Abstract:
In accordance with the invention, the spring constant of a scanning probe microscope cantilever mechanically coupled to a microelectromechanical system (MEMS) actuator may be determined in-situ using a frequency resonance method.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application relates to the co-pending application Ser. No. ______ (Attorney Reference No. 10060541-1), filed on the same day entitled “Displacement Method for Determining the Spring Constant of Scanning Probe Microscope Cantilevers using MEMS Actuators” by Workman, Hoen and Clifford, and Ser. No. ______ (Attorney Reference No. 10060542-1), filed on the same day entitled “Force Method for Determining the Spring Constant of Scanning Probe Microscope Cantilevers using MEMS Actuators” by Workman, Hoen and Clifford, both owned by the assignee of this application and both incorporated herein by reference. 
     
     BACKGROUND 
       [0002]    Typically, it is difficult to measure the vertical and lateral spring constant of scanning probe microscope cantilevers accurately. The typical method of calibrating scanning probe microscope (SPM) cantilevers is the “Sader method”, described, for example, by Sader, Chon and Mulvaney in “Calibration of rectangular atomic force microscopy cantilevers”, Review of Scientific Instruments, 70(10), p. 3967, 1999 or by Cain et al. in “Force calibration in lateral force microscopy”, Journal of Colloid and Interface Science 227, p. 55, 2000. The “Sader method uses the length, width, resonance frequency, and quality factor, Q, of the scanning probe microscope cantilever to determine the spring constant. The “Sader method” does not depend on the optical lever sensitivity calibration. 
         [0003]    Other methods for determining the spring constant include the thermal power spectral density method described by Hutter and Bechhoefer in “Calibration of atomic-force microscope tips”, Review of Scientific Instruments, 64(7), p. 1868, 1993; the “Cleveland method”, described by Cleveland in “A non-destructive method for determining the spring constant of cantilevers for scanning force microscopy”, Review of Scientific Instruments, 64, p. 403, 1993; and the torsional MEMS method, described by Cumpson et al. in “Microelectromechanical system device for calibration of atomic force microscope cantilever spring constants between 0.01 and 4 N/m”, Journal of Vacuum Science and Technology A, 22(4), p. 1444, 2004. 
       SUMMARY 
       [0004]    In accordance with the invention, the spring constant of a scanning probe microscope cantilever mechanically coupled to a microelectromechanical system (MEMS) actuator may be determined in-situ using a frequency resonance method. 
     
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIGS. 1   a - b  shows an electrostatic MEMS motor. 
           [0006]      FIGS. 1   c - d  show an electrostatic comb drive. 
           [0007]      FIG. 1   e  shows the sensor position versus vertical probe position. 
           [0008]      FIG. 2   a  shows force versus position curves for an electrostatic comb drive in accordance with the invention. 
           [0009]      FIG. 2   b  shows force versus position curves for an electrostatic MEMS motor in accordance with the invention. 
           [0010]      FIG. 3  shows an embodiment in accordance with the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0011]      FIG. 1   a  shows scanning probe microscope cantilever  150  attached to electrostatic MEMS motor rotor  130  of electrostatic MEMS motor  135  in an embodiment in accordance with the invention. Scanning probe microscope cantilever  150  is attached to electrostatic MEMS motor rotor  130  such that scanning probe microscope cantilever  150  extends past the boundary of electrostatic MEMS motor rotor  130  to allow the use of, for example, an optical lever technique to monitor the vertical position of scanning probe tip  155 . The resonance frequency, ω 0 , of electrostatic MEMS motor rotor  130  is determined while scanning probe tip  155  is not in contact with surface  120 . The resonance frequency, ω new , of electrostatic MEMS motor rotor  130  is then measured with scanning probe tip  155  in contact with surface  120  as shown in  FIG. 1   b  and scanning probe microscope cantilever  150  is at the zero sensor position. Surface  120  is assumed to be sufficiently “hard” that scanning probe tip  155  moves less than about 10 percent as much as electrostatic MEMS motor rotor  130  or electrostatic comb drive rotor  182  (see  FIG. 1   c ) when scanning probe tip  155  is brought into contact with surface  120 . Methods for determining the resonance frequency are discussed below. 
         [0012]    For the purposes of this description, contact between scanning probe tip  155  and surface  120  is defined as when the vertical position of scanning probe tip  155  is to the left of inflection point  199  of probe-surface interaction force  198  as shown in  FIG. 1   e . Note that the sensor position is proportional to probe-surface interaction force  198 . The term “sensor position” refers to the position of the reflected optical beam on the bi-cell photodetector as described, for example, in U.S. Pat. No. 5,587,523 and incorporated herein by reference. The position of the reflected optical beam can be used to determine the vertical position of scanning probe tip  155 . To simplify the discussion, the sensor is positioned so the zero of the sensor position readout corresponds to the situation when there are no surface forces acting on scanning probe tip  155  and corresponds to point  197  in  FIG. 1   e.    
         [0013]    If the resonant frequency of electrostatic MEMS motor rotor  130  is less than the resonant frequency of cantilever  150  then the additional spring force provided by the force of scanning probe microscope cantilever  150  pushing on surface  120  raises the resonance frequency of electrostatic MEMS motor rotor  130  by an amount proportional to the spring constant of scanning probe microscope cantilever  150 . The new resonance frequency of electrostatic MEMS motor rotor  130  is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     new 
                   
