Abstract:
A calibration device and method for lateral force calibration in small force measuring devices such as atomic force microscopes is disclosed. A platform has a substantially planar surface including a slot for accommodating at least part of the AFM cantilever tip, one or more supporting legs arranged to provide sprung resistance to the platform and a capacitive drive means for driving the platform laterally with respect to the AFM cantilever tip.

Description:
FIELD OF THE INVENTION  
       [0001]     The present invention relates to a calibration device and methods suitable for lateral forces in calibrating small force measuring devices. In particular, the present invention relates to a lateral calibration device and methods in which accurate measurements under, and traceable to standards such as the SI system can be obtained.  
       BACKGROUND OF THE INVENTION  
       [0002]     Measurements of small forces, in the nanonewton and piconewton range, have become important in recent years due to the widespread use of the Atomic Force Microscope (AFM) and associated instruments. There is a need to measure such small forces accurately, for example, protein-protein interactions or materials properties via the small force applied to an indenting tip.  
         [0003]     The quantification of interaction forces is much more problematic. Force on the tip is inferred from the deflection of the cantilever, using an assumed value for the cantilever spring constant. The accuracy to which the spring constant is known is the limiting factor in the accuracy of a force measurement. Many methods have been proposed for calibrating the stiffness of an AFM probe, but none are traceable, and typical accuracy is only about 20-30%.  
         [0004]     Reference artifacts for dimensional calibration of AFM have been available from many sources for ten years or more, but calibration of the force constant of AFM cantilevers is more troublesome. Uncalibrated cantilevers lead to very large errors in the measurement of nanonewton forces, such as in direct experiments to break individual covalent bonds by AFM, or the measurement of protein interaction forces. Commercial reference artifacts are available, but offer no traceability to the SI measurement system. This is important because there are two important methods of measuring nanoscale forces, AFM and optical tweezers. AFM is most conveniently calibrated using reference cantilevers, whereas optical tweezer forces are estimated based on the rate of change of photon momentum. Both methods are used, for example, in measuring bond-breaking forces. They must both have a common force scale, or burgeoning work in both areas will be difficult to build-upon. What is more, a traceable calibration method is now timely.  
         [0005]     AFMs measure topography accurately, and are calibrated for this purpose quite easily using step-height standards. Some AFM instruments even incorporate laser interferometry to make traceable height measurements. However accuracy is rarely mentioned for AFM force measurements. There is an increasing need for the accurate measurement of small lateral forces by AFM, in the mechanical analysis of contamination on semiconductor surfaces, polymer blends, functional thin films, recording media and measuring adhesion of nanoparticulates at surfaces. The lateral force signal is useful for identifying surface composition where the materials are relatively flat but have significantly different friction characteristics. When combined with the use of chemically functionalised AFM tips, lateral force imaging can reveal contrast between different surface species where none can be seen in any other scanned probe mode. Many existing and future applications use the lateral force signal only to provide image contrast, but in many other applications the quantitative comparison of lateral force measurements is essential. This has been difficult so far, due to the wide range of torsional constants seen in even supposedly similar cantilevers. Cantilever coatings, to improve reflectivity, or chemically functionalise the tip, can have a significant effect on spring and torsional constants that are difficult to model. A calibration method is required.  
         [0006]     A wide variety of methods have been used to calibrate normal spring constant, including thermal vibrations, reference cantilevers of measured dimensions, and radiation pressure. Commonly assigned co-pending patent application No. PCT/GB2004/002134, which is herein incorporated by reference discloses a MEMS device designed for the calibration of normal forces in AFMs, allowing piconewton and nanonewton force measurements to be made traceable to the SI system. However, calibration of lateral forces is more of a problem. Thermal vibrations can be useful, but there are fewer other options. Many of the methods that have been tried for the purpose of normal force calibration have extensions to allow the calibration of lateral forces, but some have no obvious method of being extended in this way, and are likely to be limited to the calibration of normal forces only. Those existing methods able to measure the torsional constant typically require accurate dimensional measurements (e.g. in an SEM) or high frequency power spectrum measurement that is beyond the bandwidth of the signal amplifiers supplied as part of the AFM electronics. In other words, these methods require additional facilities the AFM user may not have access to, and even if available, requires special training to achieve the accuracy needed.  
       STATEMENT OF INVENTION  
       [0007]     According to an aspect of the present invention, there is provided a calibration device for lateral calibration of a small force measuring device&#39;s tip comprising a platform having a substantially planar surface including a slot for accommodating at least part of the tip, one or more supporting legs arranged to provide sprung resistance to the platform and a capacitive drive means for driving the platform laterally with respect to the tip to enable measurement of a torsional constant for said small force measuring device.  
         [0008]     The capacitive drive means may include one or more interdigital comb drive capacitive actuators. The capacitive drive means may include a Watt Balance device.  
         [0009]     The calibration device may further comprising imaging means for enabling optical access to the calibration device when in use. The imaging means may include an optical prism and/or one or more mirrors.  
         [0010]     The slot may include one or more substantially tapered sides.  
         [0011]     The small force measuring device preferably includes an atomic force microscope.  
