Принципы нанометрологии (Раздаточные материалы от преподавателя), страница 67

This displacement, measuredFIGURE 10.2Schema of the NPLlow-force balance.Low-force measurementby a plane mirror differential interferometer (see section 5.2.6), is used bya control system to create a deflection-nulling feedback force. The feedbackforce is generated by a potential difference across a system of verticallyoriented capacitor plates, V in equation (10.4), and acts vertically on themoving dielectric vane.10.3.4 Low-force transfer artefactsDue to the size of the primary low-force balances and their associatedinstrumentation, their requirement for vibration isolation and their sensitivity to changes in orientation, it is not possible to connect anything butsmall items to the balance for force measurement.

From this, and from thelogistics of moving each target instrument to the balance’s vicinity, stems theneed for transfer artefacts.10.3.4.1 Deadweight force productionThe most intuitive method of force production makes use of the Earth’sgravitational field acting on an object of finite mass: a deadweight. Deadweights have traditionally been, and are still, used routinely for maintainingforce traceability in the millinewton to meganewton range (see section 2.5).However, below 10 mN at the higher end of the low-force balance (LFB) scale,handling difficulties, contamination and independent testing issues lead tohigh relative uncertainties in weight measurement. The trend is for therelative uncertainty to increase in inverse proportion to the decrease in mass.Deadweights are, therefore, unsuitable for use as transfer artefacts, althoughuseful for comparison purposes at the higher end of the force scale of typicalLFBs [35].10.3.4.2 Elastic element methodsApart from gravitational forces from calibrated masses, the next most intuitive and common technology used for calibrated force production is anelastic element with a known spring constant.

The element, such asa cantilever or helical spring, is deflected by a test force. The deflection ismeasured, either by an external system such as an interferometer, or by anon-board MEMS device such as a piezoelectric element. With the springconstant previously determined by a traceable instrument such as an electrostatic force balance, the magnitude of the test force can be calculated. Inthis way a force calibration is transferred.Several examples of elastic elements use modified AFM cantilevers, asthese are of the appropriate size and elasticity, a simpler geometry thancustom designs and thus more reliably modelled, and generally well301302C H A P T ER 1 0: Mass and force measurementFIGURE 10.3 Experimental prototype reference cantilever array – plan view.understood by those working in the industry.

Very thin cantilevers, themanufacture of which is now possible, have low enough spring constants toallow, in principle, force measurement at the nanonewton level.The calibration of the spring constant of an AFM cantilever is discussedin section 7.3.6. Other elastic element methods will be described here thatare not necessarily AFM-specific. In order to provide suitable performanceacross a working range, usually one spring constant is insufficient.

It iscommon to design devices containing elements with a range of springconstants. This may be achieved in two ways with cantilever arrangements.Either an array of cantilevers with attached probes or single defined probingpoints is used, or one cantilever with multiple defined probing points is used.An example of the former, called an ‘array of reference cantilevers’, has beendeveloped at NIST [36] and is shown in Figure 10.3. The arrays, microfabricated from single-crystal silicon, contain cantilevers with estimatednominal spring constants in the range 0.02 N$m1 to 0.2 N$m1. Variationsin resonant frequency of less than 1 % are reported for the same cantileversacross manufactured batches, as an indication of uniformity.

The springconstants were verified on the NIST electrostatic force balance. Cantileverarrays are commercially available for AFM non-traceable calibration.However, their route to traceability puts a much lower ceiling on theiraccuracy and the uncertainties specified.As the simple devices described in this section are passive, they wouldrequire pushing into a LFB by an actuator system and some external means ofmeasuring deflection. This second requirement is significant as it relies onthe displacement metrology of the target instrument. The working uncertainty of these devices is higher than active-type cantilevers and may bebetter calibrated by such an active-type artefact.Low-force measurementThe alternative to the arrays of high-quality passive cantilevers discussedabove is a single cantilever with onboard deflection metrology.

These can beused to calibrate target instruments or indeed cheaper, lower-accuracy,disposable transfer artefacts. One of the first examples of an AFM probe withon-board piezoresistive deflection sensing is discussed in [37]. The devicewas fabricated as a single piezoresistive strain element with pointed-tipcantilever geometry. The researchers claim a 0.01 nm vertical resolution,which is equivalent to 1 nN with a spring constant of 10 N$m1 for thisproof-of-concept device.A number of piezoresistive cantilevers have been developed by severalNMIs.

NPL has developed the C-MARS (cantilever microfabricated array ofreference springs) device as part of a set of microfabricated elastic elementdevices intended for traceable AFM calibration [38]. The relatively largecantilever (150 mm wide by 1600 mm long) is marked with fiducials that inprinciple allow precise alignment of the contact point for a cantilever-oncantilever calibration. The size of the fiducials is influenced by the 100 mmby 100 mm field of view of typical AFMs.

