10.1.1 Manufacture of the kilogram weight and the original copies

After many attempts in France, Johnson Matthey of London made a successful casting of a 90 % platinum 10 % iridium alloy ingot in 1879. Three cylindrical pieces were delivered to St-Claire Deville metallurgists in France where they were hammered in a press to eliminate voids, rough machined and polished, and finally adjusted against the kilogram des archives [3]. One of these kilograms was designated K and became the International Prototype Kilogram. Forty further kilogram weights were produced using the same techniques and delivered in 1884. Twenty of these were allocated to the signatories of the convention of the metre as national standards.

The International Prototype Kilogram (commonly known as the (International) Kilogram or just K) is a cylinder of approximate dimensions 39 mm diameter×39 mm height [4] (see Figure 2.5). The design of the artefact minimises its surface area while making it easy to handle and machine (a sphere would give the minimum surface area but presents difficulties in manufacture and use). Platinum–iridium was chosen as the material for the kilogram for a number of reasons. Its high density (approximately 21.5 g·cm⁻³) means that the artefact has a small surface area and, therefore, the potential for surface contamination is minimised. The relatively inert nature of the material also minimises surface contamination and enhances the mass stability of the artefact. The high density of the material also means that it displaces a smaller amount of air than a kilogram of less dense material (stainless steel or brass for example). The weight-in-air of the kilogram (or any mass standard) depends on the density of the air in which it is weighed because the air (or any fluid in which it is weighed) exerts a buoyancy effect proportional to the volume of the artefact. Minimising the volume of the weight minimises the effect of changing air density on the weight of the artefact. Platinum and its alloys are reasonably easy to machine [5], enabling a good surface finish to be achieved on the artefact, again reducing the effect of surface contamination. The addition of 10 % iridium to the platinum greatly increases its hardness and so reduces wear.

10.1.2 Surface texture of mass standards

The surface texture of the kilogram standards has a major effect on their stability. Early copies of the International Prototype (and the Kilogram itself) were finished by hand polishing using gradually finer polishing grains, concluding finally by polishing with a grain diameter of 0.25 μm [6]. More recent copies (since 1960) have been diamond turned, producing a visibly better finish on the surface. Measurements using coherence scanning interferometry (see Section 6.7.3.4) have shown typical surface roughness (Ra) values of 65–85 nm for hand-polished weights, compared with 10–15 nm achieved by diamond turning [7].

10.1.3 Dissemination of the kilogram

The BIPM is responsible for the dissemination of the unit of mass worldwide. Dissemination is achieved via official copies of the International Prototype Kilogram, known as national prototypes, held by all countries that are signatories to the Metre Convention. These are periodically compared, at the BIPM, with the International Prototype. The official copies of the kilogram are, like the original, made of platinum–iridium alloy, and the final machining and adjustment is done at BIPM. At present, there are approximately 100 official copies of the kilogram.

Periodic verification of the national kilogram copies takes place approximately every 30 to 40 years [8]. Each time the national copies are returned to the BIPM, they are cleaned and washed by a process known as nettoyage–lavage [9], which theoretically returns them to a reference value. All kilograms, including the International Prototype, are subject to nettoyage–lavage prior to the periodic verification exercise. The BIPM justify the use of this cleaning process because of the wide spread in the contamination levels of the returning national prototypes and the need to return K to its reference value. Surface contamination varies between national copies and ranges from those which are not used at all (some are returned to the BIPM with the seal on the container still intact from the last verification) to those that are used on a regular basis and have collected many tens of micrograms worth of accreted material on their surfaces.

10.1.4 Post nettoyage–lavage stability

Although the gravimetric effects of the nettoyage–lavage process have been studied by various NMIs [7,10,11] and the (variable) reproducibility of the method is documented, there has not been much work done to link the actual effect on the surface of the weight (measured by a reliable surface analysis technique) with either the mechanical cleaning method or the observed weight loss (but see Refs. [12,13]).

Furthermore, while the BIPM has made studies of the mass gain over the first 3 months after cleaning based on the behaviour of all the national prototypes, the return of the prototypes to their NMIs after this period means no longer term studies have been made (but see [14]).

