Research Article System-Level Design Considerations for Carbon...

Hindawi Publishing CorporationJournal of SensorsVolume 2013 Article ID 384643 12 pageshttpdxdoiorg1011552013384643

Research ArticleSystem-Level Design Considerations forCarbon Nanotube Electromechanical Resonators

Christian Kauth Marc Pastre Jean-Michel Sallese and Maher Kayal

Electronics Laboratory Ecole Polytechnique Federale de Lausanne 1015 Lausanne Switzerland

Correspondence should be addressed to Christian Kauth christiankauthepflch

Received 24 May 2013 Accepted 19 September 2013

Academic Editor Andrea Cusano

Copyright copy 2013 Christian Kauth et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Despite an evermore complete plethora of complex domain-specific semiempiricalmodels no succinct recipe for large-scale carbonnanotube electromechanical systems design has been formulated To combine the benefits of these highly sensitive miniaturizedmechanical sensors with the vast functionalities available in electronics we identify a reduced key parameter set of carbonnanotube properties nanoelectromechanical system design and operation that steers the sensorrsquos performance towards systemapplications based on open- and closed-loop topologies Suspended single-walled carbon nanotubes are reviewed in terms oftheir electromechanical properties with the objective of evaluating orders of magnitude of the electrical actuation and detectionmechanisms Open-loop time-averaging and 1120596 or 2120596 mixing methods are completed by a new 4120596 actuation and detectiontechnique A discussion on their extension to closed-loop topologies and system applications concludes the analysis coveringsignal-to-noise ratio and the capability to spectrally isolate themotional information fromparasitical feedthrough by contemporaryelectronic read-out techniques

1 Introduction

Since their discovery [1] and tremendous boost in popularitytwo decades ago [2] carbon nanotubes (CNTs) incitedresearchers from various domains to investigate amongothers their electrical and mechanical properties Their highintegrity quality factor and small dimensions are white hopefor the single-walled carbon nanotubesrsquo (SWNTs) advanceto applications such as electromechanical resonators for RFtransmission and reception voltage-controlled oscillators orsingle molecule weighing [3] First models emerged and keptrefining up to reach an impressive complexity sometimesbeyond the scope of circuit design that generally prefers totrade model complexity for simplicity and clarity To datethe library of models that describe in detail partial CNTbehaviour under specific conditions has reached a criticalvolume Focusing on strictly relevant parameters is not a triv-ial task for nanoelectromechanical system (NEMS) designersanymoreThis is an unfortunate fact when one considers thatthe relatively low device yield and rare occasions to observe

the desired phenomena already protract the CNT-NEMSrsquoadvancement to system-level industrial applications

In this scope our work reviews the state-of-the-art CNT-NEMS devices from a system-level point of view and drawsclear guidelines on CNT parameter selection and device bias-ing to foster the CNTrsquos operation as a mechanical componentwithin electronic circuits that implement versatile function-alities Never measured orders of magnitude of the signalsencoding motional and parasitical information are extractedfor those deliberate designsThis analysis leads to a systematiccompletion of the recent open-loop readout techniques basedon time-averaging [4] and 1120596 [5] or 2120596 [6] mixing by a new4120596 mixing technique To pave the way to appealing sensorapplications closed-loop oscillators locking on the NEMSrsquomotional signal is introduced and discussed with respect tothe capability of circuit-level frontends to amplify and filterthese signals out of the noisy background for further process-ing Crucial effects steering the overall system behaviour areidentified others are shown to be negligible The approachremains at high level with major emphasis on principles and

2 Journal of Sensors

orders of magnitude while the underlying physics is brieflyhighlighted referring to the related state-of-the-art literature

Section 2 formalizes the tunable suspended clamped-clamped SWNT [7] with its electrodes as an electromechan-ical multiport core component of any system applicationA possible operation of this multiport within a closed-looposcillator structure is outlined The functionality of electrical(Section 3) andmechanical (Section 4) ports is first examinedisolatedly in their respective energy domains then in con-junction (Section 5) opening the gate to electronic actuationand sensing of mechanical phenomena The strength andfrequency of the dominant effects are assessed and mayserve as reference work to future NEMS application design-ers The capability of low-noise electronic circuits to senseand amplify the NEMSrsquo motional information in terms ofspectral signal separation from parasitical feed-through andminimum detectable signal is assessed for open-loop res-onators (Section 6) and all-purpose closed-loop topologies(Section 7) leading to promising conclusions (Section 8) foremerging NEMS sensors

2 The NEMS as anElectromechanical Multiport

Motivated by remarkable electromechanical sensing proper-ties the system of interest operates the SWNT as channelmaterial contacted in a transistor configuration (Figure 1)Mechanical degrees of freedom occur when the channel issuspended and allow for frequency tuning via straining [8]The CNTsrsquo current source behaviour (Figure 2) lends itselfto current-mode readout Maximal motional signal strength(piezoresistive and motional field effect) minimal parasiticsignal strength (gate-drain capacitance noise) and mini-mum intrinsic signal loss (output impedance) are desirablefrom a system-level point of view and become NEMS devicedesign objectives

With force and voltage (potential difference) as across-variables velocity and current as through-variables in themechanical and electrical energy domains respectively thiselectromechanical system presents two electrical (gate andsource potential) and two mechanical (source and channelpositions) degrees of freedom to steer interdomain energytransfer if the drain is chosen to be the electrical (poten-tial) and mechanical (position) reference This decisionroots in the fact that the movable frequency tuning elec-trode (source) presents an elevated parasitic capacitancewhich hampers high frequency readout necessary for closed-loop applications as depicted in Figure 1 The CNT is actu-ated via an electrostatic force pulling from the gate elec-trode Its motion modulates the current flowing throughit by nanoamperes This current is sensed at the drainand amplified by a low-noise frontend Given the CNTsrsquoultra-high resonance frequencies noise is integrated overa considerable bandwidth and a bandpass filter is requiredto put things right (Section 7) This filter also attenuatesundesired parasitical feed-through signals from the actuatingelectrode (Section 6) Gain and phase regulation close theloop by ensuring proper oscillation buildup and stabilizationMolecules binding to the tube or strain induced via the source

Current sensing

frontend

Voltage actuation backend

Bandpass filter

Phase and gain

regulationS

D

GCNT

Figure 1 Sensing resonator NEMS in a generic closed-loop oscilla-tor topology

CNT

G

S

DR = b

C = 1k

L = m

ΔVp

119986 (ΔVp Vg) iPiezo iFET iNoise

iCapa

iLoss

Figure 2 NEMS as electromechanical two- motional contributions(piezoresistive and motional-FET) to the drain current

may alter the CNTrsquos resonance frequency entailing sensorand voltage controlled oscillator (VCO) applications [3]

With the mechanical properties of SWNTs being pri-marily defined by the strong covalent in-plane 1199041199012 bonds(120590-bonds) while the electronic properties depend almostexclusively on the delocalized 120587-states the system qualifiesfor a preliminary isolated study in the purely electrical andmechanical domains The combination of electromechanicaleffects will be studied subsequently and reveals the efficientoperation of SWNTNEMS as electromechanical transducersand sensors

3 Carbon Nanotube Field Effect Transistor

This section highlights the most pertinent system-level pa-rameters of carbon nanotube field effect transistors (CNFET)the CNTrsquos projection into the purely electrical domain Sincetheir first demonstration at room temperature [9] a heapof related work has been reported on SWNT electronics[10] ballistic effects [11] their advance to GHz frequen-cies [12] and circuit models [13] To countersteer systemdesignerrsquos resentment felt against the plethora of state-of-the-art findings which reveal that apparently similar devices maybehave very differently orders of magnitude of the differentCNFET phenomena are extracted and intend to guide systemdesigners through proper device selection DC bias and ACoperation summarized at the end of this section

31 CNT Properties

311 Electronic Nature (Affected by Chirality) Themetallic orsemiconducting nature of SWNTs is defined by their chiralityThe lack of chirality control [14] in the growth processes cur-rently constrains CNT-based circuits to small sizes for yield

Journal of Sensors 3

considerationsWithout any selection [15] only two tubes outof three can be expected to possess the desirable conductancecontrollability property The electronic properties of SWNTsoriginate from the band structure of graphene confined toa 2D rolled-up stripe Under the nearest neighbour tightbinding approximation [16] the valence and conduction 120587-bands of graphene intersect in 6 singular K-points withinthe first Brillouin zone awarding graphene the semimetaldesignation Via zone folding the Brillouin zone for SWNTsis quantized and its dispersion relation presents a bandgap ifnone of the K-points belongs to the quantized zone which isstatistically the case for two SWNTs out of three A refinedmodel [17] predicts for small diameter tubes the opening ofa narrow bandgap in otherwise metallic tubes but thermalnoise restricts their exploitation to cryogenic temperatures

312 Bandgap (Affected by Diameter) While the electricalnature is defined by the chirality the bandgap details ofsemiconducting tubes are mainly imposed by the diameter 119889Tremendous progress has beenmade on diameter control [18]and allows for synthesis of large (119889 gt 2 nm) or narrow (119889 lt1 nm) diameter tube distributionswith standard deviations ofless than 01 nm translating into precisely controlled energygaps in the 05ndash08 eV range [18] The SWNT diameterwill enter the design parameter set and fixes the bandgapapproximately via [19]

119864119892 [

eV] = 08

119889 [nm] (1)

implying larger bandgaps for narrow tubes As a consequencelarge diameter tubes are generally favoured for their lowercontact resistance and higher current drive capabilities whilenarrow diameter tubes are beneficial for low-power targets

313 The Ultimate Resistance in CNT and Saturation Current(Affected by Length) The intrinsic resistance of metallicSWNTs is composed of a quantum resistance described by theLandauer-Buttiker formula [20] completed by a Drude-likeresistanceThe quantum resistance originates from the severereduction of the large number of modes in the macroscopiccontacts to solely two quasidegenerate bands with van Hovesingularities each of which can carry spin up or down in the1D SWNTThis limits the maximum conductance of SWNTsto 1198660= 41198902ℎ asymp 155 120583S Under the hypothesis of perfect

ohmic contacts which will be discussed in Section 321semiconducting SWNTs present close to zero conductanceif the Fermi level falls into the bandgap and can reach 119866