         [0012]     The calibration device is preferably a micro-electro-mechanical system (MEMS). The calibration device is preferably a silicon-on-insulator micro-machined device.  
         [0013]     The calibration device may be fabricated on a die including one or more other calibration devices.  
         [0014]     According to another aspect of the present invention, there is provided a calibration method for determining the torsional constant of a small force measuring device comprising:  
         [0015]     providing a calibration device comprising a platform having a substantially planar surface including a slot for accommodating at least part of the tip, one or more supporting legs arranged to provide sprung resistance to the platform and a capacitive drive means for driving the platform laterally with respect to the tip;  
         [0016]     placing the tip of said small force measuring device in contact with at least one side of the slot of the platform of the calibration device;  
         [0017]     measuring lateral force applied to said at least one side using the capacitive drive means; and,  
         [0018]     dividing the lateral spring constant of the calibration device by the measured lateral force.  
         [0019]     The method may further comprise obtaining the lateral spring constant of the calibration device.  
         [0020]     The method may further comprise performing said calibration after use of the small force measuring device.  
         [0021]     According to another aspect of the present invention, there is provided a method of determining the spring constant of a calibration device comprising:  
         [0022]     providing a calibration device comprising a platform having a substantially planar surface including a slot for accommodating at least part of the tip, one or more supporting legs arranged to provide sprung resistance to the platform and a capacitive drive means for driving the platform laterally with respect to the tip;  
         [0023]     measuring equilibrium lateral displacement of the calibration device as a function of applied voltage;  
         [0024]     measuring current to earth passing through the calibration device whilst substantially simultaneously measuring vibration velocity;  
         [0025]     measuring the spring constant of the one or more supporting legs; and, calculating the spring constant in dependence on the measurements.  
         [0026]     The step of measuring current to earth passing through the calibration device whilst substantially simultaneously measuring vibration velocity may include:  
         [0027]     applying a predetermined vibration to the calibration device and simultaneously measuring the velocity of the platform; and,  
         [0028]     calculating the gradient of capacitance of the calibration device in dependence on the measured velocity.  
         [0029]     The step of measuring equilibrium lateral displacement of the calibration device as a function of applied voltage may include:  
         [0030]     applying a predetermined voltage to the capacitive drive means and simultaneously measuring the static displacement of the platform.  
         [0031]     Measurement of the static displacement may use white-light interferometry.  
         [0032]     Measurement of the velocity may use Doppler velocimetry.  
         [0033]     Embodiments of the present invention are directed to a microfabricated device for the calibration of torsional spring constant, potentially traceable to SI standards. This will be particularly useful in the measurement of small frictional forces with near nanometre resolution.  
         [0034]     Lateral force comparisons can be performed easily in the user&#39;s AFM, with a precision of better than ±5%.  
         [0035]     For those AFM tips that have been chemically functionalised, the calibration is best performed retrospectively, after any experimental measurements that may depend on tip functionalisation.  
         [0036]     Calibration of lateral force microscopy (LFM) cantilevers is necessary for the measurement of nanonewton and piconewton frictional forces, which are critical to analytical applications of polymer surfaces, biological structures and organic molecules at nanoscale lateral resolution.  
         [0037]     In an embodiment of the present invention, a compact and easy-to-use reference calibration device is used for calibration.  
         [0038]     The calibration device allows measurements to be made that are traceable to the SI standards.  
         [0039]     A non-contact method enables measurement of the spring constant of these calibration devices, by a combination of electrical measurements and Doppler velocimetry. Traceability is important to ensure that force measurements by AFM are comparable to those made by optical tweezers and other methods.  
         [0040]     In preferred embodiments, the calibration device is a MEMS device fabricated by silicon-on-insulator (SOI) micromachining, and therefore has extremely low mass and good immunity to vibration.  
         [0041]     In an embodiment of the present invention, a method of calibrating the torsional constant of an AFM cantilever using a calibration device is disclosed. Importantly, the method requires no special equipment beyond the calibration device and that already present in the majority of AFM instruments, and familiar to the AFM practitioner. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0042]     Embodiments of the present invention will now be described in detail, by way of example only, with reference to the accompanying drawings in which:  
         [0043]      FIG. 1  is an optical micrograph of a calibration device according to an embodiment of the present invention;  
         [0044]      FIG. 2  is a block diagram showing how the spring constant of the calibration device of  FIG. 1  is measured;  
         [0045]      FIG. 3  is a graph illustrating fundamental mechanical resonance of the calibration device of  FIG. 1  in air at atmospheric pressure, measured by Doppler velocimetry;  
         [0046]      FIG. 4  is a diagram illustrating lateral force calibration by “Continuous Contact” (CC) using the calibration device of  FIG. 1 ;  
         [0047]      FIG. 5  is a diagram illustrating the sequence of events during “Non Continuous Contact” (NCC) lateral force calibration of an AFM cantilever;  
         [0048]      FIG. 6  is a graph of a measured lateral force signal for Continuous Contact (CC) lateral force calibration of an AFM cantilever;  
         [0049]      FIG. 7  is a schematic diagram of an arrangement for AFM lateral force calibration according to an embodiment of the present invention; and,  
         [0050]      FIGS. 8 and 9  are views of an arrangement for AFM lateral force calibration according to another embodiment of the present invention. 
     