Only an NMI with at least two other platinum–iridium kilograms, against which the stability of the national prototype could be monitored, would be able to carry out such work and even so the stability of the other 3 kg would affect the results. Due to the lack of data on the stability of national standards after returning from BIPM (approximately 3 to 4 months after cleaning and so still relatively unstable), a wide variety of algorithms are used to predict the longer term mass gain of the kilogram standards. Some algorithms are expressed as a function of time; for example, National Physical Laboratory (NPL) has used the following expression to predict the value of kilogram 18 after cleaning at the BIPM

$\mathrm{\Delta }V=0.356097\times t0.511678\mathrm{\mu }\mathrm{g},$ (10.1)

(10.1)

where ΔV is the measured difference from nominal in micrograms directly after cleaning (as measured by the BIPM) and t is the time after cleaning in days.

The most commonly used algorithm is that the national standard has the value assigned on leaving BIPM (approximately 3 months after cleaning) plus 1 μg per year. Some NMIs modify this by using a 0.22 μg per month gain for the first 2 years. Other NMIs assume that their national kilogram is perfectly stable on return from the BIPM and the mass gain is zero.

10.1.5 Limitations of the current definition of the kilogram

The kilogram is unique among the seven base SI units in that it is the only one that is still defined in terms of a physical artefact. As an artefact definition, its realisation and dissemination present a unique set of practical problems.

While the theoretical uncertainty associated with the value of K is zero (it is, by definition, exactly 1 kg), the practical accuracy with which the kilogram can be realised is limited by the stability of the artefact and the repeatability of the nettoyage–lavage cleaning process. Although the BIPM monitor the stability of K against a number of official copies (témoins) it keeps, the practical limit of the uncertainty in its value is, at best, about ±2 μg. Additionally, the value of platinum–iridium kilograms has been seen to drift by up to 2 μg per year, although K is probably more stable than this.

The fact that one artefact provides traceability for the entire worldwide mass scale also presents difficulties. The calibration of the national prototypes presents a problem for the BIPM as it involves a large number of measurements. The use of the nettoyage–lavage cleaning process to return the kilograms to a ‘base value’ is not only user dependent, time consuming and arduous in itself but greatly increases the number of weighings which must be made on the artefacts. Values of the kilograms before and after cleaning are calculated, as is the weight gain of the kilograms immediately after the cleaning process, from measurements made over a period of several weeks. Thus, not only is the work load of the BIPM very high, but the national prototype kilograms are not available to their NMIs for up to 6 months.

Most NMIs around the world hold only one official copy of the kilogram and thus their entire national mass measurement system is dependent on the value of their national prototype. This means that the handling and storage of this weight is very important and any damage means it would at least have to be returned to the BIPM for re-calibration, and at worst replaced.

10.1.6 Investigations into an alternative definition of the kilogram

For the last 20 years, there has been a considerable amount of work undertaken looking for an alternative, more fundamental, definition for the SI unit of the kilogram [14]. This work has been driven by two main assumptions. The limitations of the stability, realisation and dissemination of the kilogram have been discussed in Section 2.4. The other reason for the redefinition work currently being performed is the perception of the definition using an artefact as ‘low tech’ when compared with the definitions of the other six SI base units. For this reason, the approaches to a fundamental redefinition have, in some ways, been forced rather than being logical solutions to the problem. The other base units have more simple definitions based on one measurement (such as the speed of light for the metre), whereas any of the current proposals for the redefinition of the kilogram involve a number of complicated measurements. In the same way, the timescale for the redefinition of the other base units was defined by the discovery of a suitable phenomenon or piece of equipment (e.g. the laser used to define the metre). A similar method for redefinition of the kilogram has yet to be found.

At present there are two main methods being investigated with a view to providing a new fundamental definition for the SI unit of the kilogram. Even from these brief descriptions of the two approaches given in Sections 10.1.6.1 and 10.1.6.2, it can be seen that they involve a number of demanding measurements. Almost all of these measurements must be performed at uncertainties which represent the state of the art (and in some cases much better than those currently achievable) to realise the target overall uncertainty of 1 part in 10⁸ set for this work. The absolute cost of the equipment also means that the ultimate goal of all NMIs being able to realise the SI unit of the kilogram independently will, on purely financial grounds, not be achievable.

Both approaches require traceability to a mass in vacuum, both for their initial determination and for dissemination. The significance of the work described in this book, therefore, extends not only to improving knowledge of the stability of the current definition of the kilogram but also to facilitating the practical use of any of the currently considered methods of redefinition.

