Принципы нанометрологии (1027506), страница 65
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Physical constants take their standard values: e0 ¼ 8.854 1012C2$N1$m2; h ¼ 1.055 1034 m2$kg$s1 and c ¼ 3 108 m$s1reason it is more common to have a force balance that gains its traceabilitythrough electrical and length measurements.The current force traceability route is at least a two-stage process. Thefirst stage is to develop a primary force standard instrument deriving traceability directly from the base unit definitions realized at the world’s NMIs.These primary instruments will typically sacrifice practicalities in order toobtain the best possible metrological performance. Various groups havedeveloped such instruments, with the current best performance held byexamples at NIST, PTB and NPL.The second stage in the traceability route is to design a transfer artefact,or sequence of artefacts, to transfer the force calibration to target instrumentsin the field.
These artefacts may sacrifice uncertainties, resolution or range offorce measurement, in exchange for cost reductions, portability or compliance with other physical constraints, such as size or environmentaltolerance.10.3.3 Primary low-force balancesThe leading examples of force measurement instruments, operating in themillinewton to nanonewton range, are based on the electrostatic forcebalance principle. The force to be measured is exerted on a flexure system,which deflects.
This deflection is measured using an interferometer. Thedeflection of the flexure also changes the capacitance of a set of parallelcapacitor plates in the instrument. This is usually achieved either by299300C H A P T ER 1 0: Mass and force measurementchanging the plate overlap, or by changing the position of a dielectric, withflexure deflection. In this way the capacitance changes linearly with deflection. The interferometer signal is used in a closed-loop controller to generatea potential difference across the capacitor generating an electrostatic forcethat servos the flexure back to zero deflection.
Measurement of the forceexerted is derived from traceable measurements of length, capacitance andpotential difference. The exerted force is calculated using equation (10.4), inwhich z is the flexure displacement, and C and V the capacitance ofand voltage across the parallel plates respectively. The capacitance gradient,dC/dz, must be determined prior to use.1dCF ¼ V22dz(10.4)The first electrostatic force balance primarily designed with the traceability for low-force measurements in mind was developed at NIST [31].Subsequently, balances have been developed at The Korea Research Instituteof Standards and Science (KRISS) [32], PTB [33] and NPL [34].The NPL balance will be discussed in some detail as an example and isshown schematically in Figure 10.2. A vertical force applied to the platendisplaces the connected flexure and dielectric.
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.
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