    
     DETAILED DESCRIPTION  
       [0051]      FIG. 1  is an optical micrograph of a calibration device  10  according to an embodiment of the present invention. The inset  100  shows a cross section of the device  10 .  
         [0052]     The calibration device  10  includes a gold-coated silicon platform  20  suspended on supporting legs, in this embodiment in the form of four cantilever beams  40 . A capacitive drive means in the form of an electrostatic comb drive  30  allows the platform  20  to be moved laterally by the application of an electrical potential. Etched into the platform  20  is a slot  25  of width 3 μm, which an AFM tip will at least partially enter if scanned across the surface of the platform  20 . The calibration device  10  is preferably fabricated from a single crystal “silicon on insulator” (SOI) layer of nominal thickness 10±1 μm. This is patterned from the front by deep reactive ion etching (DRIE) leading to almost vertical side walls. A 400 μm thick silicon handling wafer is separated from the SOI structural layer by a 1 μm insulating oxide layer.  
         [0053]     The handling wafer was etched from the back side (i.e. the opposite side to the SOI layer) to completely remove a section of it below the resonator, while retaining enough mechanical robustness to allow electrical connections to be made on the front side by conventional gold wirebonding.  
         [0000]     Calibration of the Calibration Device  
         [0054]     The calibration device  10  realises a known nanonewton force in terms of traceable measurements of electrical quantities and linear displacement and velocity. This is performed in two measurement steps. 
        1. Static measurement. This consists of measuring the equilibrium lateral displacement of the calibration device  10  as a function of applied voltage. We measure this static displacement using white-light interferometry using a Zygo NewView 5020 interferometer (Zygo Corporation, Middlefield, Conn. 06455-0448, USA). The DRIE etched face of the calibration device  10  is sufficiently perpendicular to the plane of motion for optical fringes to be located and measured, allowing accurate measurement of static displacement.     2. Dynamic measurement, illustrated schematically in  FIG. 2 . This consists of measuring the current to earth passing through the device, while simultaneously measuring its instantaneous vibration velocity using Doppler velocimetry. The extremely sharp resonance of the calibration device, even when operating in air, allows us to separate the change in capacitance of the device due to mechanical displacement from the inevitable parasitic capacitances elsewhere in the circuit.        
 
