GB2401945A - Atomic force microscope (AFM) cantilever calibration - Google Patents

Abstract

A micro-machined capacitive atomic force microscope (AFM) cantilever calibration device has a platform for landing the AFM cantilever tip and one or more supporting legs to provide sprung resistance to the platform. A capacitive sensor measures the combined spring constant of the one or more supporting legs with respect to displacement perpendicular to the platform surface. A method of determining the spring constant of a structure (such as the landing platform of the calibration device) is also disclosed, along with an AFM cantilever having a capacitive sensor to determine the spring constant of the cantilever.

Description

Calibration Device

Field of the Invention

The present invention relates to a calibration device sutab]e for calibrating small force rneasurir.g devices. In particular, the present,. vention relates to a calibration device m which accurate measurements wider, and traceable to, the SI system can be obtained.

Background to the Invention

Measurements of small forces, in the nanonewton and pconewton 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 sma]1 forces accurately, for example, protein-protem interactions or materials properties via the small force applied to an indenting tip.

Accuracy is rarely mentioned for AFM force measurements. AFMs measure displacement accurately, and are calibrated quite easily using stepheight standards.

Some AFM instruments even incorporate laser nterferometry to make traceable height measurements.

The quantification of Attraction forces is much more prob]ematc. 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 Smiting factor in the accuracy of a force measurement. Many methods have been proposed for calibrating the stuffiness of an AFM probe, but noise are traceable, and typical accuracy Is only about 20-30%.

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 m direct experiments to break individual covalent bonds by AFM, or the measurement of protein ulteracton forces.

Commercial reference artifacts are available, but offer no traceability to the ST neasurcment system. This is important because tlerc 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 word in botl1 areas will be difficult to bul]d-upom What is more, a traceable calibration method is now timely. If we wait, careful but untraceable measurements Will appear which Will not receive the citations they deserve.

The reason is that the thickness of AFM cantilevers is difficl]t to control to the tolerance required to give them repeatable spring constants. The force constant for cantilevers from two different batches can vary by almost a factor of ten, because force constant is proportional to the cube of cantilever thcluless, and the equipment for making them comes from the microelectronics industry where Si film thickness to I this exqusito level is not a priority. This situation seems set to remain for the foreseeable future, mcamng that an easy and accurate method of calibrating cantilevers would be very useful.

Tn addition, tip coatings can have a significant effect on spring constant that is difficult to model or predict.

A number of different methods have been suggested for the calibration of AFM cantilcvcrs. Particularly effective is the use of a reference cantilever. Those methods that depend on measurement of the resonant frequency (or frequencies) of the cantilever are also attractive, because they are simple and do not risk damage to the cant]ever. Unfortunately these resonance methods are problematic, for the following reasons; À Resonant frequencies give us only the ratio of the spring constant to an ulertal teen. The inertial team is often Occult to estimate because the mass of the cantilever Is dstrbuted making the theory complex, especially due to the non-trivial shapes of AFM cantilevers designed for increased imagmg performance.

À Methods that rely on calibrating the cantilever by comparing the thermal vibration of the cantilever at a known temperature with the predictions of statistical themlodynainics are attractive due to the apparent experimental simplicity of this measurement, but are also problematic. It is often unclear, for example, how much OT the power ire tine supposedly "thermal" vibration Is m fact acoustic or mecllancal noise from the environment.

This can lead to the spring constant of cantilevers being systematically underestimated. Mechanical and acoustic noise must be excluded much more rigorously than is required for normal AFM operation, which spoils the apparent simplicity for the user.

Table I lists a number of known methods for Detenninaton of AFM Spring Constant, their respective accuracy and problems: __ __.

Class of 1\ lethod Accuracy Demerits Method

_

Resonance 1 0% Positioning and calibration frequency with of load difficult; potentially added mass destructive Dynamic Thermal 10- 20% Temperature control response fluctuations essential; only suitable for methods soft levers; requires analysis of resonance curve Smple scaling 5-10% Depends on dimensional from resonance accuracy and determmaton _ frequency of eifectve mass Fmite difference 10%+ Depends on dimensional calculation accuracy and Young's Theoretical modulus methods Parallel beam 10%+ Depends on dimensional I I approximation I accuracy and Young's l | modulus Static deflection 15% Positioning and with added mass determination of load dfficult; potentially Static destructive response Response to 30-40% Compiox and tiine metl1ods pendulum force _ consuming procedure Static deflection 15-40% Requires accurate external with external standard standard _ Reference cantilevers are commercially available. Flo\vever, these are small and difficult to land the AFM tip on. One needs a separate optical microscope to measure the distance frown the base of the reference cantilever to tl1e tip - perhaps half of all AFM instruments do not have this. Even if they do, the accuracy with which this distance can be measured is not sufficient, because the spring constant of the reference cantilever depends on the cube of this distance: If one measures it to 3% uncertainty, the uncertainty in the calibration of the AFM cantilever is around] 0%.

I0 In the]980's and 1990's a method of comparing mass standards to the SI system via e]ectncal units, known as a Watt balance, was developed. Dimensional uncertainties are cancelled through the connation of static and dynamic experiments.

The apparatus used in Watt balance is non1lally a moving-col inductive device. These ] 5 Watt balances are sophisticated devices designed to achieve accuracies of around one part m 108, for forces In the r egion of 1- I (IN, a very demanding requirement requiring metrological work of the highest order.