0

under ballistic transportWithmean free paths up to120583mscale[10] SWNTs are ballistic [11] under low bias and length scalesbelow hundreds of nm Shorter devices will switch fasterdue to time-of-flight considerations but will never exceedthis upper conductance bound In longer channels localizedlattice defects [21] and long range potential fluctuations inthe oxide may cause elastic scattering Combined to inelasticlow-bias acoustical electron-phonon scattering these reflec-tions translate into a series Drude-like resistance Conse-quently the voltage starts to drop along the SWNT channeland parameters such as charge mobility and resistivity can be

definedwith best of breed values of 104 sdot sdot sdot 105 cm2Vs [22 23]and 10minus6Ωcm [10] As the channel resistance grows largerthan the contact resistance very long devices might behavemore like bulk-switchingMOSFETs where transport is dom-inated by drift of carriers inside a charge gradientThe contactresistance to CNTs shows to be rather high in practice oftenabove MΩ This can be mediated by proper choice of theelectrode material to have conduction states which extendthrough both the CNT and the metal Further defects canbe intentionally introduced either on the CNT or the metalto cause scattering at the interface [24] which results in areduced contact resistance of about 100 kΩ [25] This sumof the linear regime on-resistance and the contact resistancecan be directly measured under low drain-source bias Whileconductances close to the upper bound can be achieved byfreezing out the Drude resistance room-temperature valuesof 10 and 50 of this upper limit have been demonstratedfor long (119871 = 3 120583m) and short (119871 = 300 nm) devices[11] respectively Under high bias optical electron-phononexcitation sets in resulting in a linear increase of the Druderesistance [26] and implying a saturation asymptote Themaximum current through SWNTs was indeed shown tobe sim25120583A [26 27] This asymptote was circumvented inextremely short devices (119871 sim 10 nm) with current values of60120583A and no sign of saturation [28] Saturation currents of120583A are considered an indicator of acceptable device selectiondesign and operation for current mode sensor applicationsIt is noteworthy tomention that this current saturation booststheCNTrsquos output impedance and transforms theCNTrsquos linearregime resistive behaviour (under small 119881

119889119904) into an active

controllable current source behaviour (under larger119881119889119904) [10]

32 CNFET Design Although careful choice of the relevantSWNT properties is essential it is not sufficient for properdevice functionality

321 Contact Type (Affected by ContactMaterial) TheSchot-tky or ohmic nature of the contacts is greatly determinedby the contact material Due to the unique 1D structure anda quasi 0D interface of SWNTs the interface states arenot strong enough to pin the Fermi level [29] The heightof the Schottky barrier at the metal-CNT interface thereforedepends strongly on the metal work function Desired ohmiccontacts for high device performance can be achieved foramong others Al Cr or Pd contacts [11] Smaller bandgapsfavour the formation of ohmic contacts for at least one type ofcarrier In accordance with (1) a clear diameter and Schottky-barrier height dependence of the apparent on-state resistancewere demonstrated [30] SWNT diameters well below 1 nmhave bandgaps that approach 1 eV making the formation ofSchottky barriers at the interfacemore likely and boosting theapparent SWNT resistance to MΩ values [31]

322 Controllability (Affected by Oxide Thickness and Gate-CNT Distance) The devicersquos controllability via the gate elec-trode strongly depends on the effective electrode overlapwith the channel the dielectric constant of the insulator andthe gatersquos distance to the possibly suspended channel While


cylindrical all-around thin-oxide [32] and electrolytic [33]gates provide best controllability [34] electrostatic actuationfor motion claims for an asymmetrical gate structure Thinhigh-k oxides acting on a wide section of the device enhancecontrollability [35] but such nm thick oxides might leak pA[36] The exact mechanisms of channel control are discussedsubsequently

33 Device Bias Once the design is accomplished thereremain two electrical parameters to regulate the device char-acteristics gate and source potentials (with respect to thefixed drain potential)

331 Gate Potential Device conductance can be modulatedvia the gate potential mainly through modulation of theSchottky barrier width The latter being fixed by the semi-conducting material and its doping level in 3D structuresfield screening in 1D SWNTs is weak and band bendingdue to gate potential variations can be used to change thebarrier thickness and hence the tunnelling probability Itwas shown [37] that thermally assisted tunnelling and notthermionic emission dominates carrier injection into 1Dsemiconductors For nicely controllable devices a variationof 1V on the gate potential is sufficient to switch the devicefrom the off- to the on-state [30 36] and sweep throughsix decades of drain current Thick oxides can require gatevoltage excursions of tens of volts [38] Although precisevalues of the derivative of the conductance with respect tothe gate potential depend on the exact zero-bias Fermi levelposition with respect to the valence and conduction bands aswell as on the controllability efficiency the onoff transitiongenerally happens in the plusmn5V range Consequently CNFETsshow ambipolar characteristics which can be suppressed viagate structure [39] and contact engineering Section 5 revealsthe optimal bias point for electromechanical operation to liesomewhere between the transition and the on-state

332 Source Potential The drain-source voltage 119881119889119904

influ-ences the device performance to a lesser extent than the gatepotential but must obey some constraints A nonzero 119881

119889119904

being necessary to cause charges to flow the current increaseslinearly with 119881

119889119904until saturation occurs Beyond a critical

value of 119881119889119904 minority carrier injection sets in leading to

nonnegligible off-state currents and an exponential increasein on-current beyond the 25 120583A saturation limit with risk ofdevice destruction A CNFET acts as two Schottky barriersconnected via a low-field Drude resistance Assuming amidgap lineup of the Fermi level with respect to the bandgapand keeping 119881

119892close to 119881

119904while increasing 119881

119889lead to band

bending at the drain and enhanced hole injection Sweeping119881119892towards119881

119889now causes the drain Schottky barrier for holes

to widen and the Schottky barrier width for electrons at thesource to shrink At119881

119892= (119881119889+119881119904)2 the current through the

SWNTbecomesminimal then increases again with electronsas the majority carriers Thus larger 119881

119889119904imply the difficulty

to maintain wide Schottky barriers simultaneously for bothtypes of carriers If proper transistor behaviour is defined viaan onoff current ratio of at least 104 an upper limit for 119881

119889119904

is given [36] as a function of controllability (oxide thickness119905ox) and bandgap (diameter)

119881119889119904max [V] = (119864

13

119892[eV] minus 02)radic119905ox [nm] (2)

For CNFETs a reasonable range of 119881119889119904spans from 001 V for

highly controllable small bandgap designs to some volts forlarge bandgap designs with reduced controllability

34 Device Operation Once properly biased small-signalvariations can be superposed on the different terminals to usethe CNFET as a capacitor transistor or mixer

341 Similarities with the FET The gate voltagersquos 119881119892effi-

ciency of modulation of the drain current 119868119889is expressed as

a transconductance 119892119898= 120597119868119889120597119881119892 reaching peak values of

30 120583S [12 26 27] at the onset of conduction and decreases inthe saturation regime The inferred CNFET current modula-tion is

120575119868120596infet = 119892119898 sdot 120575119881

120596in119892 (3)

The electrical FET current has the same frequency 120596in (indi-cated by superscript notation throughout this paper) as thedriving voltage and can take values up to 120583A in highlycontrollable geometries As the current is mainly controlledvia the Schottky barrier widths CNFETs can be controlledequivalently through the gate and source (for electrons asmajority carriers) or drain (for holes as majority carriers)Average transconductances of 15120583S for long and 12 120583S forshort devices were found [12] to be independent of frequencyWith gate-drain capacitances of sim100 aF such transcon-ductances lead to state-of-the-art unity-gain frequencies of50GHz [12] imposed by 119891

119879= 1198921198981198901198972120587119862119892119889 To read the GHz

operation with these transconductances the tracks must beof sufficiently low resistance and minimal capacitance to thesubstrate and other signals to avoid low-pass filtering of thesignal The critical track RC product RCtrack = 12120587119891

119879asymp

100 ps requires very careful signal routing Suspension of thechannel sacrifices part of this performance and the lessercontrollability leads to typical transconductances of tens ofnS resulting in current amplitudes up to tens of nA

342 Signal Mixing CNFETs can also be driven simultane-ously from the source and the gate and hence be operated asmicrowave mixers As readout happens at lower frequenciesthe corresponding track design is less crucial and the result-ing low frequency current writes

119868Δ120596

mix =1

2

119892119898120575119881120596inminusΔ120596119904

120575119881120596in119892

(4)

and has been measured up to mixing signals of 10GHz [40]The theoretical upper limit given by the quantumcapacitance(119862119889= 119862quant) predicts unity gain to scrape terahertz [41]

343 Capacitive Feedthrough and Miller Effect Figure 1 sug-gests that any ac-signal capacitively bridges gate and drainThe coupling capacitance 119862G-CNT comprises the intrinsic


device gate-drain capacitance (sim100 aF) and the track-to-track capacitances whichmight contribute up to femtofaradsThe resulting gate-induced current modulation in the draincan easily reach 120583A amplitudes at GHz

120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (5)

This capacitance not only feeds forward part of the signalwithout amplification by the FET effect but also has evena Miller effect on the gate signal Multiplied by the voltagegain of the stage this capacitance must be minimized toprevent a severe degradation of themaximum intrinsic devicefrequency

Before projecting the NEMS into the purely mechanicaldomain to evaluate its dynamics let us put on record theNEMS design strategy proper tube selection (119871 = 01ndash1 120583m119889 = 2ndash5 nm) device (Al Cr or Pd electrodes minimal gate-drain coupling) and readout (minimal RCtrack) design alongwith appropriate bias (119881