         [0057]     The velocity of the calibration device  10  was measured, edge-on, using a Polytec OPV1 Doppler velocimeter (Polytec GmbH, Waldbronn, Germany), and this signal recorded using a HP 3562A Dynamic Signal Analyser (Agilent Technologies, Palo Alto, Calif.). These data were downloaded from the Dynamic Signal Analyser to a personal computer. Current through the calibration device  10  was measured using a CyberAmp 320 Signal conditioner with type 403 preamplifier (Molecular Devices Corporation, Union City, Calif.). By using it in “virtual-earth” configuration, any parasitic capacitance across the input of the amplifier (or between the moving part of the actuator and the die substrate) connects virtual earth  60  to earth, so its influence on the circuit operation is insignificant. In addition, the signal path from the calibration device  10  was carefully surrounded on the printed circuit board (PCB) by an earthed “guard” track  50 , to minimize the effect of small stray currents across the bare PCB surface, for example due to any small surface contamination by electrolytes.  
         [0058]     We measure the spring constant of the four supporting cantilever  40  springs by a method described in detail in co-pending commonly assigned patent application No. PCT/GB2004/002134, which is hereby incorporated by reference in its entirety. The current through the comb drive 30, for a potential V p  applied to it is given by;  
             i   =         ⅆ     (     CV   p     )         ⅆ   t       .             (   1   )             
 
         [0059]     We separate the capacitance of the calibration device  10  into two parts; the dynamic capacitance, C(x), which changes as the platform  20  is displaced laterally parallel to the x axis, and the static or parasitic part, C para , the capacitance between fixed parts of the calibration device  10 , for (example adjacent tracks and pads on the silicon die). If we measure the response of the calibration device  10  over a narrow frequency interval around the mechanical resonance, we expect the static capacitance to be constant, but the dynamic capacitance will vary with the motion of the platform. We apply a d.c. potential of V 0  to the stationary part of the comb drives  30 , together with a small a.c. component ν(t), so that 
 
 V   p ( t )= V   0 +ν(   t ).   (2) 
 
         [0060]     The purpose of the small a.c. component is to apply a small drive to the calibration device  10 , which, if this drive voltage is close to its mechanical resonant frequency, will cause it to vibrate mechanically with significant amplitude. Typically V 0  is chosen in the range 0.5 to 2V, and ν(t) is a sinusoid of amplitude chosen in the range 100 μV to 1 mV peak-to-peak. 
 
ν( t )=ν 0  cos(ω t )   (3) 
 
         [0061]     The velocity  20  of the platform is measured by Doppler velocimetry, in a configuration illustrated schematically in  FIG. 2 . For a particular bias voltage V 0 , and an a.c. component amplitude ν 0  sufficiently small, the capacitance C(x) varies linearly over the range of mechanical vibration. The lateral motion of the comb drive  30  makes it easy to fulfill this condition of linearity for larger amplitudes than possible with the normal force calibration device described in PCT/GB2004/002134, where the comb drives are operated in levitation mode.  
         [0062]     The static deflection of the platform  20  is the result of the balance between the elastic restoring force applied by the folded springs and the electrostatic force from the comb-drives  30 . The stored electrostatic field energy is  
             E   =       1   2     ⁢     CV   p   2               (   4   )             
 
 so that the electrostatic force is  
               F   elec     =       1   2     ⁢       ∂   C       ∂   x       ⁢     V   p   2               (   5   )             
 
 which balances an elastic force of; 
 
F elastic =k x{overscore (x)}   (6) 
 
 where {overscore (x)} is the measured static deflection. We equate the forces F elastic =F elec  in Eqns (5) and (6), and obtain the measured capacitance gradient ∂C/∂x from the dynamic measurements illustrated in  FIG. 2 . This allows us to determine the lateral spring constant k x . 
 
