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A novel approach to high-speed high-resolution on-chip mass sensing
Microelectronics Journal 45 (2014) 1648–1655
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Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
A novel approach to high-speed high-resolution on-chip mass sensing C. Kauth n, M. Pastre, M. Kayal STI-IEL Electronics Laboratory, Ecole Polytechnique Fédérale de Lausanne, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Received 26 December 2013 Accepted 14 July 2014 Available online 2 August 2014
The state-of-the-art mass sensing so far has been rather developed along the resolution axis, reaching atomic-scale detection, than into the direction of high-speed. This paper reports a novel self-calibrating technique, making high-speed inertial mass sensors capable of instant high-resolution particle detection and weighing. The sensing nanoelectromechanical resonator is embedded into a phase-locked loop and the sensor-inherent nonlinear phase–frequency relation is exploited for auto-calibration. A tunable onchip carbon nanotube based mass balance serves as a case study of small-size and low-cost environmental and healthcare applications. Tunability and a phase-locked loop topology make the system widely universal and invariant to nanotube characteristics. Operational for tube eigenfrequencies up to 385 MHz, the circuit integration in a 180 nm technology achieves instantaneous zeptogram resolution, while yoctogram precision is obtained within the tenth of a second. These figures of merit range at the physical limits of carbon nanotube resonators, in both mass- and time-resolution. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Analytical models Carbon nanotubes Closed loop systems Nanoelectromechanical systems Nonlinear systems Oscillators Phase locked loops System analysis and design
1. Introduction The state-of-the-art mass sensing is predicated on the detection of a shift in a resonator's eigenfrequency when an additional mass binds to its surface. Evaluation of the theoretical ultimate limits to such inertial mass sensing [1] suggests that single proton weighing is feasible at room temperature. This yoctogram mass resolution has been achieved [2] at liquid-helium temperature with an ultrasensitive carbon nanotube resonator, operated in a mixer setup [3]. The equivalent room-temperature record ranges at a hundredfold worse resolution [4]. These high-resolution sensors find applications in atomic physics, biology and life science as molecule identifiers, gas detectors and cell weighing to name but a few. In contrast to this impressive mass resolution, the temporal resolution of such sensors appears unspectacular. Frequency sweeps that last several seconds [5] and feedback loops of tens of milliseconds [2] set the benchmark hitherto. By comparing this sensing speed to the quasi-gigahertz NEMS eigenfrequency, it becomes apparent that there is immense potential to improve sensing speed by several orders of magnitude. A novel approach to simultaneous high-speed and high-resolution nanomechanical mass sensing is presented in this contribution. Today's state-of-the-art mass sensors do not combine these two desirable properties yet: high-resolution methods [2] rely on frequency sweeps or mixing, which makes them inherently slow, while high-speed methods [6] present non-quantifiable offsets, which lead
n
Corresponding author. E-mail address: christian.kauth@epfl.ch (C. Kauth).
http://dx.doi.org/10.1016/j.mejo.2014.07.004 0026-2692/& 2014 Elsevier Ltd. All rights reserved.
to erroneous mass inference. An auto-calibrating system is shown to compensate offsets and hence enables high-resolution high-speed sensing. Such precise and fast mass detection allows observation of phenomena that so far are not observable because they are either too subtle (insufficient sensor resolution) or happen too fast (insufficient sensor speed), like chemical and biochemical reactions. The concept condensates into an implementable topology and suspended carbon nanotubes (CNT) are the resonator of choice, favoured by their low effective mass and acceptable quality factor. Their high elasticity allows us to tune the CNT for maximal compatibility with the interface circuitry via mechanical strain. Feedback electronics, implemented in a 180 nm technology, form a phase-locked loop (PLL) around the CNT-NEMS resonator in order to drive and sustain the latter's motion at resonance. Metamorphosed into small-size and low-cost sensor nodes for large-scale healthcare or environmental applications, this on-chip sensor circuit presents unprecedented speed, and mass resolution down to the physical limits of nanoelectromechanical systems. Section 2 brings forward the key elements for high-speed and highresolution mass sensing, and presents a novel topology combining both features. The physical phenomenon at the basis of this self-calibrating sensor topology is investigated in Section 3. Section 4 demonstrates that this nonlinear phenomenon can indeed be exploited in practice for calibration and the operation principle of the resulting high-resolution high-speed mass sensor is explained in Section 5. Section 6 starts a case study illustrating how suspended carbon nanotubes can fulfill the role of the sensing NEMS. The motional information processing system and its integration are the topics of Section 7. Section 8 identifies which CNTs are adequate for the proposed chip and translates the latter's jitter into sensor performance in terms of resolution and speed.