Surface piezoresistors near thebase of the cantilever allow the monitoring of displacement and vibrationsof the cantilever, if required. Detail of the device is shown in Figure 10.4.Spring constants are quoted for interaction at each fiducial, providinga range of 25 N$m1 to 0.03 N$m1. NIST have also developed a cantileverFIGURE 10.4Images of the NPLC-MARS device, withdetail of its fiducialmarkings; the 10 mmoxide squares forma binary numberingsystem along the axis ofsymmetry.303304C H A P T ER 1 0: Mass and force measurementdevice that has thin legs at the root to concentrate bending in this rootregion and fiducial markings along its length [39].Researchers at PTB have created a slightly larger piezoresistive cantilever,of one millimetre width by a few millimetres length, for use in nanoindentation and surface texture work [40]. PTB has also created a two-legsphere-probe example and a single-leg tip-probe example.

The prototypes,manufactured using standard silicon bulk micromachining technology, havea stiffness range of 0.66 N$m1 to 7.7 N$m1. A highly linear relationshipbetween the gauge output voltage and the probing force in the micronewtonrange has been reported.In continuous scanning mode, the probing tip of a piezoresistive cantilever, such as the NIST device, may be moved slowly down the cantileverbeam, with beam deflection and external force values regularly recorded.Notches with well-defined positions show up as discontinuities in therecorded force-displacement curve, and act as a scale for accurate probe tipposition determination from the data.

The result is a function that describesthe spring constant of the transfer artefact, after probing with a LFB. Forinteraction with an electrostatic force balance operating in position-nulledmode, such a device needs to be pushed into the balance tip.10.3.4.3 Miniature electrostatic balance methodsNPL have developed a novel comb-drive device for force calibration. Oneexample, the ‘Electrical Nanobalance’ device [41,42], is shown inFigure 10.5.

A vertical asymmetry in the fields generated in a pair of combdrives levitates a landing stage against an internal elastic element.Measurements of the driving electrical signal and resultant deflection lead toa spring constant value potentially traceable to SI. At end-use, the devicebecomes a passive, calibrated, elastic device requiring no electrical connections and producing no interacting fields. The authors report a landing stagecentre-point spring constant of 0.195 N$m1 0.01 N$m1 and suitabilityfor calibration of AFM cantilevers in the range 0.03 N$m1 to 1 N$m1. Thedevice, calibrated dynamically, must be operated in vacuum to avoid dustcontamination of the key working elements.

A similar technique is used inNPL’s Lateral Electrical Nanobalance designed to measure lateral forces suchas friction in AFM [43].10.3.4.4 Resonant methodsChanges in the tension of a stretched string can be detected via relatedchanges in its resonant frequency. If a force is exerted on one of the stringanchor points along the string axis, the tension in the string will decrease. Fora well-characterized string the force exerted can be calculated from anLow-force measurementFIGURE 10.5Computer model of theNPL ElectricalNanobalance device.The area shown is980 mm 560 mm.Dimensionsperpendicular to theplane have beenexpanded by a factor oftwenty for clarity.accurate determination of the frequency shift.

In this way a low-forcemeasurement device is created.One example of a resonance force sensor is the ‘nanoguitar’ [44], shownschematically in Figure 10.6. Operating in vacuum, an AFM tip is pressedagainst the sample cantilever, changing the tension in the oscillating string.The beam is required to be soft compared to the string to transmit theinteraction force, improving sensitivity.

The set-up allows micrometres ofstring oscillation amplitude without significant amplitude of parasiticoscillations in the connected cantilever beam. The prototype used a carbonfibre with a diameter of 5 mm and a length of 4 mm, oscillating at 4 kHz. Asstring tension is decreased, force sensitivity rises but the response timedrops. The force resolution is limited by thermal noise in the string oscillation. The authors report a force resolution of 2.5 nN, achieved in vacuumfor a response time of 1 ms and a sensor stiffness of 160 N$m1.

The sensorperformance was limited by a low Q-factor and required precise fibre tensionadjustments. Vibration damping was significant because the string was gluedto the cantilever. Initial tension was set by sliding one anchor relative to theother using a stick-slip mechanism.The double-ended tuning fork concept forms an alternative highsensitivity force sensor, and has been studied by various groups. In oneexample [45] a vertical force acting on a sample cantilever beam changes305306C H A P T ER 1 0: Mass and force measurementFIGURE 10.6Schema of a resonantforce sensor – thenanoguitar.the resonant frequency of the fork ‘prong’ beams. The beams are vibratedby an external electromagnet and the amplitude is measured with a laserDoppler velocimeter.