may be used for this purpose. 
     Step B: The suspension sweeping V_bias is moved and V_comp as well as V_sense (*) are measured. The V_bias and V_sense values associated with the first peak of V_comp (either a max or min value) are found (it can be more accurate to do this measurement by curve-fitting). Corrections can be made when the output is shifted (due to imperfectly matched capacitances). The values for V_comp may be shifted such that the max and min are equidistant from the origin. Vc_bias is adjusted until the maxima and minima are known to the desired accuracy. The shuttle moves a distance of half the distance (x) between successive complementary-comb teeth. The change of capacitance (dC) can then be determined from the calibration data and V_sense. The force (F) exerted on the suspension is then V_bias 2 dC/(2*x). 
     Step C: By using a model of the comb-drive force (F) allows the determination of the cut error (e) from two different comb-drive sets. F is related to the number of fingers (N) of the comb-drive side with fewer comb fingers, finger gap (g) (for simplicity, let g e =g−e), finger width (w), layer thickness (h), and fringing field factor (α) as 
     
       
         
           
             F 
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                   ɛ 
                   0 
                 
                 ⁢ 
                 
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                   air 
                 
                 ⁢ 
                 h 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   V 
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                 g 
                 e 
               
             
           
         
       
     
     Assuming α and e remain nearly constant, two different comb-drive sets can be used with varying layout gaps, g 1  and g 2 , and number of fingers, N 1  and N 2 . Using step B, the V_bias bias voltages, V 1  and V 2 , are found for the two sets that generate the same force (same displacement). Cut error is then computed as 
     
       
         
           
             e 
             = 
             
               
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                     V 
                     1 
                     2 
                   
                   ⁢ 
                   
                     g 
                     2 
                   
                 
                 - 
                 
                   
                     N 
                     2 
                   
                   ⁢ 
                   
                     V 
                     2 
                     2 
                   
                   ⁢ 
                   
                     g 
                     1 
                   
                 
               
               
                 
                   
                     N 
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                     V 
                     2 
                     2 
                   
                 
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                     N 
                     1 
                   
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                     V 
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     Step D: The effective Young&#39;s Modulus may be computed as 
             E   =         F   ⁡     (     l   +   e     )       3       2   ⁢           ⁢   x   ⁢           ⁢       h   ⁡     (     w   -   e     )       3               
where 1 is the cantilever length, and w is the cantilever width.
 
     It should be noted that in step B, if and when the complementary teeth are tightly packed in the configuration in  FIG. 4 , then set Vc_bias=0 and measure the complementary-comb signal, V_comp 0 , as V_bias is swept. As Vc_bias is adjusted, the sweep data is subtracted from the original sweep (this subtraction will remove global effects), (V_comp−V_comp 0 ). 
     To measure the actual Young&#39;s Modulus, two separate suspensions may be used, one with a differing cantilever width. The stiffness due to beam-anchor compliance and webbing effects can then be determined using the relationship: K=Kweb_compliance+Ksuspension. 
     Correcting for the Levitation Effect 
     To correct for levitation effects, the optional comb-sets (B in  FIG. 7 ) can be used. For each biasing voltage, V_bias, the following steps are taken to remove levitation effects.
         A. Apply bias to V 2 _bias until |V_comp| is maximized.   B. Apply bias to Vdc until |V_comp| is maximized.       

     Refining the Cut Error Measurements 
     The cut error for differing geometries can also be measured using a gap-closing actuator shown in  FIG. 6 . This arrangement can provide for more accuracy. The measurement includes the following steps:
         A. As V_bias is swept, measure V_gap.   B. Use the calibration data to convert V_gap to capacitance, C_gap.   C. Use the complementary-comb data to convert V_bias to the translation, x.   D. Fit the data to C_gap=αε 0 ε air A/(g model −x). (αε 0 ε air A is constant)   E. Cut error is then g model −g.   F. A short set (3-5 microns in length) of gap-closing sensors may be used as gap stops to allow the determination of the sidewall angle. The cut-error determined in step E is near the midpoint of the layer. Actuate the shuttle until the gap is reached. Measure the displacement, g bottom , using the V_sense data. The sidewall angle is approximately (g bottom −g model )/(h/2).   G. Similarly, as the gap is closed, the fringing-field effect is reduced. At the point where the fringing-field effect is sufficiently reduced, the layer thickness, h, may be measured by finding the area A (from step D) and dividing by the layout protruding length (it is not affected by cut error).       

     Refining the Complementary-Comb Measurements 
     When the alternative approach shown in  FIG. 5  is taken, then refinements can be made to the comb-drive sensing device. V_comp and V_comp 2  will be 2 periodic functions (of displacement which is proportional to V_bias 2 ). They will also be 90 degrees out of phase with one another. The minima and maxima can be found when the function atan(V_comp 2 /V_comp) crosses multiples of pi. Near these regions, the data can be linearly fit to determine this crossing value precisely. Note that the atan function must be unwrapped (when values cross pi, they jump to −pi) and at that point, 2*pi is added to the results to make them continuous). 
     Fabrication 
     The MEMS metrology device in accordance with the embodiments of the present invention can be fabricated using a silicon-on-insulator substrate from a single mask, as shown in  FIGS. 10A-E . As is shown in  FIGS. 10A-E , the fabrication process includes the following steps. The fabrication of the device starts with a handling layer  1002  that has disposed on it an oxide layer  1004  that in turn has a device layer  1006  disposed on it ( FIG. 10A ). Next, a photoresist layer  1008  is deposited on the device layer  1006  ( FIG. 10B ). Next, the photoresist layer  1008  is patterned ( FIG. 10C ). Then the device layer  1006  and the oxide layer  1004  are etched ( FIG. 10D ) and then the device layer  1006  is released by the removal of the oxide layer  1004  ( FIG. 10E ), to form the finger-like structures of  FIG. 1 . For example, the device shown in  FIG. 1  is approximately 25-50 μm thick, has a foot print of approximately 1.5 mm by 1.0 mm, or preferably less than 1.0 mm by 1.0 mm. The MEMS metrology device in accordance with the embodiments of the present invention can be fabricated during the same fabrication process and formed adjacent to the MEMS device which will be monitored by the metrology device. The metrology device can then be used to measure both the under or over-cut of the device and also monitor its material, process, geometry and dynamic properties of MEMS device. Prototype MEMS metrology devices have been demonstrated to be inexpensive, reliable and accurate, having resolutions better than or similar to the resolution of optical microscopy device and nearing SEM device resolutions. Furthermore, due to the symmetric nature of over- or under-cut the deflection measurements of the MEMS metrology devices are resilient to the variation of comb tips. Such comb-tip variations include variations due to cut error resulting in narrower or wider combs; variations due to the filleting where combs are shortened and rounded; and cross-sectional variations. 
     As will be understood by those skilled in the art, the present invention may be embodied in other specific forms without departing from the essential characteristics thereof. These other embodiments are intended to be included within the scope of the present invention, which is set forth in the following claims.