The essence of the Watt balance concept Is to realse a known force In terms of traceable measurements of electrical quantities and linear displacement and velocity so that the physical mass defining, the kilogram coward be replaced with a repeatable eXpenmeilt fiOill W]lIC]1 the welt could be accurately generated. A Watt Balance apparatus Is used at the National Physical Laboratory in the United Kingdom. In particular, the apparatus is used for comparing masses of around Ikg. A solenoid is used to apply a force by induction, and the current through the solenoid is measured.

The solenoid is then moved, at a measured velocity, through a magnetic field of measured strength, and the voltage generated in the solenoid is measured. Combming these two sets of measurements - for the static and dynamic parts of the experiment - allows one so eliminate The uncertainty due to the pc,orly-Llovvn size of the solenoid (and even the partrcu]ar path within conductors taken by currents flowing through the solenoid).

An electrical method of generating small forces would be much more practical than methods dependent on microscopic deadweights. However, to date no known device has been produced to the required dimensional accuracy suitable for use in this field.

Summary of the Invention

Accordmg to one aspect of the present invention, there is provided a calibration device comprising a platfonn1 suitable for the landing of an AFM cantilever tip, one or more supporting legs ananged to provide sprung resistance to the landing of an AFM cantilever tip and a capacitive sensor for measurrmg the combined spring constant (with respect to vertical displacement) of the one or more supporting legs.

The capacitive sensor may optionally also be used as an actuator, capable of setting 2() the device into resonance.

In addition, the device may include a piezoelectric or other vibrator capable of setting the device Into resonance.

A capacitive analogue of the "Watt Balance" method Is used to produce a reference spring for the calibration of AFM cantilevers. The need to find a way of eliminating uncertainties due to the limited manufacturing tolerance of the actuator Is even more important here than it Is in the case of the large inductive version; though microfabricated devices are nnade with excellent dimensional tolerances in absolute teens, in fractional temls these are typically worse than for nacroscale devices.

The device we have demonstrated has two "folded-beam" legs, and two interdgital comb drive capacitive actuators. Versions having three legs may be more mecllamcally stable and therefore easier for the AFM user to use, and are under development. In addition, more than two comb drives can also be used. The device is fabricated on a silicon die by successive deposition of layers of polyerystalline silicon and siDeon dioxide by chemical vapour deposition (CVD). The silicon dioxide Is removed m a subsequent I-IF etch, leaving the released polycrystalline silicon structure. This process Is commonly called surface mcromacllnng m silicon.

The present invention seeks to pro-v-ice a calibration device in which accurate measurements traceable to the ST measurement system can be made.

Methods of weighing masses of around one kilogram using an electromagnet are known. These methods use large instruments aimed at weighing masses of around ]0 100g to lkg as a primary standard of mass to replace the international prototype kilogram and are not suitable for use m AFM or with small forces. l:lowever, these methods are accurate and traceable to the SI system.

According to another aspect of the present invention, there Is provided a method of ] 5 determining a spring constant of an AFM cantilever comprising determining deflection of the cantilever when pressed against a solid surface, determmng deflection of the cantilever when pressed against a ca] braton device having a predetermined spring constant and calculating the spring constant of the AFM cantilever from tile raho of the two deflections multiplied by the predetermined spring constant.

The method may Include the step of detennmng the predetermined spring constant using a Watt balance on board the calibration device.

Combining static and dynamic measurements of the resonance of the calibration device including a Watt balance it Is possible to measure the spring-constant of the legs that support it. This device can then be used as a reference spring to calibrate AFM cantilevers.

Particular,dvanta;e.s of the ealibrahon device according to the present invention Include higher precision in spring constant caLbraton than Is possible with existing reference cantilevers, precision Increasing by up to a factor of about ten and accurate traceability to the SI system.

The device allows AFM cantilevers to be cahbrated quite easily, to an uncertainty of +5% at one standard deviation. A simple substitution of the analogue velocimeter used in this work with a digital model should reduce this uncertainty to around +2%.

Both are significant Improvements on current practice, and allow traceability to the Sl system. It is envisaged that the method of calibrating the spring constant described here, rather than be used to produce a calibrated reference spring against which AFM cant]evers may be calibrated, will ultn1ate]y be "bult-in" to AFM cantilevers themselves (for example by adding comb drives at each side of the cantilever). An entire wafer of microfabricated AFM cantlevcrs, perhaps containing hundreds of cantilevers, could be calibrated using our non-contact method (combining electrical and n1terferometric measurements) quickly and easily by the manufacturer.

The calibration devices according to the present invention are very robust with respect to mechanical shock and vibration because of their exceptionally sma]] criteria. Such ]5 devices could, for example, be sent through the post without any problems. By comparison, if a large device was to be manufactured (perhaps greater than a cubic centm1etrc no volume) for applying smal] forces to an AI;M cantilever, the large mass of this device would require better vibration isolation than required for general AFM use, and therefore would not appeal to the AFM practitioner as a practical method of spring constant calibration. However, the microfabrcated device of the present mventon is just as fragile as AFM cantilevers themselves when it comes to handling - both would be destroyed if accidentally touched. In addition, the Watt Balance device should be protected from the ingress of dust patic]es under the reference springs. Protection Is preferably achieved by covering the Watt Balance chip with a glass cover slip when not in use.