10.1.6.2 The Avogadro approach

The Avogadro project will define a kilogram based on a fixed number of atoms of silicon [20,21]. The mass of a sphere of silicon will be related to its molar mass and the Avogadro constant by the following equation

$m=\frac{M_{\mathrm{m}}}{N_{\mathrm{A}}}\frac{V}{v_0},$ (10.2)

(10.2)

where m is the calculated mass of the sphere, M_m is the molar mass of the silicon isotopes measured by spectrometry, N_A is the Avogadro constant, V is the volume of the sphere measured by interferometry and v₀ is the volume occupied by a silicon atom.

To calculate v₀, the lattice spacing of a silicon crystal must be measured by X-ray interferometry [22] (see Section 5.7.2). The practical realisation of this definition relies on the calculation of a value for N_A from an initial value for the mass of the sphere [23]. This value is then set and used subsequently to give values for the mass of the sphere, m. An added complication with this definition is the growth of oxides of silicon on the surface of the spheres. The thickness of the layer needs to be monitored (probably by ellipsometry) and used to correct the value of mass, m. As with the Watt balance, the latest results from the Avogadro project, produced by the International Avogadro Coordination, are approaching the uncertainty of 2 in 10⁸ [24].

It is likely that the kilogram will be redefined with relation to the Planck constant within the next 5 years but, since the Watt balance result (for the Planck constant, h) and the Avogadro (constant) result can be linked with a high degree of accuracy by the fine structure constant, the value for h can be reached by these two independent experiments, thus giving additional confidence in the final value.

Other experiments to realise a ‘fundamental’ kilogram have been investigated but could not reach the level of uncertainty required.

10.1.6.3 The ion accumulation approach

A third approach to the redefinition of the kilogram involves the accumulation of a known number of gold atoms [25,26]. Ions of Au¹⁹⁷ are released from an ion source into a mass separator and accumulated in a receptor suspended from a mass comparator. The number of ions collected is related to the current required to neutralise them supplied by an irradiated Josephson junction voltage source. The mass of ions, M, is then given by the equation

$M=\frac{n_1.n_2.m_a}{2}{\int }_0^{t}f(t)\mathrm{d}t,$ (10.3)

(10.3)

where n₁ and n₂ are integers, m_a is the atomic mass of gold, f(t) is the frequency of the microwave radiation irradiated onto the Josephson junction and m_a=197 u, for gold isotope Au¹⁹⁷, where u is the atomic mass unit (equal to 1/12 of the mass of C¹²).

10.1.7 Mass comparator technology

From the earliest days of mass calibration, the measurements have been made by comparison, each weight or quantity being compared with a standard of theoretically better accuracy. A series of comparisons would thus allow all measurements to be eventually related back to a primary standard, whether it was a naturally occurring standard (such as a grain of wheat) or an artefact standard (such as the current International Prototype Kilogram).

Until recently these comparisons have been performed using two-pan balances. From the earliest incarnations to the present day, the technology has relied on a balance beam swinging about a pivot normally at the centre of the beam. The mechanical quality of the beam, and in particular the pivot, has been refined until modern two-pan mechanical balances are capable of resolutions of the order of 1 part in 10⁹, equivalent to 1 μg on a 1 kg mass standard.

10.2.1 Weighing by subdivision

Normally the calibration of an unknown weight is done by direct comparison with one or more standards of the same nominal value. However, for the most demanding mass calibration applications, a subdivision calibration process is used. This involves the use of standards of one or more values to assign values to weights across a wide range of mass values. A typical example of this would be to use two or three 1 kg standards to calibrate a 20 kg to 1 mg weight set. Equally it would be possible to use a 1 kg and a 100 g standard for such a calibration.

Weighing by subdivision is most easily illustrated by considering how values would be assigned to a weight set using a single standard. In reality, the weighing scheme would be extended to involve at least two standards. The standard is compared with any weights from the set of the same nominal value and also with various combinations of weights from the set that sum to the same nominal value. A check weight, which is a standard treated in the same manner as any of the test weights, is added in each decade of the calibration so that it is possible to verify the values assigned to the weight set.