         [0063]     Note that the fact that, the dynamic measurements are made while the calibration device  10  resonates in a lateral mode means it has a much higher quality factor in air (Q≈230, as shown in  FIG. 3 . This indicates a resonance quality factor of around 230, high enough to allow calibration in air) than the calibration device disclosed in PCT/GB2004/002134, which presented a much greater cross-section. Therefore the static and dynamic steps of the calibration can both be carried out in air.  
         [0000]     AFM Cantilever Calibration  
         [0064]     Calibration of the torsional constant of an AFM cantilever  200  against a calibration device according to an embodiment of the present invention is illustrated in  FIGS. 4 and 5 . An imaging scan of the surface of the calibration device  10  is performed, centered on the slot  25 . Depending on the dimensions of the tip  210  of the cantilever  200  compared to the width of the slot  25 , and the setpoint for AFM topography feedback, two types of lateral force curve may be observed, both of which allow an AFM cantilever  200  to be calibrated. 
        (a) Continuous contact (“CC”), in which the tip  210  is in contact with the calibration device  10  at all times, and     (b) Non-continuous contact (“NCC”), in which the tip  210  breaks contact with the calibration device  10  for some distance while the tip  210  is inside the slot  25 .        
 
         [0067]      FIG. 4  and  FIG. 5  show schematic cross-sections of the path of the scanning tip in CC and NCC cases respectively.  
         [0068]     We can balance the forces on the AFM tip at the point in the scan illustrated in  FIG. 5  stage (c) (this point is also a feature of in continuous contact in  FIG. 4 , between the illustrated stages (d) and (e)). At this point in the scan the very tip  210  of the AFM cantilever  200  is in contact with the corner of the slot  25  in the platform  20  of the calibration device  10 , so that the lateral force on the tip  210  is just the product of the lateral displacement of the calibration device  10  and its lateral spring constant, k x , described above. Thus 
 
F x =k x Δx=sV L-R    (7) 
 
 where F x  is the lateral force on the tip, Δ x  is the lateral displacement of the calibration device  10  caused by the contact with the AFM tip  210 , V L-R  is the “left-minus-right” signal from the split photodiode, and s is the torsional spring constant that we need to measure for this AFM cantilever  100 . We have assumed a linear relationship, which is true of almost all AFM instruments over a certain range of deflection of the optical lever. In many instruments this range is very wide, and this linear relationship reliable. In other designs of AFM instrument, perhaps through design optimisation for imaging of topography at a single setpoint rather than force measurement, the range over which linearity can be assumed is small. Rearranging and differentiating we obtain,  
             s   =           k   x     ⁡     (       ∂     V     L   -   R           ∂   x       )         -   1       .             (   8   )             
 
         [0069]     Therefore the torsional constant of the cantilever  200  is simply the known lateral spring constant of the calibration device  10 , divided by the slope of the lateral force signal when scanning perpendicular to the slot  25 , at the point illustrated in  FIG. 5  stage (c).  
         [0070]     Since k x  is known, s is easily evaluated.  FIG. 6  shows experimental results from a Park Autoprobe CP Atomic Force Microscope with a cantilever  200  of nominal normal spring constant 30 N/m quoted by the cantilever supplier. In this case the lateral force signal varies linearly over the region in which the AFM tip  210  lies within the slot  25 , so that we can perform a linear fit to this region, giving in this particular case;  
               (       ∂     V     L   -   R           ∂   x       )     =     4520   ±     210   ⁢           ⁢   V   ⁢     /     ⁢   m               (   9   )             
 