A number of the calibration devices with differing spring constants may be provided on an array to allow calibration of a wide range of devices.

lo, id Desc. itior. of the Dra'!'g' Examples of the present invention will now be described In detail, by way of example only, with reference to the accompanying drawings in which: Figure I is a three-dimensional computer model of a calibration device according to an embodiment of the present invention; Figure 2 is a cross section taken diagonally across the cahbraton device of Figure I Figure 3 is a schematic diagram Illustrating aspects of the device of Figure 1 when in operation; Figrure 4 illustrates the use of the calibration device of Figure 1; Figure S illustrates a system used to obtain more accurate calibration measurements; Figure 6 is a graph showing Vertical displacement of microfabricated Watt Balance platfonn as a function of potential apphed to the fixed comb fingers. The platform Is at earth potential; Figure 7 is an optical monograph of talc Watt balance actuator; Figure 8 Is a graph plotting peak-to-peak velocity (as measured by the Doppler method) and current through the device (including parasitic capacitances) in the vicinity of the mechanical resonance of the Watt Balance; Figure 9 Is a graph plotting ratio S against frequency in the vicinity of the mechanical resonance of the Watt Balance; Figure 10 is a growls showing tl1e data plotted n1 Fig 8, smoothed over a 20Hz interval; and, Figure 11 Illustrates a method of calibrating an AFM cantilever using a calibration device according to the present Invention.

Detailed Description

l;igure I Is a three-dmensonal computer mode] of a calibration device according to an embodiment of the present invention. The area shown Is 980 by 560 microns.

Dimensions perpendicular to the plane have been expanded by a factor of 20 for clarity.

In this embodiment, the Watt balance principle Is applied n1 an entirely different context. The caLbraton device Includes a mcrofabricated capacitive Watt balance for 3n use in AFT sprinb-constant calibration.

Figure 2 is a cross section taken diagonally across the caLbratoil device of Figure I hc measurement of the spnng-coilstant of these two legs represents the calibration required.

In tlis embodiment, the substrate 10 is a 250 microns thick So layer. There is then a silicon nitride layer 20 about 0.5 microns thick followed by a layer 30 of hg}lly-doped (and therefore conductive) polycrystalDne silicon. Comb drives 40 (one of which is illustrated in l;igure 2, although there could be any number) are also fowled from the highly doped polycrystalline silicon, as is the calibration device 50. The calibration device 50 includes a plurality of flexible legs 55, of which two are shown. Finally, in a preferred embodiment, right at the top of the device is a r ectangular gold mirror 60.

Figure 3 is a schematic diagram illustrating aspects of the device of Figure] when in operation. The AFM Landing Stage is "levitated" due to asymmetry in the electric field surrounding the interdigital electrodes due to the earthed, doped po]ysilicon groundp]ane. Field dines are shown continuous Ones, while isopotentia]s are shown as broken lmes.

Figure 4 illustrates a calbratoil method using the above described device. In steps I and TI, static and dynamic measurements of the displacement of a moveab]e capacitor plate (the comb drive(s)), together with electrical measurements, a]]ow the spring constant of the spring supporting that moveable plate to be measured, potentia]]y traceable to the SI. In step III this spring Is then used as a reference spring within the AFM to calibrate the spring constant of the cantilever under test, without further electrical or other measurement.

The device is essenta]]y a capacitor with one fixed electrode and one moveab]e electrode. The moveable e]eckode is suspended on a spring having a spring constant similar to that of the AEM cantilever to he caDhrated. The calibration comprises three steps. Steps I and II require special electrical and interferometrc measurements, and will typically be performed In a ca]bration. The results of steps I and II give the spring constant of the spring supporting the moveable electrode. The entire device is then earthed, and sent to the AFM user. To the AFIRE user this is simply a reference spring, and the static deflection of the AFM cant] ever under test is then used to measure the spring constant of that cantilever.

An important point to make at this stage is that the method involves no physical contact - the calibration of the reference spring requires only electrical and mterferometrc measurements.

An applied voltage leads to an increased separation between substrate and the AFM landing-stage. A simple parallel-plate capacitor could be used, but suffers the danger of the plates bemg attracted and sticking together.

The substrate groundplane under the device is always earthed, and is connected directly to the moving part of the device via the two legs. Therefore the comb drive fingers that move up and down are always at 0V. As we apply a voltage to the fixed fingers the field around them is not symmetrical above and below them - because of the earthed groundplane. This asymmetry means that the movable fingers see more of the field above them than below them, and are attracted upwards.

The comb drives are only used during calibration of the calibration device itself and tile AFM user, UpOil purchasing; the calibration device does no electrical measurement or even any electrical connection To the user, the device is just a platform which, when pressed with an AFM tip, responds with a known force per unit distance of downward displacement - he. a reference spring.

The Watt Balance in electrical terms is a two-terminal device: the fixed outer digits of the comb-drives are at a fixed potential Vp, while curTent to earth is measured from the structure foamed by the movable frame and fixed groundplane under it, which are In electrical contact During calibration of the calibration device, the comb doves are used as follows.