                   = 
                   
                     
                       
                         
                           κ 
                           tip 
                         
                         
                           m 
                           motor 
                         
                       
                       + 
                       
                         ω 
                         0 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where m motor  is the mass of electrostatic MEMS motor rotor  130 , cantilever  150  and scanning probe tip  155  and κ tip  is the spring constant of scanning probe microscope cantilever  150 . The spring constant of scanning probe microscope cantilever  150  is then given by: 
         [0000]      κ tip   =m   motor (ω new   2 −ω 0   2 )  (2) 
         [0000]    which can be approximated by: 
         [0000]      κ tip =2 m   motor ω 0 (ω new ω 0 )  (3) 
         [0000]    for ω 0   2 &gt;&gt;κ tip /m motor . 
         [0014]    For a conventional piezoelectric actuator used with typical scanning probe microscopes, typical values are m motor ≈0.1 kg, ω 0 ≈400π Hz and κ tip ≈0.1N/m which gives a frequency difference, (ω new −ω 0 ), of 0.0004 Hz. A frequency difference of this magnitude is typically difficult to accurately measure. On the other hand, using electrostatic MEMS motor rotor  130  with m motor ≈10 −6  kg typically gives a measurable frequency difference of about 2 Hz. Typical values for ω 0  are in the range from about 4π kHz to about 80π kHz and typical values for κ tip  are in the range from about 0.05 N/m to about 100 N/m. This method in accordance with the invention allows determination of the spring constant, κ tip , independently of the geometry, optical lever calibration or other properties of scanning probe microscope cantilever  150 . 
         [0015]    Other MEMS actuators may be used in accordance with the invention. For example, an electrostatic comb drive may be used in place of electrostatic MEMS motor  135 .  FIG. 1   c  shows scanning probe microscope cantilever  150  attached to electrostatic comb drive rotor  182  of electrostatic comb drive  180  in an embodiment in accordance with the invention. Scanning probe microscope cantilever  150  is attached to electrostatic comb drive rotor  130  such that scanning probe microscope cantilever  150  extends past the boundary of electrostatic comb drive rotor  182  to allow the use of, for example, an optical lever technique to monitor the vertical position of scanning probe tip  155 . The resonance frequency, ω 0 , of electrostatic comb drive rotor  182  is determined while scanning probe tip  155  is not in contact with surface  120  (see  FIG. 1   d ). The resonance frequency, ω new , of electrostatic comb drive rotor  182  is measured with scanning probe tip  155  in contact with surface  120  as shown in  FIG. 1   d  and scanning probe microscope cantilever  150  is at the zero sensor position. The analysis for determining the spring constant, κ tip , of scanning probe microscope cantilever  150  using the resonance frequency of electrostatic comb drive rotor  182  follows the discussion above for scanning probe microscope cantilever  150  using electrostatic MEMS motor  135 . 
         [0016]    The particular electrostatic MEMS actuator selected effects the relationship between the measured frequencies and the spring constant, κ tip , of scanning microscope cantilever  150 .  FIGS. 1   a  and  1   b  show electrostatic MEMS motor  135  which is a surface drive actuator while  FIGS. 1   c  and  1   d  show electrostatic comb drive  180 .  FIGS. 2   a  and  2   b  show force versus position curves for electrostatic comb drive  180  and electrostatic MEMS motor rotor  135 , respectively. For both electrostatic comb drive  180  and electrostatic MEMS motor  135 , the force versus position curves are the sum of three components: the force of springs  140  or  186  that constrain electrostatic MEMS motor rotor  130  or electrostatic comb drive rotor  182 , the electrostatic force generated by electrostatic MEMS motor rotor  130  or electrostatic comb drive rotor  182  and the force from scanning probe microscope cantilever  150  or scanning probe microscope cantilever  150 , respectively. The force from scanning probe microscope cantilever  150  is present only if scanning probe tip  155  is in contact with surface  120 . 
         [0017]    For electrostatic comb drive  180  as described by, for example, R. Legtenberg, A. W. Groeneveld and M. Elwenspoek in “Comb-drive actuators for large displacements”, Journal of Micromechanics and Microengineering, 6, pp. 320-329, 1996, incorporated herein by reference, the electrostatic force can be approximated as follows: 
         [0000]    
       