10.3.1 Relative magnitude of low forces

A full derivation of the surface interaction forces significant at the micro- and nanotechnogy scale is beyond the scope of this book and, indeed, has been presented by various groups previously. Nevertheless, the basic force separation dependencies are worthy of consideration by the reader and a selection is presented in Table 10.1. Equations obtained from referenced works have, where necessary, been adapted to use common nomenclature. To simplify comparison, the interaction of a sphere and flat plate is considered where possible. Since the tips of most probes can be adequately modelled as a (hemi-) sphere, this is a suitable approach. The sphere–plate separation is assumed to be much less than the sphere radius. Figure 10.2 is a comparative plot using typical values for the given parameters. Section 7.3.7 also discusses surface forces in terms of the atomic force microscope.

Table 10.1

Summary of Surface Interaction Force Equations

Interaction	Equation
Electrostatic	$F=-{\epsilon }_0{U}^2\pi R^2/D^2$ [32]
Capillary	$F=4\pi \gamma R\left(1-\frac{h-2e}{2r}\right)u(-h+L)$ [32,33]
Van der Waals	$F=-\frac{HR}{6D^2}$ for non-retarded, attractive forces [34]
Casimir effect	$F=-\frac{R{\pi }^3\hslash c}{360D^3}$ [35]

In these equations, F is a force component, U the work function difference between the materials, D the sphere-flat separation, g the free surface energies at state boundaries, H the Hamaker constant and θ the contact angle of in-interface liquid on the opposing solid surfaces. In the capillary force, the step function u(.) describes the breaking separation; e is the liquid layer thickness and r the radius of meniscus curvature in the gap.

10.3.2 Traceability of low-force measurements

Traceability for force measurement is usually carried out by comparing to a calibrated mass in a known gravitational field (see Section 2.4). However, as the forces (and hence masses) being measured decrease below around 10 μN (approximately equivalent to 1 mg), the uncertainty in the mass measurement becomes too large and the masses become difficult to handle. For this reason, it is more common to have a force balance that gains its traceability through electrical and length measurements.

An electronic force producer may be made intrinsically traceable. Given suitably defined geometry, the electrostatic force in a capacitor or the electromagnetic force in a voice coil may be calculated very accurately, in terms of the volt, farad or ampere. These units are in turn traceable to fundamental constants of nature, as shown in Figure 10.3. This means that, in principle, uncertainties can be decreased arbitrarily, and traceability is not reliant on a changeable kilogram. The technical approach and more philosophical motivation for an electronic newton relate to the Watt balance (see Section 10.1.6.1), and the redefinition of the kilogram, the volt balance [36], and the general trend to redefine the SI units and their hierarchy in terms of fundamental constants of nature [37].

The current force traceability route is at least a two-stage process. The first stage is to develop a primary force standard instrument deriving traceability directly from the base unit definitions realised at the world’s NMIs. These primary instruments will typically sacrifice practicalities in order to obtain the best possible metrological performance. Various groups have developed such instruments, with the current best performance held by examples at NIST and PTB.

The second stage in the traceability route is to design a transfer artefact, or sequence of artefacts, to transfer the force calibration to target instruments in the field. These artefacts may sacrifice uncertainties, resolution or range of force measurement, in exchange for cost reductions, portability or compliance with other physical constraints, such as size or environmental tolerance.

10.3.3 Primary low-force balances

The leading examples of force measurement instruments, operating in the millinewton to nanonewton range, are based on the electrostatic force balance principle. The force to be measured is exerted on a flexure system, which deflects. This deflection is measured using an interferometer. The deflection of the flexure also changes the capacitance of a set of parallel capacitor plates in the instrument. This is usually achieved by changing either the plate overlap or the position of a dielectric, with flexure deflection. In this way, the capacitance changes linearly with deflection. The interferometer signal is used in a closed-loop controller to generate a potential difference across the capacitor generating an electrostatic force that servos the flexure back to zero deflection. Measurement of the force exerted is derived from traceable measurements of length, capacitance and potential difference. The exerted force is calculated using Eq. (10.4), in which z is the flexure displacement, and C and V the capacitance of and voltage across the parallel plates, respectively. The capacitance gradient, dC/dz, must be determined prior to use.