Firstly, a small AC potential applied to them at the resonant frequency of the Watt Balance (for example, 4.2kHz) sets the Watt Balance into resonance. Small AC voltages are used giving a vibration amplitude of about 70nm. As well as being used as an actuator m this way (and in fact one can use an external piezo shaker If one does not want to use the comb drives as actuators) we use the comb drives to sense the gradient of capacitance of the Watt Balance, by placing a DC voltage (Vp in Fig 3) on the fixed comb fingers and monitoring the current to earth *om the moving fingers as the capacitance changes in step with the displacement of the moving fingers. Together with a separate measurement of static displacement as a function of voltage, this capacitance gradient gives us tile force required to displace the platform upwards by a known distance and hence the spring constant of the legs supporting it.

Although the comb drive is geometrically complex, it is just a twoterminal capacitor e]ectrica]]y. One particular advantage lies m the fact that it separates when a voltage Is applied, whereas a paraile]-plate capacitor (above a certain "pull in" voltage) snaps ]0 together destructively. Given the layer-wise constraints of MEMS fabrication, the comb drive method described above Is one of the few structures robust to this "pu]]- in" effect, at least for moderate voltages.

Preferably, the device incorporates a nin-or on the AFM landing-stage to simplify measurement of vertical displacement and velocity by optical nterfcrometry and ] Doppler velocinetry respectively. In the present embodiment, the device was fabricated using the three-layer polysflcon surface micronacllning MUMPs (Mu]t User MEMS) process.

Prior to manufacture, the spring constant of the calibration device, Of is selected to be approxinate]y equal to the spring constants of the AFM cantilevers to be cahlrated, since this leads to the greatest accuracy when comparing the spring constant of the AFM cantilever with that of a reference. After manufacture the exact value of this spring constant Is measured by a combination of electrca] measurements and mterferometry (preferably Doppicr interferometry) using the Watt Balance method we describe here. This ca]ibraton wi]] typically be carried-out by a speciahsed calibration department or laboratory. The ca]brated device can then be sent out to the AFM user, typically with a calibration certificate stating the value of the spring constant. Often it will] be useful to distribute a number of such devices together on the same chip, covering a range of different spring constants. The caiibraion device Is then used by the AFM practitioner as a reference surface for AFM spring constant cahbration The cantilever of the AFM is apphed against a hard surface such as tile neghbourng area of the chip holdmg the calibration device consisting of silicon 0.4mm thick. Since this surface Is rigid, the increased deflection of the tip dunug acquisition of a force-distance curve is equa] to the downward displacement of the cantilever by the tube scanner of the AFM.

The cantilever is then applied against the reference surface of the cahbratioi1 device.

The spring constant of the AFM cantilever can be calculated from the ratio of the two displacement slopes and the known value of the calibrated reference, he; . As the spring constant of the calibration device (kref) is traceable to realsatons of Sl electrical quantities, the spring constant of the AFM cantilever can also be determined to the same level oftraceablty.

Detcnnination of spring constant of the calibration device Ellis is the calibration step typically carried out In a specalsed laboratory or calibration facility. As with the macroscopic Watt Balance, there are static and ] 5 dynamic measurements to be made. Both nest be performed in vacuum to avoid air- damping (the dynamic measurement operates at low Reynolds number) and to avoid attracting dust particles to critical parts of the electrostatic comb-drive.

The most Important quantity to be measured Is the gradient of capacitance as the landng-stage Is displaced. If we know tle gradient of capacitance, we can calculate the force on the comb-drives, and therefore the balancing mechanical force exerted by the supporting folded springs. We can measure the displacement, and so we can calculate the ratio of applied force to displacement - he. the spring constant. This static measurement is relatively straightforward, but we still need to measure the gradient of capacitance, since for all but very special geometries (such as that of the Tho!llpson-Lanlpard capacitor) it cannot be calculated from the geometry of the device with sufficient accuracy. Indeed the constraints and relatively poor fractional dimensional accuracy of surface micromachning make this calculation more difficult and inaccurate than for many conceivable macroscopic versions. One could measure ti1C capacitance gradient in two ways; À Using a sensitive capacitance bridge to make direct measurements of the device capacitance at a number of static displacements. This would give very precise capacitance values, Including a constant stray capacitance originating from the fixed parts of the device. To obtain the gradient of capacitance one would need to differentiate with respect to the measured displacement, mcreasmg tile uncertainty budget, but capacitance bridges have ilOW reached such a high level of precision that this approach is viable, even for the very small capacitance of a surface mieromaehned comb drive.

À Alternatively, one can "dither" the displacement of the landing-stage, either mechanically (e.g. using a small piezo actuator under it), by superposing a very small ate. drive on the d.e. potential applied to achieve a particular static displacement. The mechanical vbraion causes a time variation in eapaetanee leading to a measurable ate. current. By simultaneously measuring the velocity of the landing-stage one can calculate the gradient of eapaetanee required.

The second of these two methods corresponds to the Watt Balance approaell. We chose this approach because (a) one can take advantage of the sharp mechanical resonance of MEMS devices to retake the "dither" procedure distinguish very clearly between tile nuisance of stray electrical capacitance and the important displacement related capacitance gradient, and (b) capacitance bridges of sufficient sensitivity also capable of dealing with a range of d.c. bias were not commercially available.