119889119904= 001ndash1V119881

119892= minus5 sdot sdot sdot+5V inside

the transitionon-state 119868on = 1 120583A) Orders of magnitudeof the strength and frequency of the three purely electricalcontributions to the drain current are summarized in Table 1indicating that capacitive feed-through starts masking thetransistor effect at GHz frequencies and higher

4 Mechanical Properties of CNT Resonators

Proper model selection is the key to accurate results It hasbeen shown [42] that nonlinear continuum models yieldgood match with the more complicated molecular dynamicsmodels We here describe the CNT by an Euler-Bernoullibeammodel that accounts for the geometric nonlinearity butneither buckling nor slack Slack is anyhow an obstacle tohigh quality resonance and can be eliminated by prestrainingthe tubeThe partial differential equation of motion in termsof Youngrsquos modulus 119864 areal moment of inertia 119868 cross-sectional area 119860 stress at rest 119904

0 and damping coefficient 119888

119865ext = 1198641198681205974119908

1205971199094+ 120588119860

1205972119908

1205971199052+ 119888

120597119908

120597119905

minus (1199040119860 +

119864119860

2119871

int

119871

0

(

120597119908

120597119909

)

2

119889119909)

1205972119908

1205971199092

(6)

expresses the CNTrsquos displacement119908(119909) caused by an externalforce 119865ext [Nm]The first two terms describe the equilibriumbetween strain and kinetic energy followed by the damp-ing term and the geometric nonlinearity due to mid-planestretching The response to an external force (6) is solvedin the clamped-clamped configuration via a reduced-ordermodel based on the Galerkin procedure which is a goodcompromise between finite elements and a lumped model

41 CNT Properties and Device Design The discussion willbe limited to the design-parameter set whose values thedesigner can influence such as tube length and diameterLarge youngmoduli andmechanical quality factors of defect-free tubes are favoured

Table 1 Expected orders of magnitude of the contributions to thedrain current (for 120575119881120596in

119892= 1V)

Effect Information Frequency Amplitude (A)Mixer Electrical Δ120596 10

minus7

Mixer Motional 10038161003816100381610038161205960minus 120596in minus Δ120596

1003816100381610038161003816

10minus9

Fet Electrical 120596in 10minus7

Capacitive Electrical 120596in 10minus15120596in

Fet Motional 1205960

10minus9

Piezo Motional 1205960or 2120596

010minus8

411 Static Behaviour Figure 3 provides insight into thesteady force homogeneously distributed along the tubelength necessary to deflect the CNT transversally Midplanestretching translates into a nonconstant stiffness and theforces may span several orders of magnitude as a function ofthe tubersquos diameter With small- and medium-diameter tubesbeing the most interesting for electromechanical applications(1) forces of tens of nN will always push the tube to its elasticlimits of roughly 5 strain [43] Contrariwise a minimalforce is required to overcome the incoherent sum of allstochastic processes driving the resonator By the fluctuation-dissipation theorem and regardless of the origin of thedissipation mechanism the motion of the NEMS ultimatelythermalizes into heat Given that quantum fluctuations arenegligible at ambient temperature and radio-frequencies(119896119861119879 ≫ ℎ120596

0) [44] the classical equipartition law predicts

an average energy of 119896119861119879 per mode with 119879 being the

physical temperature of the NEMS This established ther-momechanical noise energy [45] may infer an upper boundon the thermal fluctuations 119908

119909along the tube Hypothesiz-

ing a homogeneously distributed force the fluctuations areimplicitly defined by the systemrsquos energy or explicitly by itscoenergy

119896119879 = ⟨119864119909⟩ = ⟨int

119908119909

119911=0

119865119909(119911) d119911⟩

= 119865 ⟨119908119909(119865)⟩ minus int

119865

119891=0

⟨119908119909(119891)⟩ d119891

(7)

This thermomechanical noise energy is reported inFigure 3 and illustrates the narrow linear dynamic range ofhigh aspect-ratio tubes [46] Similar displacements can bereached by driving the tube harmonically at its resonancefrequency with a force that in the linear regime is 119876 timessmaller with119876 being themechanical quality factor Althoughquality factors of 105 have been observed at cryogenic temper-atures [47] ambient temperature reduces them to about 100[5]

412 Dynamic Behaviour Theaforementioned fundamentalresonance frequency is predicted by the Euler-Bernoullimodel to scale as

119891res =1

radic3120588

radic11986412058721199032

1198714+ 1199040

1

1198712 (8)


Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100

10minus510minus610minus710minus810minus910minus1010minus1110minus1210minus4

10minus3

10minus2

10minus1

10minus1

100

L = 1120583m

L = 100nm

d = 148 nm

Figure 3 Force-displacement (continuous line) and force-strain(dashed line) relation for suspended nonprestrained CNTs withinthe elastic limit and above the thermomechanical noise floor(300K)

where 119903 is the radius 119871 is the length and 120588 is the density ofthe CNT As can be concluded from Figure 4 the mere straininduced during oscillation might be sufficient to stiffen thetube and increase its resonance frequency leading the linearprediction into considerable error The dynamic behaviourof high aspect ratio tubes is once again shown to be severelyconfined by the thermomechanical noise and the onset of thenonlinear regime

42 Mechanical Tuning Controllability of the source posi-tion opens the option to prestrain the tubeThe induced strainrelates to the applied force via the stress-strain curve with itslinear regime expression being

1199040=

Δ119871

119871

=

1

1198641205871199032119865 (9)

Although such a straining technique might be slightly lessefficient from a force-strain perspective its advantage istwofold In contrary to the transversal force [5] this longitu-dinal force can be appliedmechanically [8] during resonancemeaning that higher forces are available with no impacton the electronic terminal potentials setting the electronicoperation regime highly sensitive to the bias Additionallyslack can be compensated by pulling the tube till the onset ofstrain while preserving symmetric oscillation (Section 6)

421 Resonance Frequency Tuning and LinearizationFigure 5 illustrates what formula (8) predicts By prestrainingthe tube sufficiently the resonance frequency can be tunedover a couple of decades and turns independent of thetubersquos diameter Note that the force necessary to induce thisprestraining remains very well a function of the diameter (9)The designer has to trade off between the wider tuning rangesof long (10MHzndash1GHz) tubes and the larger linear dynamicrange of short (1 GHzndash10GHz) tubes Besides frequencytuning prestraining also allows to weaken the resonancefrequencyrsquos sensitivity to the oscillation amplitude enablingtrivial oscillation start-up at system level

422 Tube Stiffening Although this tuning option mightlook tempting designers should keep in mind that the

Oscillation half-amplitude (nm)

Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)

10210110010minus1

d = 1 2 4 6 8 nmL = 1120583m

d = 1 2 4 6 8 nmL = 100nm

1010

109

108

107

Figure 4 Resonance frequency for nonprestrainedCNTswith oscil-lation amplitudes from thermomechanical noise floor up to themaximumstrain limit (5)Theonset ofmid-plane stretching trans-lates into increasing resonance frequencies

Prestrain ()

Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)

10minus4 10minus110minus210minus3

1010

109

108

107d = 1 2 4 6 8 nm

L = 1120583m

d = 1 2 4 6 8 nmL = 100nm

100

Figure 5 The fundamental resonance frequency as a function ofprestrain is tunable over 1 (119871 = 100 nm) or 2 (119871 = 1 120583m) decades

pulling force on the tube has to be increased appropriatelyto overcome the prestraining force and deflect the tube asdepicted in Figure 6The thermo-mechanical fluctuations arereduced by the same principle It is exactly this demand forhigh force that will limit the tuning of tubes (Section 5)

The presented force-displacement relations for electri-cally interesting tubes (1) reveal that the necessary drivingforces span a wide range from 119901119873119876 to 120583119873119876 The force-strain relations impact the detection mechanisms studied inSection 5 Linear resonance frequencies range from tens ofMHz (119871 = 1 120583m) to tens of GHz (119871 = 100 nm) and tubestraining allows for tuning over a couple of decades alongwith an increase of the linear dynamic range This featurecomes at the expense of larger minimal driving forces (seeFigure 6)

5 Carbon NanotubeElectromechanical Resonators

For the CNT to serve as NEMS and the circuit to readmotional information the signal has to flow from the back-endrsquos electrical to the NEMSrsquo mechanical back into the


Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100

10minus710minus810minus910minus1010minus1110minus1210minus4

10minus2

10minus3

10minus1

10minus1

100

L = 1120583md = 1nm

L = 100nmd = 1nm

s0 = 0 01 1

Figure 6 Force-displacement (continuous line) and force-strain(dashed line) relation for suspended prestrained CNTs within theelastic limit and above the thermomechanical noise floor (300K)

frontendrsquos electrical domain Any shortcut bypassing themechanical world constitutes an undesirable parasitical feed-through

51 Actuation The roots of actuation lie in the two-portcapacitor formed by the gate electrode and the CNT itself(Figure 2) As the energy stored in this capacitor can bemodulated via the charge in the electrical domain and via thegate-CNT distance in the mechanical domain transdomainsignal flow becomes possible Neglecting the contribution ofthe density of states in the CNT and approximating the deviceas a long equipotential cylinder above an infinite plate thegate-CNT capacitance writes

119862G-CNT = int119871

0

2120587120598

acosh ((ℎ minus 119908 (119909)) 119903)d119909 (10)

where 119908(119909) is the transversal motion as a function of theposition along the tube and 120598 is the gap permittivity As weare controlling the voltage 119881 rather than the charge ourreasoning shall be based on the coenergy119882lowast(119881 119908) = 11986211988122entailing the expression of the force on the tube suspendedat height ℎ over the gate electrode

119865 = minus

120597119882lowast(119881 119908)

120597119908

= int

119871

0

1205871205981198812

radicℎ minus 119908radicℎ minus 119908 + 2119903 sdot acosh2 ((ℎ + 119903 minus 119908) 119903)d119909

(11)