         [0071]     This value, substituted into Eq. (8), then gives us the torsional constant of the cantilever  200 .  
         [0072]     Clearly however, we should not expect that all possible tip profiles give rise to a linear lateral force signal. In particular, if the aspect ratio of the tip  210  is high, while the slot width is large, the lateral force applied to the AFM tip  210  by the calibration device  10  in  FIG. 5  (c) is further from the torsional axis of the cantilever  200  than in (b), and hence we would expect a larger gradient at point (c). We have seen this nonlinearity experimentally for some cantilevers  200 . Two approaches are likely to be useful in these cases; 
        (i) Fitting an analytical model to the lateral force signal that takes account of the height of the tip (i.e. the “top-minus-bottom” signal from the split photodiode) so as to account for the increasing couple applied to the tip as it rises out -of the slot  25  in the calibration device  10 , and     (ii) Use of a calibration device  10  having a tapered slot  25   a  (as is shown in  FIG. 7 ), so that in a raster scan image of the surface of the platform  20  one can always find a linescan in which the tip penetration into the slot  25   a  is very small, and hence the increase in couple as the tip rises is negligible. This gives a range of slot widths within a single lateral force image, so that a slot sufficiently narrow to match the aspect ratio of the tip is always available.        
 
         [0075]     One advantage of the second approach is that the topographical image acquired simultaneously contains information on the shape of the AFM tip  210 , which is often also an objective in AFM calibration.  
         [0076]     It should be noted that a disadvantage of the above approach to AFM lateral force calibration, which it shares with other methods involving mechanical contact with the tip, is that the functionalised surface of the tip  10  may be damaged or modified during the calibration. Therefore calibration is best performed retrospectively, after experimental measurements that may depend on tip functionalisation.  
         [0000]     Measuring Lateral Displacement and Velocity  
         [0077]     Fabrication of calibration devices  10  according to embodiments of the present invention may be made near the centre of a 10 mm square die, which is simply broken the die in two to expose the calibration device  10  to interferometry edge-on. As an alternative to the methods for measuring the displacement and velocity of the calibration device  10  during the calibration measurements described above, a better approach may be to include imaging means to enable optical access to the edge of the calibration device  10  in situ.  
         [0078]     For example:  
         [0079]     (a) Using a suspended optical prism  300  to address the edge of the calibration device  10  by reflection, as shown in  FIG. 7 . The 1 mm high prism is attached to a microscope cover-slip using UV-curable optical adhesive. For imaging purposes this works well, indeed the inset image of the 10 μm thick calibration device  10  shown in  FIG. 1  was taken using this prism. For white-light interferometry, however, the extra path-length introduced by the prism is a problem for the Michaelson interferometer, and corrective optics are required. Corrective optics for the Zygo objective should allow calibration of lateral force calibration devices via a suspended prism.  
         [0080]     (b) Use of a calibration device  10  fabricated in surface micromachined polycrystalline silicon, shown in  FIGS. 8 and 9 . This version of the calibration device  10  shares the comb-drive feature of the silicon-on-insulator (SOI) design described earlier, but has two mirrors  400 ,  410 , one ( 400 ) fixed and inclined near to 45° to vertical, and a second ( 410 ), near vertical mirror attached to the platform  20 . Lateral displacement of the calibration device  10  can be measured by interferometry on the vertical mirror  410 , using an optical path from above, by reflection from the 45° mirror  400  before and after the vertical one. The mirrors  400 ,  410  preferably have a highly reflective gold surface, and have microfabricated hinges; initially fabricated flat on the surface and lifted into position using a micromanipulator, or one of the various MEMS techniques for erecting optical component structures. One of the advantages of this surface micromachined polysilicon calibration device is that it can be made using the same process used to fabricate a normal force calibration device, as is described in commonly assigned co-pending patent application No. PCT/GB200/002134. This should allow the fabrication of both normal and lateral force calibration devices on the same die.