In the present embodiment, displacement and velocity measurements are made usmg an instrument that has not been calibrated traceably, but comes from a class of Doppler veloeimeter that can be made traceable: Doppler vibrometers using digital demodulation are accepted for traceable primary velocity calibrations aeeordmg to the ISO 16063: Methods for the calibration of vibration and shock transducers -- Part l 1: Primary vibration calibration by laser nterferometry. It will however be appreciated that other instruments could be used.

Pliers are statue and dynamic measurements to be made. Both are preferably 30;,erurineu hi vacuum to avoid ar-dampmg (the dynamic measurement operates at low Reynolds number) and to avoid attracting dust particles to critical parts of tile electrostatic comb-drive.

1. Static measurement (shown schematically In Fig. 4, step 1). This consists of measuring the static displacement of the AFM landng-stage as a function of applied voltage.

2. Dynamic measurement (shown schematically in Fig 4, Step II). This consists of measuring the current to earth passing through tle device, while sunultaneous]y measuring its vibration velocity using Doppler velocimetry.

The extremely sharp resonance of the Watt Balance platform, when operating in vacuum, allows us to separate the change In capacitance of the device due to mechanical displacement from the inevitable parasitic capacitances elsewhere in the circuit.

By applying a DC voltage across the comb drives of the device, and setting it Into resonance in its fundamental mode (either by adding a small AC component, or external mechanical shaking), one can simultaneously measure the velocity amplitude of vibration (typically <I mm/s) and the electrical current through the device (typically cl 00pA) . Together these allow one to detente the grradient of the capacitance of the device at this DC bias.

Static measurement ?0 This consists of measuring the static displacementof the AFM landng- stage as a function of DC bias vo]tagc. We measured this static displacement using white-lght interferometry using a Zygo NcwView 5020 Interferometer. This measurement allows us to balance the electrostatic force on the comb drives (which depends on the capacitance gradient measured in the Dynamic Experiment, and which would otherwise not be known to sufficient accuracy) with the mechanical force arising from a measured displacement of the legs. This allows the spring-constant of the folded sprang legs to be found.

The static deflection of the platform Is the result of the balance between the elastic restoring force applied by the folded sp^ngs and the electrostatic foi-ce from the comb-drives. The stored electrostatic field energy is E =-CVp ]4 where C is the capacitance, and Vp is the potential difference across it. The electrostatic force, :/c, Is 1 bC F. =--V whch balances an elastic force, FCa;c, of; F. - kz elastic where z is the static deflection, so that If we plot S against Irequency in the region of the resonance, where

-

-

- thlmlt,, 17>ppe/. is determined m the dynan1c expcrin1ent discussed below, and Is the average of the current amplitude far bc]ow and far above the resonance (typically between 5 and 10 times the full-width at half maximum height of the resonance). We expect a sigmodal curve of step height 2k, where k is the spring constant we English to measure.

Figure 6 is a graph showing Vertical displacement of microfabncated Watt Balance p]atfonn as a function of potential applied to the fixed comb fingers. The platoon is at earth potential.

D,vnamlc measurement This consists of measuring the current to earth passing through the device, wh]e snnultaneous]y measuring its vibration velocity using Doppler ve]ocimetry. The extremely shard resonance of the Watt Balance platform, wYcn operating In vacuum, allows us to separate the change in capacitance of the device due to mechanical displacement from the nevitab]e parasitic capacitances elsewhere n1 the circuit The cunent through the Watt Balance and the height of the mirror were recorded simu]tancously and averaged to reduce noise using a Hew]ett- Packard 3562A Dynamic Signa] Ana]yser. These data were downloaded Tom to a PC computer.

Current through the device was measure using a CyberAmp 320 Signal conditioner with type 403 preamplifier (Axon Instruments Inc.). By using it in "vrtual-earth" configuration, any parasitic capacitance across the input of the amplifier (or between tl-le Solving part OT the actuator and the die substrate) connects virtual earth to earth, so its influence on the circuit operation is insignificant. In addition, the signal path from the Watt Balance actuator was carefully surrounded on the printed circuit board (PCB) by an earthed "guard" track, to minimize the effect of Pinball stray currents across the bare PCB surface, for example due to any small surface contamination by l O electrolytes.

The Watt Balance devices tend to resonate at around 4.2kiTz, with a Full Width at Half Maximum (FWIlM) of around 7Hz at a background pressure of 2. 3Pa. This corresponds to a quality factor for this resonance of Q 600. Since the device has a large cross-sectional area in the direction of displacement, this quality factor is rapidly reduced by air-damping at higher pressures. Therefore the ca]braton of the Watt balance springs must be performed In vacuum. Of course, the subsequent use of these springs to calibrate AFM cantilever, performed by the AFM user, will typically be in air or Squid.

The current through the device at resonance Is equal to the rate of change of the product of its capacitance and the voltage across it.

it) d (CVp) dt We consider also "stray" or "parasitic" capacitances, which we will call Cpar, . Figure 5 Illustrates a system used to obtain accurate calibration measurements.