511 Electrostatic Force Amplitude For small oscillationamplitude (119908(119909) ≪ ℎ) and to first order this force scales withthe device geometry as 119871ℎ with 119871 being the tube suspensionlength and ℎ its distance to the gate electrode This force isrelatively insensitive to the tube radius 119903 (supposing ℎ ≫ 119903)Formula (12) provides a good estimate on the effective forcepulling on the SWNT

log10(119865eq)

=

minus12 + log10(

119871

ℎ

) + 2log10(119881) + log

10(119876) 120596

0

minus12 + log10(

119871

ℎ

) + 2log10(V) else

(12)

Energy conservation at resonance leads to larger apparentforces compared to the static case Figures 3 and 6 translatethe equivalent mechanical force 119865eq directly into an oscil-lation amplitude and the related induced strain At roomtemperature and depending on the prestrain a pristine tube(119876 = 100) in a good setup (119871ℎ = 10) would require avoltage amplitude of 01 V to 1 V to sustain a 10minus5 to 10minus3strain variation at resonance Larger driving voltages arenot acceptable as they firstly would impact the electricaloperating point eventually switching the device from on- tooff-state and secondly might increase the electrostatic forceto a value that the elastic restoring force can no longer resistleading to a sudden collapse of the structure named dynamicpull-in [48]

512 Electrostatic Force Frequency Decomposing the drivingvoltage into a continuous 1198810

119892and an harmonic 119881120596

119892cos(120596119905)

excitation reveals that the force proportional to the voltagesquared possesses three frequency components at 0 120596 and2120596

119865eq sim ((1198810

119892)

2

+

1

2

(119881120596

119892)

2

) + (21198810

119892119881120596

119892) sdot cos (120596119905)

+ (

1

2

(119881120596

119892)

2

) sdot cos (2120596119905) (13)

If there is no accumulated charge (1198810119892= 0) or this charge

is trapped in defect states the electrostatic force only displaysa 2120596 component in the Fourier spectrum For the device to bein the on-state a nonzero 119881

119863119862might nevertheless be indis-

pensable leading to an increasingly strong 1120596 responseassuming that the excess charges have sufficient mobilityto follow the RF gate signal Consequently to excite theCNTrsquosmechanical resonance at120596

0 the necessary gate driving

frequency must be

120596in =

1205960 with excess charges

1205960

2

without excess charges(14)

52 Motion Detection The motional information can beinferred via two different physical phenomena One is theCNTrsquos conductance in a potential field and the other isthe piezoresistive property of CNTs For the investigationof both detection mechanisms we suppose the tube to bevibrating harmonically at a frequency 120596

0 while it is driven

at a frequency 120596in

521 Field Effect The channel motion in a potential fieldmodulates the Schottky barrier width and the charge inducedon the tube By the fact that the conductance change forsemiconducting [9] and small-bandgap [49] SWNTs is pro-portional to the charge variation on the tube the motion in achanging potential field influences the conductance 119866 as

120575119866 =

119892119898

119881119863119878

(120575119881120596in119892+

119881119892

119862119892

1205751198621205960

119892) (15)


Electrical Field EffectThe conductance change due to the gatevoltage variation is the purely electrical field effect analyzedin Section 34 entailing a parasitical feed-through expressedby (3) of tens of nA at the driving frequency 120596in

Motional Field effect The useful component of the field effectoriginates from the displacement 120575119911 of the tube yielding acurrent smaller than its electrical counterpart 120575119868120596infet

1205751198681205960

fet =119892119898119881119866

radicℎ minus 119908radicℎ minus 119908 + 2119903 sdot 119886 cosh ((ℎ + 119903 minus 119908) 119903)120575119911

(16)

and not exceeding the nA floor at the vibrating frequencyThis is subjected to the condition that the gate potentialoriginally reserved for biasing creates the required potentialfield

522 Piezoresistivity The piezoresistive transduction princi-ple can be traced back to the bandgap sensitivity of CNTs tostrain While axial strain moves the K-points of the rolled-up graphene sheet the Poisson ratio caters for a reductionin the tube diameter and new boundary conditions entailinga strain-dependent bandgap This strain dependence is mostpronounced in metallic zig-zag SWNTs while totally absentin metallic armchair SWNTs All other chiralities find theirbandgap sensitivity to a strain 120576 between those two extremeswith maximum sensitivities 119889119864

119892119889120576 predicted to reach up

to 100meV depending on the model [50ndash53] This quasiomnipresence turns piezoresistive component detection intoa reliable readout strategy Besides contact strain modulatingthe tunnelling barrier width and hence contact resistance theinduced strain can be sensed indirectly through the changein resistance it causes characterized through the gauge factorGF = (Δ119877119877)(1120576) With thermally activated transport beingmost sensitive on the bandgap the largest GFs are measuredin the device off-state where transport is dominated byexactly this phenomenon At symmetrical oscillation aroundthe tubersquos rest position the piezoresistive current has twice thefrequency of the mechanical vibration while their frequencyis identical otherwise

12057511986821205960|1205960

piezo = GF1205761198680119889

(17)

and reveals the existence of an optimal bias point Whilethe off-state is favourable to large GF acceptable drain biascurrent 1198680

119889requires the device to be in the on-state The

tradeoff lies in the transition state close to the maximumtransconductance bias Prestraining was shown to enhancethe GF from 856 to 2900 [54] and the larger off-currentsof large-diameter tubes (119889 gt 2 nm) are beneficial It wasshown that GFs of 100 may coexist with bias currents of 1120583Ain small-gap semiconducting SWNTs [55] leading to piezo-resistive currents of 1 to 100nA in the targeted 10minus5 to 10minus3strain region

53 Parasitical Feed-Through Finally the parasitical feed-through from the driving electrode analyzed in Section 34and scaling with frequency

120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)

comes with a strength overshadowing the motional compo-nents at frequencies higher than 100MHz While device andtrack designs minimizing the gate-drain capacitive couplingare an asset the only loophole lies in frequency separationand signal filtering as will be discussed subsequently

Among all contributions to the drain current the mo-tional field effect (120596

0) and the piezoresistive component

(1205960or 2120596

0) were shown to encode information on the

CNTrsquos motion In accordance with [6] we find the quasiomnipresent piezoresistive response more pronounced thanthe motional field effect The electrical field effect (120596in) andcapacitive coupling (120596in) on the other hand constitute the par-asitical feed-through With the latter dominating the outputsignal at frequencies above hundreds ofMHz it is essential toselect the useful frequencies via filtering Table 1 summarizesorders of magnitude of the strength and frequency of thesesignals

6 Open-Loop Resonators

The NEMSrsquo electromechanical characterisation uses to hap-pen in a laboratory context allowing the rich use of sophis-ticated equipment such as high magnetic fields [56] opticalinterferometry [57] spectrum- and network analyzers Time-averaging and mixing techniques seem to be the trend forCNT-NEMS resonator characterisation This section reviewsthe most common techniques analyzes from which drain-current component (Table 1) they infer motional informa-tion and reveals the existence of a yet unexploited 4120596mixingtechnique

61 Time-Averaging Time-averaging techniques stimulatethe NEMS with a slowly varying frequency ramp at the gatewhile a constantDCbias fixes the source and drain potentialsUpon motion the drain current is instantly modulated andif it has a nonlinear dependence on the gate voltage thevariation does not cancel out over one oscillation meaningthat the average drain-current holds precious informationabout the oscillation amplitude A short-term integration ofthe drain-current allows thus to detect resonance based onnonlinear piezoresistive [4] or electrostatic interactions in thevicinity of Coulomb oscillations within quantum dots [47]Simultaneously purely electrical contributions to the draincurrent must either react linearly to the gate voltage or befrequency independent [12]

62 Signal Mixing Similar to its electrical counterpart(Section 34) the device can also be operated as an elec-tromechanical mixer For this purpose the source terminalis driven at a frequency 119899 sdot 120596in plusmn Δ120596 while the gateis driven at 120596in In such a mixing setup the gate signaldefines the actuation while the source signal selects thedrain-current contribution to be detected via lock-in at Δ120596


Defining the CNTrsquos resonance frequency as 1205960 equation (14)

leads to resonance for an 120596in = 12059602 gate frequency in theabsence of RF modulable static charges and for a 120596in = 120596

0

actuation if such charges invade the CNT Device bias andthe presence or absence of mechanical prestrain may forcethe CNT to oscillate symmetrically or asymmetrically aroundits least-strain position with the former causing the CNTto bend twice per oscillation cycle yielding a piezoresistivedrain-current contribution at twice the oscillation frequencyTable 2 highlights the different drain-current contributionsrsquofrequency for each of these four possible situations Thesignalsrsquo strength can be read fromTable 1Three values for theratio 119899 between indirect detection and actuation frequencyallow to read motional information of which two have beensuccessfully tested and reported in the literature so far the 1120596[10] and the 2120596 [6] techniquesWe here point out that a never-mentioned 4120596 technique exists which detects CNT motionvia the piezoresistive contribution allows to determine theresonance frequency uniquely and further separates themotional from the parasitical information by a factor fourin frequency The full advantages of this new technique willstick out in closed-loop topologies (Section 7)While it seemstempting to shift the signal to low frequencies in order tocircumvent the afore mentioned high-frequency obstaclesthe price to pay is phase information loss and consequentlythe impossibility to operate the NEMS in a self-regulatingclosed-loop configuration

7 Closed-Loop Oscillators

The way towards future closed-loop operation poses twochallenges On one hand the motional information must beisolated from the parasitical one while on the other hand thesignal must be detectable from the background noise