$F=-\frac{1}{2}{V}^2\frac{\mathrm{d}C}{\mathrm{d}z}$ (10.4)

(10.4)

The first electrostatic force balance primarily designed with the traceability for low-force measurements in mind was developed at NIST [38]. Subsequently, balances have been developed at The Korea Research Institute of Standards and Science (KRISS) [39], PTB [40], NPL [41] and Centre for Measurement Science – International Technology Research Institute, Taiwan [42].

The NPL balance will be discussed in some detail as an example and is shown schematically in Figure 10.4. A vertical force applied to the platen displaces the connected flexure and dielectric. This displacement, measured by a plane mirror differential interferometer (see Section 5.2.6), is used by a control system to create a deflection-nulling feedback force. The feedback force is generated by a potential difference across a system of vertically oriented capacitor plates, V in Eq. (10.4), and acts vertically on the moving dielectric vane.

10.3.4 Low-force transfer artefacts

Due to the size of the primary LFBs and their associated instrumentation, their requirement for vibration isolation and their sensitivity to changes in orientation, it is not possible to connect anything but small items to the balance for force measurement. From this, and from the logistics of moving each target instrument to the balance’s vicinity, stems the need for transfer artefacts.

10.3.4.2 Elastic element methods

Apart from gravitational forces from calibrated masses, the next most intuitive and common technology used for calibrated force production is an elastic element with a known spring constant. The element, such as a cantilever or helical spring, is deflected by a test force. The deflection is measured, either by an external system such as an interferometer or by an on-board microelectromechanical systems (MEMS) device such as a piezoelectric element. With the spring constant previously determined by a traceable instrument, such as an electrostatic force balance, the magnitude of the test force can be calculated. In this way a force calibration is transferred.

Several examples of elastic elements use modified AFM cantilevers, as these are of the appropriate size and elasticity, a simpler geometry than custom designs and thus more reliably modelled, and generally well understood by those working in the industry. Very thin cantilevers, the manufacture of which is now possible, have low enough spring constants to allow, in principle, force measurement at the nanonewton level.

The calibration of the spring constant of an AFM cantilever is discussed in Section 7.3.6. Other elastic element methods will be described here that are not necessarily AFM-specific. In order to provide suitable performance across a working range, usually one spring constant is insufficient. It is common to design devices containing elements with a range of spring constants. This may be achieved in two ways with cantilever arrangements. Either an array of cantilevers with attached probes or single-defined probing points is used, or one cantilever with multiple defined probing points is used. An example of the former, called an ‘array of reference cantilevers’, has been developed at NIST [44] and is shown in Figure 10.5. The arrays, microfabricated from single-crystal silicon, contain cantilevers with estimated nominal spring constants in the range of 0.02–0.2 N·m⁻¹. Variations in resonant frequency of less than 1 % are reported for the same cantilevers across manufactured batches, as an indication of uniformity. The spring constants were verified on the NIST electrostatic force balance. Cantilever arrays are commercially available for AFM non-traceable calibration. However, their route to traceability puts a much lower ceiling on their accuracy and the uncertainties specified.

As the simple devices described in this section are passive, they would require pushing into an LFB by an actuator system and some external means of measuring deflection. This second requirement is significant as it relies on the displacement metrology of the target instrument. The working uncertainty of these devices is higher than active-type cantilevers and may be better calibrated by such an active-type artefact.

The alternative to the arrays of high-quality passive cantilevers discussed above is a single cantilever with on-board deflection metrology. These can be used to calibrate target instruments or indeed cheaper, lower accuracy, disposable transfer artefacts. One of the first examples of an AFM probe with on-board piezoresistive deflection sensing is discussed in Ref. [45]. The device was fabricated as a single piezoresistive strain element with pointed-tip cantilever geometry. The researchers claim a 0.01 nm vertical resolution, which is equivalent to 1 nN with a spring constant of 10 N·m⁻¹ for this proof-of-concept device.

A number of piezoresistive cantilevers have been developed by several NMIs. NPL has developed the cantilever microfabricated array of reference springs (C-MARS) device as part of a set of microfabricated elastic element devices intended for traceable AFM calibration [46]. The relatively large cantilever (150 μm wide×1600 μm long) is marked with fiducials that in principle allow precise alignment of the contact point for a cantilever-on-cantilever calibration. The size of the fiducials is influenced by the 100 μm×100 μm field of view of typical AFMs. Surface piezoresistors near the base of the cantilever allow the monitoring of displacement and vibrations of the cantilever, if required. Detail of the device is shown in Figure 10.6. Spring constants are quoted for interaction at each fiducial, providing a range of 25–0.03 N·m⁻¹. NIST has also developed a cantilever device that has thin legs at the root to concentrate bending in this root region and fiducial markings along its length [47].