Separating the capacitance of the device Into two parts: a. The dynamic capacitance, C(z), which changes as the platform is displaced, and b. The static or parasitic part, Cpnn. This Is the capacitance between fixed parts of the device, for example adjacent tracks and pads on the silicon die. s

Measunng the response of the device over a narrow frequency interval around th mechanical resonance, the static capacitance is expected to be constant, but the dynamic capacitance Will vary with the moron of the platfonn.

i(t)=[C(z)+C:,aa]-cat-+Vp (t) oz dt Applying a d.c. potential of V, to the stationary part of the comb drives, together with a small a c. component aft), so that 1 5 Up (t) = To + N1(t) The purpose of the small a.c. component is to apply a small drive to the device, WhiC}I, if this drive voltage is close to its mechanical resonant frequency, will cause it to vibrate mechanically with significant amplitude. Typically 4bo Is chosen In the range I to 4V, and lilt) Is a sinusoid of amplitude and vO chosen In the range 25OIlV to 2.5mV v(t) = v;'sln(rot) Now the velocity of the platform can be measured by Doppler velocimetry. At each Instant we have a measurement of the velocity V(t) = V, cos('t + O) of the platform.

For a gavels amplitude of a.c. drive, both the amplitude loo and phase with respect to that dove (-/2) vary as the drive frequency passes through resonance. We Identify the Doppler velocity with the velocity (dz I aft) that appears above to give 30](t) = [C(Z) + Cpara d( ) + 0 + too so)] (TV) For a particular bias voltage (0, and an a.c. component amplitude JO sufficiently small that the capacitance C(z) varies linearly over the range of mechanical vibration, we obtain, i(t) = [C(z) + Cpara 1VOO cost) + ó0: V(t) The first term on the right hand side of the above equation represents a parasitic capacitive current that is constant in amplitude for frequencies near the mechanca] resonance, and (;T/2)radans In advance of the a.c. drive signa] The second tend Is the Interesting one, because it Is proportional to the capacitance gradient we wish to measure. This term has the same phase as the ve]octy of the mirror p]atfom (and comb doves). At low frequencies iDe mirror displacement is in phase with the drive signal, whereas far above the resonance it lags by Radians. Therefore the ve]octy is (A / 2) radians n1 advance of the a.c. drive voltage far below the resonance, ) [C(Z) + Cpara, COS(6lJt) + To Vo COS(.9t), for << ' and lags by / 2 radians far above it, i(t) = [C(Z) + Cpoa jock cts(6ot)-ó0 Vn cos(wt) for >> wr Liz wl:ere 6 r = 2;, is the angular frequency of the mechanical resonance. The sharp mechanical resonance allows us to measure the magnitude of the second as this phase change occurs, since the first term Is essentially constant over this narrow frequency intei-val.

As the drying frequency Increases through the resonant frequency the two teens in the equation above are at first u1-phase, and finally out of phase, leading to the step observed m the quantity S. The magnitude current change gives us the grad ent In capacitance; PC(-) which as discussed above, is all we need to obtam a value for the spring constant of the device using the static experiment results. This is neatly summarized ',n tl e plot of the quantity S. though more sophisticated fitting to the curves shown in Fig 7 (possibly also incorporating phase information) are likely to be ultimately more accurate.

Figure 6 shows Zygo white-light.nterfe.-ometry measurements of vertical displacement of the Watt Balance platform as a function, of the potential applied to the fixed corny fingers. This was performed in vacuum, residual pressure being measured as 2.3Pa.

A complete analysis involved fitting data to an electrical equivalent circuit model, for which a signal analyzer with four channels would be necessary to achieve the most accurate results. However, since the resonance of this MEMS structure is so sharp, compared to the slow variation in parasitic capacitance with frequency, the transition from 'mphase" to "anti-phase" addition of current offers us a good way to gain insight into the measurement of the spring-constant. First, defining an expression for the spring constant k by ba]ancmg electrostatic and elastic forces on the platform, k = fo2 ac where z Is the time-average of the displacement z. An expression for the gradient of capacitance (5C/57) gves Art I'll- i,| la'>:'', , <, LIZ2ó'oVo where the current amplitudes that appear at the top right hand side of the equation represent the measured cun-ent amplitude above and below resonance respectively.

Using this to substitute for the capacitance gradient appearing in the previous equation we obtain, k = 4 ó V (id |J>>ir-iO la'<'r) A plot of a quantity as a function of frequency from this equation shows that the spring constant emerges naturally. We define the quantity S. where I O So) = ó(' [ ('(do) 'to] 4 7 VO (if) ) and As the average con ent amplitude far from resonance (we used the average of the current measured 90 Hz above the resonance and 90 Hz below it, in each case averaging over an interval of 10 Hz centred on + 90 liz). The function S has no special physical interpretation, except that when plotted as a function of frequency n the vicinity of the resonance it should exhibit a step equal to the spring-constant of the device. If we plot Sagamst frequency in the region of the resonance, we should expect a sgmodal curve of step height k, where k is the spring constant we wish to measure. Figure 9 shows this data plotted for tile current and velocity measurements of Fig. S. Tllere IS a good deal of residual noise that could be Improved by longer acquisition times than the five seconds this scan took. A 20 Hz running average smooth improves the plot considerably, as shown in Fig. 10. This gives us a spring- constant of 0.193+0.01 N/m, In reasonable agreement with an earlier, more approximate value of 0.23+0.03 N/m based on an estimate of the mass and resonant frequency of the vibrating part of the device.