71 Spectral Separation Although the electrical contributionsto the drain current overwhelm the motional ones the 1120596mixing technique allowed to detect resonance by the merefact that the motional information is frequency dependentwhile the electrical contributions depend relatively less or notat all on frequency [12] Using this small variation in a locallysteady large signal for closed-loop self-regulation seems notstraight-forward The 2120596 approach is extendable to closed-loop topologies in the sense that a factor two in terms offrequency separates the motional from the parasitical infor-mation To infer the mechanical resonance frequency fromthe motional information uniquely quantitative knowledgeof static charges or oscillation symmetry is indispensableThe4120596 technique would separate motional and parasitical infor-mation further and hence require less aggressive filteringAlso the mechanical frequency can be inferred uniquely Thedisadvantage of this approach remains to be the requirementfor symmetrical oscillation at the absence of static chargeswhich may or may not be compatible with acceptable devicebias (Section 33) depending on the device To transform the1 100 ratio between the motional and parasitical signals atGHz frequencies into a 10 1 proportion a 10th-order Butter-worth or a 6th-order Chebyshev band-pass filter is requiredin the 2120596 case while 5th and 4th order are respectivelynecessary in the 4120596 case

Table 2 Spectral components of the drain-current with or withoutstatic charges (119902

0 1199020) and symmetrical or asymmetrical (119904

0 1199040)

oscillation as detected by the 1120596 2120596 and 4120596mixing techniques

Motional Electrical DetectedFET Piezo Fet Capacitive 1120596 2120596 4120596

1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash

72 Minimum Detectable Signal To conclude this walk-through of CNT NEMS operation within electronic cir-cuits let us finally assess the ability of standard electronicdevices such as bipolar junction (BJT) or field effect (FET)transistors to sense and amplify the currents reported inTable 1 Supposing displacements exceeding the thermal fluc-tuations (Section 4) and device operation in theMHz to GHzband above the corner frequency [58] where white noisedominates the minimum detectable signal (MDS) dependson the signal-to-noise ratio (SNR) necessary for subsequentsignal processing the frontendrsquos noise figure (NF) and thecircuitrsquos bandwidth 119861 defined by the bandpass filter (seeFigure 1)

119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)

With the frontend being a cascade of stages its noise figure isexpressed via Friisrsquo formula

NFtotal = log10(1198651+

119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866

119894denote the noise factor and power gain

of stage 119894 NFtotal shall be minimized under the constraintof overall sufficient gain The resulting MDS shall be loweror equal to the signal provided by the CNT as reported inTable 1 Considered candidate circuits operate the BFP750a high linearity low noise Silicon-Germanium-Carbon NPNtransistor in a common emitter (CE) configuration Wemake the reasonable assumption that bias resistors exceedthe transistorrsquos base impedance up to GHz frequenciesmaking their noise contributions negligible The CNT isinterfaced by the discrete component frontend [59] as shownby the inset of Figure 7 which drives an integrated signal-processing CMOS feed-back loop [60] Hence the interfacecapacitances are of the order of 119862 = 1 pF [59] The frontendrsquostransimpedance writes

119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)

and must be able to convert a 1nA current variation into a10mV stimulus for the CMOS IC hence exceeding 107Ω


TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)

4-stages3-stages2-stages1-stage

Figure 7 Gain-constrained minimum detectable signal and NF asa function of frequency and number of stages for CE frontend andSNRout = 1

Given the CNTrsquos thermal current noise density 4119896119879119877CNTand the transistorsrsquo base 2119902119868

119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)

Completed by further constraints for acceptable transistorbias (1mA to 100mA) the constrained optimization problemis solved via sequential quadratic programming (SQP) andleads to Figure 7 The strikingly high NF is entirely ascribedto the NEMSrsquo high output impedance in combination witha picofarad interconnect capacitance Practice shows thatall but 1 dB come from the most noise-critical first stageWhile a single stage can provide the necessary gain up tofew MHz cascading is necessary for higher frequenciesAlthoughFigure 2 stageswork out up to 200MHz the use of 3stages allows to relax the gain constraint and provides notablybetter noise figures Close to optimal noise figures can beachieved by 4 stages up to GHz The optimal bias withsufficient number of stages to ensure gain lies at the lowerbound of 1mA for the BFP750 Combining this with the factthat most of the SNR degradation is due to the first stage theaddition of supplementary stages does not compromise theNF and an optimally biased 4-stage frontend will performnearly optimal for any signal frequency from MHz to GHzRequiring an output SNR of 10 at 1 GHz Figure 7 indicatesthat the CNT signal must exceed 25nA which is probably notthe case (see Table 1) Bandpass filtering which is limited tofilter quality factors below the NEMSrsquos quality factor (sim100 atroom temperature) if CNT-based oscillator applications asin Figure 1 are targeted may relax this constraint to 25 nA(119876filter = 100) and allow a 4-stage BFP750 common emitterfrontend to sense CNT motion up to GHz without phaseinformation loss This filter must follow the frontend butpreliminary filters for spectral separation (see Section 71)can interlace the frontendrsquos stages given the negligible SNRdegradation due to all but the first stage

8 Conclusion

In the scope of combining the carbon nanotube NEMSrsquoability of finemechanical sensing with the vast functionalitiesavailable in electronic circuit design synthetic guidelineson proper carbon nanotube selection NEMS resonator andreadout design along with appropriate bias and operationare at the outcome of an analysis of state-of-the-art results inthe respective domains Orders ofmagnitude of electrical andmechanical components forming theNEMSoutput spectrumwere extracted and tabulated Motion inference from thepiezoresistive current contribution was shown to be the mostreliable and a new 4120596-approach henceforth completes the setof readout techniques Signal isolation from parasitical feed-through and background noise has been shown to be possiblevia cascaded amplification without phase information lossand up to GHz frequencies This fact entails promises forhighly functional tunable and sensitive systems emergingfrom the combination of carbon nanotube NEMS with theestablished CMOS integrated circuits

Acknowledgments

This research is funded by Nano-Terach and evaluated bySNSF

References

[1] L Radushkevich and V Lukyanovich ldquoAbout the structure ofcarbon formed by thermal decomposition of carbon monoxideon iron substraterdquo Zhurnal Fizicheskoi Khimii vol 26 pp 88ndash95 1952

[2] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

[3] C KauthM Pastre andM Kayal ldquoOn-chipmass sensing at thephysical limits of nanoelectromechanical systemsrdquo in Proceed-ings of the Advances in Sensors and Interfaces pp 131ndash135 BariItaly June 2013

[4] H Chandrahalim C I Roman and C Hierold ldquoAnalytic mod-eling and piezoresistive detection theory of acoustic resonancesin carbon nanotubesrdquo inProceedings of the 10th IEEEConferenceon Nanotechnology (NANO rsquo10) pp 778ndash781 Seoul Republic ofKorea August 2010

[5] V Sazonova Y Yalsh I Ustunel D Roundy T A Arlas andP L McEuen ldquoA tunable carbon nanotube electrochemicaloscillatorrdquo Nature vol 431 no 7006 pp 284ndash287 2004

[6] H B Peng CW Chang S Aloni T D Yuzvinsky and A ZettlldquoUltrahigh frequency nanotube resonatorsrdquo Physical ReviewLetters vol 97 no 8 Article ID 087203 2006

[7] B Peng L Ding and Z Guo ldquoResonant modelling of two typesof tunable carbon nanotube electromechanical oscillatorsrdquoMicro and Nano Letters vol 5 no 6 pp 365ndash369 2010

[8] M Muoth S W Lee and C Hierold ldquoPlatform for strainableTEM-compatible MEMS-embedded carbon nanotube transis-torsrdquo in Proceedings of the 24th IEEE International Conferenceon Micro Electromechanical Systems (MEMS rsquo11) pp 83ndash86Cancun Mexico January 2011

[9] S J Tans A R M Verschueren and C Dekker ldquoRoom-tem-perature transistor based on a single carbon nanotuberdquo Naturevol 393 no 6680 pp 49ndash52 1998


[10] P L McEuen M S Fuhrer and H Park ldquoSingle-walled carbonnanotube electronicsrdquo IEEE Transactions on Nanotechnologyvol 1 no 1 pp 78ndash84 2002

[11] A Javey J Guo Q Wang M Lundstrom and H Dai ldquoBallisticcarbon nanotube field-effect transistorsrdquo Nature vol 424 no6949 pp 654ndash657 2003

[12] J Chaste L Lechner P Morfin et al ldquoSingle carbon nanotubetransistor at GHz frequencyrdquoNano Letters vol 8 no 2 pp 525ndash528 2008

[13] P Burke ldquoAn rf circuit model for carbon nanotubesrdquo inProceedings of the 2nd IEEE-NANO Conference pp 393ndash3962002

[14] KKoziol CDucati andAHWindle ldquoCarbonnanotubeswithcatalyst controlled chiral anglerdquo Chemistry of Materials vol 22no 17 pp 4904ndash4911 2010

[15] H Guo Z Bo P Banghua et al ldquoDirect growth of semicon-ducting single-walled carbon nanotube arrayrdquo Journal of theAmerican Chemical Society vol 131 no 41 pp 14642ndash146432009

[16] S Reich J Maultzsch C Thomsen and P Ordejon ldquoTight-binding description of graphenerdquo Physical Review B vol 66 no3 2002

[17] A Kleiner and S Eggert ldquoBand gaps of primarymetallic carbonnanotubesrdquo Physical Review B vol 63 no 7 Article ID 0734084 pages 2001

[18] W Song C Jeon Y S Kim et al ldquoSynthesis of bandgap-controlled semiconducting single-walled carbon nanotubesrdquoACS Nano vol 4 no 2 pp 1012ndash1018 2010

[19] C L Kane and E JMele ldquoSize shape and low energy electronicstructure of carbon nanotubesrdquo Physical Review Letters vol 78no 10 pp 1932ndash1935 1997

[20] S Datta Electronic Transport inMesoscopic Systems CambridgeUniversity Press Cambridge UK May 1997

[21] J-C Charlier X Blase and S Roche ldquoElectronic and transportproperties of nanotubesrdquo Reviews of Modern Physics vol 79 no2 pp 677ndash732 2007