Researchers at PTB have created a slightly larger piezoresistive cantilever, of 1 mm width by a few millimetres length, for use in nanoindentation and surface texture work [48]. PTB has also created a two-leg sphere-probe example and a single-leg tip-probe example. The prototypes, manufactured using standard silicon bulk micromachining technology, have a stiffness range of 0.66–7.7 N·m⁻¹. A highly linear relationship between the gauge output voltage and the probing force in the micronewton range 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 cantilever beam, with beam deflection and external force values regularly recorded. Notches with well-defined positions show up as discontinuities in the recorded force-displacement curve and act as a scale for accurate probe tip position determination from the data. The result is a function that describes the spring constant of the transfer artefact, after probing with an LFB. For interaction with an electrostatic force balance operating in position-nulled mode, such a device needs to be pushed into the balance tip.

10.3.4.3 Miniature electrostatic balance methods

NPL has developed a novel comb-drive device for force calibration. One example, the ‘Electrical Nanobalance’ device [49,50], is shown in Figure 10.7. A vertical asymmetry in the fields generated in a pair of comb drives levitates a landing stage against an internal elastic element. Measurements of the driving electrical signal and resultant deflection lead to a spring constant value potentially traceable to SI. At end-use, the device becomes a passive, calibrated, elastic device requiring no electrical connections and producing no interacting fields. The authors report a landing stage centre-point spring constant of 0.195±0.01 N·m⁻¹ and suitability for calibration of AFM cantilevers in the range of 0.03–1 N·m⁻¹. The device, calibrated dynamically, must be operated in vacuum to avoid dust contamination of the key working elements. A similar technique is used in NPL’s Lateral Electrical Nanobalance designed to measure lateral forces such as friction in AFM [51].

10.3.4.4 Resonant methods

Changes in the tension of a stretched string can be detected via related changes in its resonant frequency. If a force is exerted on one of the string anchor points along the string axis, the tension in the string will decrease. For a well-characterised string, the force exerted can be calculated from an accurate determination of the frequency shift. In this way, a low-force measurement device is created.

One example of a resonance force sensor is the ‘nanoguitar’ [52], shown schematically in Figure 10.8. Operating in vacuum, an AFM tip is pressed against the sample cantilever, changing the tension in the oscillating string. The beam is required to be soft compared to the string to transmit the interaction force, improving sensitivity. The set-up allows micrometres of string oscillation amplitude without significant amplitude of parasitic oscillations in the connected cantilever beam. The prototype used a carbon fibre with a diameter of 5 μm and a length of 4 mm, oscillating at 4 kHz. As string tension is decreased, force sensitivity rises but the response time drops. The force resolution is limited by thermal noise in the string oscillation. The authors report a force resolution of 2.5 nN, achieved in vacuum for a response time of 1 ms and a sensor stiffness of 160 N·m⁻¹. The sensor performance was limited by a low Q-factor and required precise fibre tension adjustments. Vibration damping was significant because the string was glued to the cantilever. Initial tension was set by sliding one anchor relative to the other using a stick-slip mechanism.

The double-ended tuning fork concept forms an alternative high-sensitivity force sensor and has been studied by various groups. In one example [53], a vertical force acting on a sample cantilever beam changes the resonant frequency of the fork ‘prong’ beams. The beams are vibrated by an external electromagnet and the amplitude is measured with a laser Doppler velocimeter. The monolithically manufactured system has an experimentally determined minimum detection force limit of 19 μN, with a theoretical value as low as 0.45 μN.

An attempt has been described to create a tuneable carbon nanotube electromechanical oscillator whose motion is both excited and detected using the electrostatic interaction with the gate electrode underneath the tube [54]. The advantages of the nanotube are highlighted: they are made of the stiffest material known, have low densities, ultra-small cross sections and can be defect free. The group report that despite great promise, they have as yet failed to realise a room-temperature, self-detecting nanotube oscillator due to practical difficulties. For example, the adhesion of the nanotube to the electrodes inevitably reduces the device’s quality factor by several orders of magnitude.