Modes of Vibration The microfahncated Watt Balance hats a small capacitance; which can be implicitly measured by montorng the current through the device as the comb drives move with respect to each other. This current is small, typically less than lOOpA, and therefore a number of precautions should be taken to make an accurate measurement. The mechanical resonance of the device assists us, by ensuring that (given the large quality factors we have measured in vacuum for this device) we can be confident that on-resoilance the amplitude of vibration of other parts of the system are small compared to the amplitude of vibration of the device itself.

The Watt Balance actuator has many vibrational modes. To ensure that we attribute the correct mode of vibration to each of its resonant frequencies, measurements were made of the phase of vcca] motion at a number of different pouts on the device.

Because the device was vibrated vertically we do not see lateral modes.

Mode Frequency/kHz

__

Fundamenta] 4.25 Higher mode 15 Note that in both of these vibrational modes the comb-drive displacement has the same phase throughout the length of the drive. Therefore both represent the same kind of vertical displacement as will occur when a normal force Is applied to the center of the device by an AFM tip Therefore, while n1 principle only one such mode would be necessary, n1 fact both modes can be used to determine the capacitance of the device as a function of the vertical dsplaccment of the comb-drives.

Figure 7 Is an optical mcrograph of the Watt balance actuator. Two combdrves at the top and bottom of the picture apply "levitation" mode forces, leading to a displacement out of the plane of the photo. A folded spring mechamsm provides a spring constant comparable to those of the AFM cantilevers to be calibrated. The central gold mirror can be seen clearly.

Figure 8 Is a graph plotting ratio S against frequency in the vcmty of the mechanical resonance of the Watt Balance. The continuous curve is an average over 20Hz.

Figure 9 is a graph plotting ratio S against frequency h1 tile variety of the mechanical resonance of the Watt Balance. The continuous curve is an average over 20Hz. . The step In this function gives the spring-constant of the device.

Figure 10 is a graph showing the data plotted In Fig 9, smoothed offer a 20Hz intervah The spring constant of the device is half the difference in S as one passes through the resonance.

The structural material used for the resonator is chemical vapourdeposited po]yCrysta]lH?e silicon. It is heavily doped to give a high conductivity, but it Is conceivable that charges trapped close to its surface, perhaps at defects or grain boundaries, can add to the measured current during mechanical resonance. This would be analogous to the operation of the electret microphone, where a much larger charge Oi? a vbratng membrane gives rise to a very easily rmeasurable potential.

To check for the presence of trapped charges we reversed the polarity of the potential applied to the fixed section of each comb drive. As before, the moveable parts of the device are earthed. If significant trapped charges are present, their polarity wild, of course, remain the same If so we should observe a significant In the magnitude, not Just the sign, of the measured current in the vicinity of the mecl?ancal resonant peak n1 the frequency spectrum.

After ibe Watt Balance springs are calibrated using the above method, they can be distributed to AFM users. AFM users need not make any electrical measurements, ! but simply use these devices as calibrated reference springs. One method of calibrating AFM cantilevers using such reference springs Is shown in Fig 11 and is i also illustrated n1 step III of Figure 4. In figuec 11, a single force-distance curve shows three distinct SCCtOilS. 'Lee spring constant of tile cantilever is simply the measured spnng-constant of the Watt Balance multphed by the ratio of the slopes of sections Tl I and II of the force-dstance curve.

The neasurement of the spring constant of an AFM cantilever, kc, is determined by comparison v;lth the Watt balance spring coistailt, k. kC earl be found from iDe ratio of the slopes of the force-distance curve in Regions II and 111, as follows, rAvI]-R kc = k A V " - - I Az 11 where VA B is a potential drffercnce represclltlilg the "A-B" signal from the four quadrant detector of an AFM, and AVIS B. AVE[A are increments in the curves in regions II and 111 corresponding to displacement increments of AZ" -B and; AZI1/A-R In the heg]lt of the peso stage, respectively. The ratios (A VHA-B / AZ" ) and (6 V H/ 4_ / If /H) are simply the slopes of the curve in Region II (where the tip is in '' contact with the movable platform) and Region III (where the platform Is also in contact with the substrate) respectively. The spring constant of the cantilever is simply the calibrated spring constant of the reference spring multp]ed by the ratio of the slope of the force-distance curve In section III to that in section 11, minus unity.

A]thoug]1 t]:e lateral r evolution of AFM is around I ()nm, it is typically very difficult to approach a target smaller than around 301lm on a surface. This Is due to the mechanics of the gross-approach mechanism of most AFM designs, relying on a stepper motor and a screw thread to lower the cantilever and tube scanner. This typically has some residual eccentricity blat makes precise positioning of the tip prior to contact rather! difficult. The Watt Balance device described above has a "landing area" of 80x 1 Em. This is sufficiently large for the AFM user to approach without difficulty. i Conversely, this means that the Watt Balance device cannot be made very much i smaller than these dimensions without making it significantly more difficult to use by the AFM practitioner. It will always need to be a micro rather than a nano device.