[22] X Zhou J-Y Park S Huang J Liu and P L McEuen ldquoBandstructure phonon scattering and the performance limit ofsingle-walled carbon nanotube transistorsrdquo Physical ReviewLetters vol 95 no 14 Article ID 146805 2005

[23] T Durkop S A Getty E Cobas and M S Fuhrer ldquoExtraor-dinary mobility in semiconducting carbon nanotubesrdquo NanoLetters vol 4 no 1 pp 35ndash39 2004

[24] J Tersoff ldquoContact resistance of carbon nanotubesrdquo AppliedPhysics Letters vol 74 no 15 pp 2122ndash2124 1999

[25] M J OrsquoConnellCarbonNanotubes Properties and ApplicationsTaylor amp Francis Oxford UK 2006

[26] Z Yao C L Kane and C Dekker ldquoHigh-field electrical trans-port in single-wall carbon nanotubesrdquo Physical Review Lettersvol 84 no 13 pp 2941ndash2944 2000

[27] A Javey J Guo D B Farmer et al ldquoSelf-aligned ballisticmolec-ular transistors and electrically parallel nanotube arraysrdquo NanoLetters vol 4 no 7 pp 1319ndash1322 2004

[28] A Javey J Guo M Paulsson et al ldquoHigh-field quasiballistictransport in short carbon nanotubesrdquo Physical Review Lettersvol 92 no 10 Article ID 106804 2004

[29] F Leonard and J Tersoff ldquoRole of fermi-level pinning innanotube schottky diodesrdquo Physical Review Letters vol 84 no20 pp 4693ndash4696 2000

[30] Z Chen J Appenzeller J Knoch Y-M Lin and P AvourisldquoThe role of metal-nanotube contact in the performance ofcarbon nanotube field-effect transistorsrdquo Nano Letters vol 5no 7 pp 1497ndash1502 2005

[31] P Avouris Z Chen andV Perebeinos ldquoCarbon-based electron-icsrdquo Nature Nanotechnology vol 2 no 10 pp 605ndash615 2007

[32] J Guo M Lundstrom and S Datta ldquoPerformance projectionsfor ballistic carbon nanotube field-effect transistorsrdquo AppliedPhysics Letters vol 80 no 17 pp 3192ndash3194 2002

[33] M Kruger M R Buitelaar T Nussbaumer C Schonenbergerand L Forro ldquoElectrochemical carbon nanotube field-effecttransistorrdquo Applied Physics Letters vol 78 no 9 pp 1291ndash12932001

[34] Z Chen D Farmer S Xu R Gordon P Avouris and J Appen-zeller ldquoExternally assembled gate-all-around carbon nanotubefield-effect transistorrdquo IEEE Electron Device Letters vol 29 no2 pp 183ndash185 2008

[35] J Appenzeller J Knoch V Derycke R Martel S Windand P Avouris ldquoField-modulated carrier transport in carbonnanotube transistorsrdquo Physical Review Letters vol 89 no 12Article ID 126801 4 pages 2002

[36] M Radosavljevic S Heinze J Tersoff and P Avouris ldquoDrainvoltage scaling in carbon nanotube transistorsrdquo Applied PhysicsLetters vol 83 no 12 pp 2435ndash2437 2003

[37] J Appenzeller M Radosavljevic J Knoch and P AvourisldquoTunneling versus thermionic emission in one-dimensionalsemiconductorsrdquo Physical Review Letters vol 92 no 4 4 pages2004

[38] J Nygard D H Cobden M Bockrath P L McEuen and P ELindelof ldquoElectrical transport measurements on single-walledcarbon nanotubesrdquo Applied Physics A vol 69 no 3 pp 297ndash304 1999

[39] Y-M Lin J Appenzeller and P Avouris ldquoAmbipolar-to-unipolar conversion of carbon nanotube transistors by gatestructure engineeringrdquo Nano Letters vol 4 no 5 pp 947ndash9502004

[40] S Rosenblatt H Lin V Sazonova S Tiwari and P L McEuenldquoMixing at 50GHz using a single-walled carbon nanotubetransistorrdquo Applied Physics Letters vol 87 no 15 Article ID153111 3 pages 2005

[41] P J Burke ldquoAC performance of nanoelectronics towards aballistic THz nanotube transistorrdquo Solid-State Electronics vol48 no 10-11 pp 1981ndash1986 2004

[42] M Dequesnes Z Tang and N R Aluru ldquoStatic and dynamicanalysis of carbon nanotube-based switchesrdquo Journal of Engi-neeringMaterials andTechnology Transactions of theASME vol126 no 3 pp 230ndash237 2004

[43] D AWalters L M Ericson M J Casavant et al ldquoElastic strainof freely suspended single-wall carbon nanotube ropesrdquoAppliedPhysics Letters vol 74 no 25 pp 3803ndash3805 1999

[44] C Stampfer S Rotter and J Burgdorfer ldquoComment on dynamicrange of nanotube- and nanowire-based electromechanicalsystemsrdquoApplied Physics Letters vol 88 no 3 Article ID 0361012006

[45] A N Cleland and M L Roukes ldquoNoise processes in nanome-chanical resonatorsrdquo Journal of Applied Physics vol 92 no 5pp 2758ndash2769 2002

[46] H W C Postma I Kozinsky A Husain and M L RoukesldquoDynamic range of nanotube- and nanowire-based electrome-chanical systemsrdquoApplied Physics Letters vol 86 no 22 ArticleID 223105 3 pages 2005


[47] A K Huttel G A Steele B Witkamp M Poot L P Kouwen-hoven and H S J Van Der Zant ldquoCarbon nanotubes asultrahigh quality factor mechanical resonatorsrdquo Nano Lettersvol 9 no 7 pp 2547ndash2552 2009

[48] HMOuakad andM I Younis ldquoNonlinear dynamics of electri-cally actuated carbon nanotube resonatorsrdquo Journal of Compu-tational and Nonlinear Dynamics vol 5 no 1 Article ID 01100913 pages 2010

[49] C Zhou J Kong and H Dai ldquoIntrinsic electrical propertiesof individual single-walled carbon nanotubes with small bandgapsrdquo Physical Review Letters vol 84 no 24 pp 5604ndash56072000

[50] L Yang and J Han ldquoElectronic structure of deformed carbonnanotubesrdquo Physical Review Letter vol 85 no 1 pp 154ndash1572000

[51] M Huang Y Wu B Chandra et al ldquoDirect measurement ofstrain-induced changes in the band structure of carbon nan-otubesrdquo Physical Review Letters vol 100 no 13 Article ID136803 2008

[52] G Y Guo L Liu K C Chu C S Jayanthi and S Y Wu ldquoElec-tromechanical responses of single-walled carbon nanotubesInterplay between the strain-induced energy-gap opening andthe pinning of the Fermi levelrdquo Journal of Applied Physics vol98 no 4 Article ID 044311 4 pages 2005

[53] Y Yoon and J Guo ldquoAnalysis of strain effects in ballistic carbonnanotube FETsrdquo IEEE Transactions on Electron Devices vol 54no 6 pp 1280ndash1287 2007

[54] C Stampfer A Jungen R Linderman D Obergfell S RothandCHierold ldquoNano-electromechanical displacement sensingbased on single-walled carbon nanotubesrdquo Nano Letters vol 6no 7 pp 1449ndash1453 2006

[55] T Helbling Carbon Nanotube Field Effect Transistors As Elec-tromechanical Transducers vol 8 of Scientific Reports on Microand Nanosystems Der Andere Tonning Germany 2010

[56] X M H Huang C A Zormant M Mehreganyt and M LRoukes ldquoNanoelectromehanical systems nanodevicemotion atmicrowave frequenciesrdquo Nature vol 421 no 6922 article 4962003

[57] D W Carr L Sekaric and H G Craighead ldquoMeasurement ofnanomechanical resonant structures in single-crystal siliconrdquoJournal of Vacuum Science and Technology B vol 16 no 6 pp3821ndash3824 1998

[58] P G CollinsM S Fuhrer andA Zettl ldquo1f noise in carbon nan-otubesrdquoApplied Physics Letters vol 76 no 7 pp 894ndash896 2000

[59] C Kauth M Pastre and M Kayal ldquoWideband low-noise rffront-end for cnt-nems sensorsrdquo in Proceedings of the MixedDesign of Integrated Circuits and Systems Conference pp 289ndash293 IEEE Warsaw Poland May 2012

[60] C Kauth M Pastre and M Kayal ldquoA self-regulating oscillatorfor sensor operation of nanoelectromechanical systemsrdquo in Pro-ceedings of the New Circuits and Systems Conference pp 1ndash4IEEE Paris France June 2013

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DistributedSensor Networks



orders of magnitude while the underlying physics is brieflyhighlighted referring to the related state-of-the-art literature

Section 2 formalizes the tunable suspended clamped-clamped SWNT [7] with its electrodes as an electromechan-ical multiport core component of any system applicationA possible operation of this multiport within a closed-looposcillator structure is outlined The functionality of electrical(Section 3) andmechanical (Section 4) ports is first examinedisolatedly in their respective energy domains then in con-junction (Section 5) opening the gate to electronic actuationand sensing of mechanical phenomena The strength andfrequency of the dominant effects are assessed and mayserve as reference work to future NEMS application design-ers The capability of low-noise electronic circuits to senseand amplify the NEMSrsquo motional information in terms ofspectral signal separation from parasitical feed-through andminimum detectable signal is assessed for open-loop res-onators (Section 6) and all-purpose closed-loop topologies(Section 7) leading to promising conclusions (Section 8) foremerging NEMS sensors

2 The NEMS as anElectromechanical Multiport

Motivated by remarkable electromechanical sensing proper-ties the system of interest operates the SWNT as channelmaterial contacted in a transistor configuration (Figure 1)Mechanical degrees of freedom occur when the channel issuspended and allow for frequency tuning via straining [8]The CNTsrsquo current source behaviour (Figure 2) lends itselfto current-mode readout Maximal motional signal strength(piezoresistive and motional field effect) minimal parasiticsignal strength (gate-drain capacitance noise) and mini-mum intrinsic signal loss (output impedance) are desirablefrom a system-level point of view and become NEMS devicedesign objectives