         
           
             
               
                 
                   F 
                   ≈ 
                   
                     
                       
                         ɛ 
                         0 
                       
                        
                       
                         LV 
                         applied 
                         2 
                       
                     
                     d 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where L is the sum of the thicknesses of all comb fingers  185  in electrostatic comb drive  180 . From Equation (4), it can be seen that the electrostatic force, F, is essentially independent of position. At equilibrium, the electrostatic force is equal to and the negative of the spring forces contributed by springs  186  and scanning probe microscope cantilever  150 . This allows the rest position of electrostatic comb drive  180  to be determined by considering where the negative of the spring forces are equal to the force generated by electrostatic comb drive  180 . In  FIG. 2   a , curve  210  shows the negative of the spring forces as a function of position when scanning probe tip  150  is not in contact with surface  120  and curve  220  shows the negative spring forces as a function of position when scanning probe tip is in contact with surface  120 . When scanning probe microscope cantilever  150  is not in contact with surface  120 , the equilibrium position of electrostatic comb drive  180  is shown by non-contact point  260  in  FIG. 2   a . When scanning probe tip  150  is in contact with surface  120 , an additional spring force is added due to the spring force contributed by scanning probe microscope cantilever  155  and the equilibrium position of electrostatic comb drive  180  moves and is shown by contact point  265  in  FIG. 2   a . Because force curve  267  for electrostatic comb drive  180  is essentially independent of position, changes in resonant frequency are due to the spring force contributed by scanning probe microscope cantilever  155  when scanning probe tip  150  is in contact with surface  120 . 
         [0018]    For electrostatic MEMS motor  135  as described in, for example, U.S. Pat. No. 5,986,381, incorporated herein by reference, the electrostatic force is not independent of position. The electrostatic force is typically periodic with the rotor position and for electrostatic MEMS motor rotor  130  the electrostatic force is a sinusoidal function of position as shown by curve  270  in  FIG. 2   b . The amplitude of the electrostatic force depends on the applied voltage and the position of the zero crossing depends on the specific voltage pattern applied to electrostatic MEMS motor  135 . In  FIG. 2   b , the force of springs  140  with position is shown by curve  280  and the force of springs  140  plus the force due to the contact of scanning probe tip  155  in contact with surface  120  with position is shown by curve  285 . Electrostatic MEMS motor  135  is at rest in equilibrium position  286  when scanning probe tip  155  is not in contact with surface  120 . Equilibrium position  286  occurs where curve  280  intersects curve  270 . When scanning probe tip  155  is in contact with surface  120 , an additional spring force due to scanning microscope cantilever  150  results in new equilibrium position  288  which is where curve  285  intersects curve  270 . Equilibrium position  286  and the associated resonance frequency depend on the functional form of the electrostatic force curve. For small changes in position as shown in  FIG. 2   b , electrostatic force curve  270  can be approximated as a straight line. 
         [0019]    The resonance frequency, ω n , may be measured by observing the response of electrostatic MEMS motor rotor  130  to a step, pulse or swept-sine forcing function. Measurement of the resonance frequency, ω n , is performed using a sensor which does not affect the result such as an optical or capacitive sensor. 