Instead of using a Watt balance, one could, in principle, build a small parallel-plate capacitor, apply a known voltage to it, and obtain a sma]] divorce traceable (via the c]ectrca] units) to the SI Newton. One plate of the capacitor would be fixed, and the opposite plate suspended above it on flexible springs having similar spring constant to those of the AFM cantr]evers one wishes to calibrate. The dynamic part of the operation of else device as a Watt balance would require d.c. voltages to be applied to the capacitor causing the plates to approach. This must be contro]]ed carefully, to avoid "snap-on", where the movable plate comes Into contact with the fixed plate, Van der N7vraals forces then making it almost impossible to separate them, effectively ending the '.soful kite of the device. However the geometrical precision with which one can make such a small capacitor Is limited, and this causes uncertainty in the forces generated by it. In addition, the Watt balance is hiLereiitly inore stable and easier to control than a simple parallel-plate capacitor.

One could measure the gradient of capacitance using a capacitance budge, eliminating the dynamic step.

JO

In the future it Will be possible to inakc AFM cantilevers with this system "built-in", so that they can be manufactured Oil the scale of a silicon wafer, calibrated in tun1 and then sold. This is more convenient than having a separate calibration device. All AFM cantilevers need a partially reflective platform to operate the optical lever system. Thus, such a platform (or some other suitable part of the AFM structure) could be used In combination with one or more comb drives as descrbcd previously to provide an in-bult calibration system.

A number- of MEMS cantilever sensors are in development around the world that may, conceivably, require calibration and the calibration device described would be suitable for such a task.

The calibration dcvce could be made by conventional machining or as a MEMS device. MEMS design has severe restrictions imposed by the layerwise]thographic processes commonly used. Convcntioi1al macl1innig is certainly capable of producing to lOmm-scale devices that can apply nanonewton forces, but they will also be susceptible to vibration due to large r atio of their weight to the forces they apply. Of course, careful vibrator isolation can help a great deal, but often the result Is a compioinise m -wl^ich soine other aspect of performance Is lost. What Is moire, the AFM user may well ask Why they need better vibration Isolation for cantilever calibration than they need for the AFM in normal use.

Tn contrast, MEMS devices can be made extremely snial], with very sma]] mass and much Hoer sensitivity to vibration.

Surface micromachning may also be used.

Electrostatic comb drives are typically the method of choice, and one can easily imagine an electrostatic Watt balance using such a comb drive. This would be suitable, for example, for the calibration of cantilevers in Lateral Force Microscopy.

However we need a force perpendicular to the surface for the calibration of AFM spring constant.

Claims (1)

1. Claims 1. A calibration device comprising a platform having a

   substantially planar surface suitable for the]andrug of an AFM cantilever tip, one or more supporting legs ananged to provide sprung resistance to the platform and a capacitive sensor for measuring the combined spring constant of the one or more supporting legs with respect to displacement substantially perpendicular to said substantially planar surface.

   2. A device according to claim 1, wherein the capacitive sensor also comprises an actuator for setting the device Into resonance.

   3. A device according to claim I or 2, further comprising a vibrator for setting the device Into resonance.

   4. A device according to claim 3, wherein the vibrator Is pezoelectric.

   S. A device according to any preceding claim, wherein the supporting legs comprise folded-beams.

   JO

   6. A device according to any of the preceding claims, wherein the capacitive sensor Includes one or more nterdgital comb drive capacitive actuators.

   7. A device according to any one of the preceding claims, wherein the capacitive sensor comprises a Watt Balance device.

   8. A method of detenninng the spring constant of a calibration device according to any of the preceding claims composing: a) applying a predetermined vibration to the device and simultaneously measuring the velocity of the platform; b) calculating the gradient of capacitance of the device in dependence on the measureci velocity; c) applying a predetermined voltage to the capacitive sensor and smlu]taneous]y measuring the static displacement of the platform; and, d) calculating the spring constant In dependence on the gradient of capacitance and the measured dsllacement.

   9. A method according to claim 8, wherein the static displacement is measured S using white-hght interferometry.

   ]O. A method according to claim 8 or 9, wherein the ve]octy is measured using Doppler velocimetry.

   11. A method according to any of claims 8 to 10, wherein steps a and c are performed in a vacuum.

   ] 2. A method of determining a spring constant of an AFM cantilever composing: determining deflection of the cantilever when pressed against a soLd surface; determining deflection of the cantilever when pressed against a calibration device as claimed In any one of claims l to 6 having a predetermined spring constant; and, calculating the spring constant of the AFM cantilever from the raho of the two deflections multiplied by the predetermined spring constant 13. A method of determining spring coefficient of a structure using a capacitive sensor Incorporated into the structure Including the steps of applying a predetermined voltage to the capacitive sensor and sunu] taneous]y measuring a static displacement parameter of the structure.

   14. A method as c]amed In claim 13, further comprising applying a predetermined Filtration to the device and smu]taneous]y neasurmg the velocity of the platform.

   15. A method according to clam i 4, further comprising calculating the gradient of capacitance of the structure In dependence on the Pleasured velocity, and calculating the spring constant m dependence on the gradient of capacitance and the measured dsp]acement.

   16. An AFM cantilever including a capacitive sensor for determining the spring constant of the cantilever.

   17. An AFM cantilever as claimed in claim 16, wherein the capacitive sensor comprises an electrostatic comb drive.

   18. An AFM cantilever as claimed in claim 16 or 17 wherein the capacitive sensor comprises a Watt balance device.

   19. An A1:;M cantilever including a calibration device as claimed in any of claims 1 to7.

   20. A test device as herein described and as illustrated iTI the accompanying drawings.

   21. A method as herein described and as illustrated in the accompanying drawings.