With force and voltage (potential difference) as across-variables velocity and current as through-variables in themechanical and electrical energy domains respectively thiselectromechanical system presents two electrical (gate andsource potential) and two mechanical (source and channelpositions) degrees of freedom to steer interdomain energytransfer if the drain is chosen to be the electrical (poten-tial) and mechanical (position) reference This decisionroots in the fact that the movable frequency tuning elec-trode (source) presents an elevated parasitic capacitancewhich hampers high frequency readout necessary for closed-loop applications as depicted in Figure 1 The CNT is actu-ated via an electrostatic force pulling from the gate elec-trode Its motion modulates the current flowing throughit by nanoamperes This current is sensed at the drainand amplified by a low-noise frontend Given the CNTsrsquoultra-high resonance frequencies noise is integrated overa considerable bandwidth and a bandpass filter is requiredto put things right (Section 7) This filter also attenuatesundesired parasitical feed-through signals from the actuatingelectrode (Section 6) Gain and phase regulation close theloop by ensuring proper oscillation buildup and stabilizationMolecules binding to the tube or strain induced via the source

Current sensing

frontend

Voltage actuation backend

Bandpass filter

Phase and gain

regulationS

D

GCNT

Figure 1 Sensing resonator NEMS in a generic closed-loop oscilla-tor topology

CNT

G

S

DR = b

C = 1k

L = m

ΔVp

119986 (ΔVp Vg) iPiezo iFET iNoise

iCapa

iLoss

Figure 2 NEMS as electromechanical two- motional contributions(piezoresistive and motional-FET) to the drain current

may alter the CNTrsquos resonance frequency entailing sensorand voltage controlled oscillator (VCO) applications [3]

With the mechanical properties of SWNTs being pri-marily defined by the strong covalent in-plane 1199041199012 bonds(120590-bonds) while the electronic properties depend almostexclusively on the delocalized 120587-states the system qualifiesfor a preliminary isolated study in the purely electrical andmechanical domains The combination of electromechanicaleffects will be studied subsequently and reveals the efficientoperation of SWNTNEMS as electromechanical transducersand sensors

3 Carbon Nanotube Field Effect Transistor

This section highlights the most pertinent system-level pa-rameters of carbon nanotube field effect transistors (CNFET)the CNTrsquos projection into the purely electrical domain Sincetheir first demonstration at room temperature [9] a heapof related work has been reported on SWNT electronics[10] ballistic effects [11] their advance to GHz frequen-cies [12] and circuit models [13] To countersteer systemdesignerrsquos resentment felt against the plethora of state-of-the-art findings which reveal that apparently similar devices maybehave very differently orders of magnitude of the differentCNFET phenomena are extracted and intend to guide systemdesigners through proper device selection DC bias and ACoperation summarized at the end of this section

31 CNT Properties

311 Electronic Nature (Affected by Chirality) Themetallic orsemiconducting nature of SWNTs is defined by their chiralityThe lack of chirality control [14] in the growth processes cur-rently constrains CNT-based circuits to small sizes for yield




119864119892 [

eV] = 08

119889 [nm] (1)




0














119889119904





119892close to 119881


119889lead to band








119889119904


119881119889119904max [V] = (119864

13









120575119868120596infet = 119892119898 sdot 120575119881

120596in119892 (3)


119879= 1198921198981198901198972120587119862119892119889 To read the GHz


119879asymp



119868Δ120596

mix =1

2

119892119898120575119881120596inminusΔ120596119904

120575119881120596in119892

(4)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (5)



119889119904= 001ndash1V119881






119865ext = 1198641198681205974119908

1205971199094+ 120588119860

1205972119908

1205971199052+ 119888

120597119908

120597119905

minus (1199040119860 +

119864119860

2119871

int

119871

0

(

120597119908

120597119909

)

2

119889119909)

1205972119908

1205971199092

(6)




119892= 1V)


minus7


1003816100381610038161003816

10minus9




10minus9


010minus8







119896119879 = ⟨119864119909⟩ = ⟨int

119908119909

119911=0

119865119909(119911) d119911⟩

= 119865 ⟨119908119909(119865)⟩ minus int

119865

119891=0

⟨119908119909(119891)⟩ d119891

(7)



119891res =1

radic3120588

radic11986412058721199032

1198714+ 1199040

1

1198712 (8)


Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus3

10minus2

10minus1

10minus1

100

L = 1120583m

L = 100nm

d = 148 nm




1199040=

Δ119871

119871

=

1

1198641205871199032119865 (9)





Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)

10210110010minus1

d = 1 2 4 6 8 nmL = 1120583m

d = 1 2 4 6 8 nmL = 100nm

1010

109

108

107


Prestrain ()

Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)


1010

109

108

107d = 1 2 4 6 8 nm

L = 1120583m

d = 1 2 4 6 8 nmL = 100nm

100







Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus2

10minus3

10minus1

10minus1

100

L = 1120583md = 1nm

L = 100nmd = 1nm

s0 = 0 01 1




119862G-CNT = int119871

0

2120587120598

acosh ((ℎ minus 119908 (119909)) 119903)d119909 (10)


119865 = minus

120597119882lowast(119881 119908)

120597119908

= int

119871

0

1205871205981198812


(11)


log10(119865eq)

=

minus12 + log10(

119871

ℎ

) + 2log10(119881) + log

10(119876) 120596

0

minus12 + log10(

119871

ℎ

) + 2log10(V) else

(12)



119892and an harmonic 119881120596

119892cos(120596119905)


119865eq sim ((1198810

119892)

2

+

1

2

(119881120596

119892)

2

) + (21198810

119892119881120596

119892) sdot cos (120596119905)

+ (

1

2

(119881120596

119892)

2

) sdot cos (2120596119905) (13)






frequency must be

120596in =


1205960

2






120575119866 =

119892119898

119881119863119878

(120575119881120596in119892+

119881119892

119862119892

1205751198621205960

119892) (15)




1205751198681205960

fet =119892119898119881119866


(16)





12057511986821205960|1205960

piezo = GF1205761198680119889

(17)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)




(1205960or 2120596










0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119864119892 [

eV] = 08

119889 [nm] (1)




0














119889119904





119892close to 119881


119889lead to band








119889119904


119881119889119904max [V] = (119864

13









120575119868120596infet = 119892119898 sdot 120575119881

120596in119892 (3)


119879= 1198921198981198901198972120587119862119892119889 To read the GHz


119879asymp



119868Δ120596

mix =1

2

119892119898120575119881120596inminusΔ120596119904

120575119881120596in119892

(4)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (5)



119889119904= 001ndash1V119881






119865ext = 1198641198681205974119908

1205971199094+ 120588119860

1205972119908

1205971199052+ 119888

120597119908

120597119905

minus (1199040119860 +

119864119860

2119871

int

119871

0

(

120597119908

120597119909

)

2

119889119909)

1205972119908

1205971199092

(6)




119892= 1V)


minus7


1003816100381610038161003816

10minus9




10minus9


010minus8







119896119879 = ⟨119864119909⟩ = ⟨int

119908119909

119911=0

119865119909(119911) d119911⟩

= 119865 ⟨119908119909(119865)⟩ minus int

119865

119891=0

⟨119908119909(119891)⟩ d119891

(7)



119891res =1

radic3120588

radic11986412058721199032

1198714+ 1199040

1

1198712 (8)


Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus3

10minus2

10minus1

10minus1

100

L = 1120583m

L = 100nm

d = 148 nm




1199040=

Δ119871

119871

=

1

1198641205871199032119865 (9)





Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)

10210110010minus1

d = 1 2 4 6 8 nmL = 1120583m

d = 1 2 4 6 8 nmL = 100nm

1010

109

108

107


Prestrain ()

Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)


1010

109

108

107d = 1 2 4 6 8 nm

L = 1120583m

d = 1 2 4 6 8 nmL = 100nm

100







Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus2

10minus3

10minus1

10minus1

100

L = 1120583md = 1nm

L = 100nmd = 1nm

s0 = 0 01 1




119862G-CNT = int119871

0

2120587120598

acosh ((ℎ minus 119908 (119909)) 119903)d119909 (10)


119865 = minus

120597119882lowast(119881 119908)

120597119908

= int

119871

0

1205871205981198812


(11)


log10(119865eq)

=

minus12 + log10(

119871

ℎ

) + 2log10(119881) + log

10(119876) 120596

0

minus12 + log10(

119871

ℎ

) + 2log10(V) else

(12)



119892and an harmonic 119881120596

119892cos(120596119905)


119865eq sim ((1198810

119892)

2

+

1

2

(119881120596

119892)

2

) + (21198810

119892119881120596

119892) sdot cos (120596119905)

+ (

1

2

(119881120596

119892)

2

) sdot cos (2120596119905) (13)






frequency must be

120596in =


1205960

2






120575119866 =

119892119898

119881119863119878

(120575119881120596in119892+

119881119892

119862119892

1205751198621205960

119892) (15)




1205751198681205960

fet =119892119898119881119866


(16)





12057511986821205960|1205960

piezo = GF1205761198680119889

(17)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)




(1205960or 2120596










0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119889119904





119892close to 119881


119889lead to band








119889119904


119881119889119904max [V] = (119864

13









120575119868120596infet = 119892119898 sdot 120575119881

120596in119892 (3)


119879= 1198921198981198901198972120587119862119892119889 To read the GHz


119879asymp



119868Δ120596

mix =1

2

119892119898120575119881120596inminusΔ120596119904

120575119881120596in119892

(4)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (5)



119889119904= 001ndash1V119881






119865ext = 1198641198681205974119908

1205971199094+ 120588119860

1205972119908

1205971199052+ 119888

120597119908

120597119905

minus (1199040119860 +

119864119860

2119871

int

119871

0

(

120597119908

120597119909

)