         [0020]    In particular, one way to determine the resonance frequency, ω n , of electrostatic MEMS rotor motor  130  in accordance with the invention is to apply a low voltage sine wave, typically about 0.025 of the overall bias voltage, to the disrupter electrode (not shown, see for example, U.S. Pat. No. 5,986,381) of electrostatic motor  135 . The voltage signal from the capacitive position sensor (not shown) is then multiplied by the applied sine wave voltage and averaged over several periods to produce a sine mixed signal. The voltage signal from the capacitive position sensor is also multiplied by a signal that is 90 degrees out of phase with the applied sine wave voltage and average over several periods to produce a cosine mixed signal. The sine mixed signal is combined in quadrature with the cosine mixed signal to give the signal magnitude. The frequency of the applied sine wave voltage is then typically varied by several Hz to determine the signal magnitude as a function of frequency. The resonant frequency occurs when the signal magnitude is a maximum. Alternatively, the resonant frequency may be found by noting the frequency where the sine mixed signal crosses zero. 
         [0021]    Similar methods for determining the resonance frequency, ω n , may be used for other MEMS actuators such as electrostatic comb drive  180  where the mass of electrostatic comb drive rotor  182  is used in place of the mass of electrostatic MEMS motor rotor  130 . Note that the resonance frequency, ω n , is not necessarily the lowest resonance frequency. For example, it is possible to design electrostatic comb drives  180  where the vibration of probe tip  155  is not parallel to the desired direction of probe tip travel at the lowest resonance frequency. Here, it is necessary to go to a higher order resonance to obtain resonance motion that is parallel to the desired direction of travel for probe tip  155 . 
         [0022]      FIG. 3  shows the steps of an embodiment in accordance with the invention where electrostatic MEMS motor  135  is used as an example. Initially, in step  310 , electrostatic MEMS motor rotor  130  is lowered until scanning probe tip  155  is in contact with surface  120  and scanning probe microscope cantilever  150  is substantially at the zero sensor position (see  FIG. 1   e ). Alternatively, surface  120  may be raised until surface  120  is in contact with scanning probe tip  155  but this typically results in a less controllable sensor position. Next, in step  320 , the resonance frequency, ω new , of electrostatic MEMS motor rotor  130  is determined while scanning probe tip  155  is in contact with surface  120 . Then, in step  330 , the resonance frequency, ω 0 , of electrostatic MEMS motor rotor  130  is determined after surface  120  has been lowered so that scanning probe tip  155  is no longer in contact with surface  120 . Then in step  340 , the spring constant, κ tip , of scanning probe microscope cantilever  150  is calculated as described above. Note that one may start with step  330  and then proceed to step  310  after electrostatic MEMS motor rotor  130  is lowered but this typically changes the applied voltage which introduces an additional variable, thereby complicating the analysis for certain electrostatic MEMS actuators such as electrostatic MEMS motor  135 . The method in accordance with the invention described above provides that the position of electrostatic MEMS motor rotor  130  relative to curves  285  and  280  is identical during the two resonance measurements because cantilever  150  is in the same position so that points  288  and  286  lie on top of one another. The example of electrostatic MEMS motor  135  is illustrative and any suitable MEMS actuator such as electrostatic comb drive  180  discussed above may be used in accordance with the invention. For electrostatic comb drive  180 , electrostatic comb drive rotor  182  replaces electrostatic MEMS motor rotor  130  in the above discussion and in  FIG. 2   
         [0023]    While the invention has been described in conjunction with specific embodiments, it is evident to those skilled in the art that many alternatives, modifications, and variations will be apparent in light of the foregoing description. Accordingly, the invention is intended to embrace all other such alternatives, modifications, and variations that fall within the spirit and scope of the appended claims.