2

119889119909)

1205972119908

1205971199092

(6)




119892= 1V)


minus7


1003816100381610038161003816

10minus9




10minus9


010minus8







119896119879 = ⟨119864119909⟩ = ⟨int

119908119909

119911=0

119865119909(119911) d119911⟩

= 119865 ⟨119908119909(119865)⟩ minus int

119865

119891=0

⟨119908119909(119891)⟩ d119891

(7)



119891res =1

radic3120588

radic11986412058721199032

1198714+ 1199040

1

1198712 (8)


Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus3

10minus2

10minus1

10minus1

100

L = 1120583m

L = 100nm

d = 148 nm




1199040=

Δ119871

119871

=

1

1198641205871199032119865 (9)





Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)

10210110010minus1

d = 1 2 4 6 8 nmL = 1120583m

d = 1 2 4 6 8 nmL = 100nm

1010

109

108

107


Prestrain ()

Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)


1010

109

108

107d = 1 2 4 6 8 nm

L = 1120583m

d = 1 2 4 6 8 nmL = 100nm

100







Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus2

10minus3

10minus1

10minus1

100

L = 1120583md = 1nm

L = 100nmd = 1nm

s0 = 0 01 1




119862G-CNT = int119871

0

2120587120598

acosh ((ℎ minus 119908 (119909)) 119903)d119909 (10)


119865 = minus

120597119882lowast(119881 119908)

120597119908

= int

119871

0

1205871205981198812


(11)


log10(119865eq)

=

minus12 + log10(

119871

ℎ

) + 2log10(119881) + log

10(119876) 120596

0

minus12 + log10(

119871

ℎ

) + 2log10(V) else

(12)



119892and an harmonic 119881120596

119892cos(120596119905)


119865eq sim ((1198810

119892)

2

+

1

2

(119881120596

119892)

2

) + (21198810

119892119881120596

119892) sdot cos (120596119905)

+ (

1

2

(119881120596

119892)

2

) sdot cos (2120596119905) (13)






frequency must be

120596in =


1205960

2






120575119866 =

119892119898

119881119863119878

(120575119881120596in119892+

119881119892

119862119892

1205751198621205960

119892) (15)




1205751198681205960

fet =119892119898119881119866


(16)





12057511986821205960|1205960

piezo = GF1205761198680119889

(17)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)




(1205960or 2120596










0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (5)



119889119904= 001ndash1V119881






119865ext = 1198641198681205974119908

1205971199094+ 120588119860

1205972119908

1205971199052+ 119888

120597119908

120597119905

minus (1199040119860 +

119864119860

2119871

int

119871

0

(

120597119908

120597119909

)

2

119889119909)

1205972119908

1205971199092

(6)




119892= 1V)


minus7


1003816100381610038161003816

10minus9




10minus9


010minus8







119896119879 = ⟨119864119909⟩ = ⟨int

119908119909

119911=0

119865119909(119911) d119911⟩

= 119865 ⟨119908119909(119865)⟩ minus int

119865

119891=0

⟨119908119909(119891)⟩ d119891

(7)



119891res =1

radic3120588

radic11986412058721199032

1198714+ 1199040

1

1198712 (8)


Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus3

10minus2

10minus1

10minus1

100

L = 1120583m

L = 100nm

d = 148 nm




1199040=

Δ119871

119871

=

1

1198641205871199032119865 (9)





Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)

10210110010minus1

d = 1 2 4 6 8 nmL = 1120583m

d = 1 2 4 6 8 nmL = 100nm

1010

109

108

107


Prestrain ()

Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)


1010

109

108

107d = 1 2 4 6 8 nm

L = 1120583m

d = 1 2 4 6 8 nmL = 100nm

100







Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus2

10minus3

10minus1

10minus1

100

L = 1120583md = 1nm

L = 100nmd = 1nm

s0 = 0 01 1




119862G-CNT = int119871

0

2120587120598

acosh ((ℎ minus 119908 (119909)) 119903)d119909 (10)


119865 = minus

120597119882lowast(119881 119908)

120597119908

= int

119871

0

1205871205981198812


(11)


log10(119865eq)

=

minus12 + log10(

119871

ℎ

) + 2log10(119881) + log

10(119876) 120596

0

minus12 + log10(

119871

ℎ

) + 2log10(V) else

(12)



119892and an harmonic 119881120596

119892cos(120596119905)


119865eq sim ((1198810

119892)

2

+

1

2

(119881120596

119892)

2

) + (21198810

119892119881120596

119892) sdot cos (120596119905)

+ (

1

2

(119881120596

119892)

2

) sdot cos (2120596119905) (13)






frequency must be

120596in =


1205960

2






120575119866 =

119892119898

119881119863119878

(120575119881120596in119892+

119881119892

119862119892

1205751198621205960

119892) (15)




1205751198681205960

fet =119892119898119881119866


(16)





12057511986821205960|1205960

piezo = GF1205761198680119889

(17)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)




(1205960or 2120596










0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus3

10minus2

10minus1

10minus1

100

L = 1120583m

L = 100nm

d = 148 nm




1199040=

Δ119871

119871

=

1

1198641205871199032119865 (9)





Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)

10210110010minus1

d = 1 2 4 6 8 nmL = 1120583m

d = 1 2 4 6 8 nmL = 100nm

1010

109

108

107


Prestrain ()

Fund

amen

tal r

eson

ance

freq

uenc

y (H

z)


1010

109

108

107d = 1 2 4 6 8 nm

L = 1120583m

d = 1 2 4 6 8 nmL = 100nm

100







Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus2

10minus3

10minus1

10minus1

100

L = 1120583md = 1nm

L = 100nmd = 1nm

s0 = 0 01 1




119862G-CNT = int119871

0

2120587120598

acosh ((ℎ minus 119908 (119909)) 119903)d119909 (10)


119865 = minus

120597119882lowast(119881 119908)

120597119908

= int

119871

0

1205871205981198812


(11)


log10(119865eq)

=

minus12 + log10(

119871

ℎ

) + 2log10(119881) + log

10(119876) 120596

0

minus12 + log10(

119871

ℎ

) + 2log10(V) else

(12)



119892and an harmonic 119881120596

119892cos(120596119905)


119865eq sim ((1198810

119892)

2

+

1

2

(119881120596

119892)

2

) + (21198810

119892119881120596

119892) sdot cos (120596119905)

+ (

1

2

(119881120596

119892)

2

) sdot cos (2120596119905) (13)






frequency must be

120596in =


1205960

2






120575119866 =

119892119898

119881119863119878

(120575119881120596in119892+

119881119892

119862119892

1205751198621205960

119892) (15)




1205751198681205960

fet =119892119898119881119866


(16)





12057511986821205960|1205960

piezo = GF1205761198680119889

(17)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)




(1205960or 2120596










0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Force (N)

Max

disp

lace

men

t (nm

)

Indu

ced

strai

n (

)

102

101

100


10minus2

10minus3

10minus1

10minus1

100

L = 1120583md = 1nm

L = 100nmd = 1nm

s0 = 0 01 1




119862G-CNT = int119871

0

2120587120598

acosh ((ℎ minus 119908 (119909)) 119903)d119909 (10)


119865 = minus

120597119882lowast(119881 119908)

120597119908

= int

119871

0

1205871205981198812


(11)


log10(119865eq)

=

minus12 + log10(

119871

ℎ

) + 2log10(119881) + log

10(119876) 120596

0

minus12 + log10(

119871

ℎ

) + 2log10(V) else

(12)



119892and an harmonic 119881120596

119892cos(120596119905)


119865eq sim ((1198810

119892)

2

+

1

2

(119881120596

119892)

2

) + (21198810

119892119881120596

119892) sdot cos (120596119905)

+ (

1

2

(119881120596

119892)

2

) sdot cos (2120596119905) (13)






frequency must be

120596in =


1205960

2






120575119866 =

119892119898

119881119863119878

(120575119881120596in119892+

119881119892

119862119892

1205751198621205960

119892) (15)




1205751198681205960

fet =119892119898119881119866


(16)





12057511986821205960|1205960

piezo = GF1205761198680119889

(17)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)




(1205960or 2120596










0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1205751198681205960

fet =119892119898119881119866


(16)





12057511986821205960|1205960

piezo = GF1205761198680119889

(17)





120575119868120596incap = 119862G-CNT120596in120575119881

120596in119892 (18)




(1205960or 2120596










0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0







0 1199040)



1199020 1199040

1205960

1205960

12059602 120596

02 120596

02 120596

0mdash

1199020 1199040

1205960

21205960

12059602 120596

02 120596

02 120596

021205960

1199020 1199040

1205960

1205960

1205960

1205960

1205960

mdash mdash1199020 1199040

1205960

21205960

1205960

1205960

1205960

21205960

mdash


119868CNTRMS ge radic4119896119879119861

119877CNTsdot 10

NFsdot SNRout (19)



119899

sum

119894=2

119865119894minus 1

prod119894minus1

119895=1119866119895

) (20)

where 119865119894and 119866



119860ΩCE =

1

1 + 119904ℎfe1 sdot 1198621198921198981sdot

119899

prod

119894=1

ℎfe119894 sdot1

119904119862

(21)



TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










TNC Rbup1

Rbdn1

Cc1

Rc1

Re1 Ce1

Stage 1 Stage n

Rbupn

CenRen

Ccn

Rcn

Rbdnn

10minus8

10minus9

10minus10

106 107 108 10920

25

30

35

40

45

I CN

Tm

inRM

S(A

)

NF C

E(d

B)




119861

NFCE = log10(1 +

119877CNT10038161003816100381610038161 + 119904ℎfe1 sdot 1198621198921198981

1003816100381610038161003816

sdot

119899

sum

119894=1

119892119898119894

prod119894

119895=1ℎfe119895

)

(22)


8 Conclusion


Acknowledgments


References

































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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