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2. Motivation and novelty Fast and precise identification of the NEMS eigenfrequency ω0 may be modelled as a search problem. A questioner interrogates an oracle 8 > < 1 if ω oω0 if ω ¼ ω0 ð1Þ O : ω↦ 0 > : þ 1 if ω 4ω 0 on whether the unknown eigenfrequency ω0 equals an arbitrary guess ω. The oracle answers with 1, 0 or þ 1 as a function of the relative position of ω and ω0. Two key parameters to high-speed high-resolution mass sensing crystallize from this formal description of the eigenfrequency search. First must the oracle provide fast and trustworthy responses, and second shall the questioner interpret the answer quickly and rapidly to improve the quality of the guess. The fastest way to determine a periodic signal's frequency is to measure exactly one period. Hence high NEMS eigenfrequencies are beneficial for the first criterion and suspended carbon nanotubes are a neat choice, as shown in Section 6. Noise may distort the oracle's answer, but can be countervailed by averaging over several periods. The resulting speed-precision trade-off is assessed in Section 8. The current section's emphasis lies on the inspection of techniques to rapidly observe the oracle's answer and formulate a precise guess on ω0. The following observations are independent of the exact NEMS nature and averaging. A generic dynamic NEMS may be modelled as a normalized damped harmonic oscillator ω0 1 ( 1 2 jHðsÞj ¼ A Q 4Q HðsÞ ¼ ; ð2Þ ω0 argðHðsÞÞ ¼ ϕ s2 þ s þ ω20 Q with eigenfrequency ω0 and quality factor Q. The eigenfrequency is observable via detection of the amplitude A peak at ωp or the phase ϕ inflection at ωi: vs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 1 t ð3Þ 4 2 1 ω p ¼ ω0 1 ; ωi ¼ ω 0 2Q 2 Q ωp ωi ω0 for sufficiently large Q. Adsorption of a particle onto the oscillating NEMS causes a relative change Δmeff =meff in the oscillator's effective mass, defined by the particle's weight and the binding position [7], and shifts the resonance characteristics as pffiffiffi pffiffiffi ðω0 ; Q Þ-ðω0 = α, αQ ), with α ¼ 1 þ ðΔm=mÞ. This shift in eigenfrequency is to be detected as fast and precise as possible. Today's techniques of doing so may be systematically categorized into
ω-controllable techniques drive the NEMS at a given frequency ω and observe its amplitude A or phase ϕ response. Such approaches benefit from a reduced noise bandwidth when
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operating the NEMS as a mixer [5]. The mixed signal out of the NEMS, however, has a much lower frequency than ω0. This makes mixing incompatible with the high-speed detection exigence, and feedback loops take tens of milliseconds [2] to improve the estimation of ω0. High-resolution weighing is enabled by the simultaneous extraction of ωp or ωi and Q from a response plot around the resonance peak [2], which takes seconds. It is concluded that ω-controllable methods are precise, but slow. ϕ-controllable techniques embed the NEMS into a phase locked loop (PLL) and loop its output onto its input [4,8], with a controllable phase shift Φ, which forces the NEMS to oscillate at a unique frequency ω. An adsorption-related shift in ω is directly observable, enabling high-speed weighing, but the error in the observed Δω0 scales as 1=Q tan ðϕÞ. If ϕ ¼ π=2, the error vanishes, but this ideal hypothesis is hard to satisfy in practice, where component delays play a role and are a priori hardly entirely corrigible. It is concluded that ϕ-controllable methods are fast, but have limited resolution.
As a matter of fact could the high-resolution and high-speed sensing hitherto be combined yet. An novel approach is shown in Fig. 1(a), where in situ calibration of the NEMS phase ϕ via a nonlinear technique adds precision to ϕ-controllable methods and opens the gate to high-speed high-resolution mass sensing. The NEMS is embedded into a phase locked loop, formed by a phase–frequency detector (PFD), a loop filter (F PLL ) and a voltage controlled oscillator (VCO). At steady state, the VCO oscillates at a frequency for which both feedback paths, one formed by the NEMS and the readout electronics [9], the other by a simple phase shifter, have exactly the same delay. Hence the NEMS is forced to oscillate at a precise frequency, at which ϕ satisfies the above constraint. This frequency depends consequently on the loop delay Φ. The additional feedback adjusts Φ so as to centre resonance at ωi, quasi-equal to ω0 for NEMS with decent Q. Fig. 1 (b) illustrates how the auto-calibration circuit adjusts the loop phase Φ so as to bring the NEMS phase ϕ to π=2, which reflects oscillation at ω0 and error-free sensing. Once the NEMS has been calibrated, Φ may simply be memorized and sensing happens precisely at the speed of ϕ-controllable methods. The underlying hypothesis for this memorization to work out is that the NEMS phase is far more sensitive to the oscillation frequency than any other circuit component. This condition is tacitly met in ϕ-controllable methods, as the PLL locks onto a frequency imposed by the circuit component with the highest quality factor, speak phase sensitivity, which must be the NEMS, if sensor operation is envisioned.
3. Physical phenomenon The technique to enable calibration of the feedback loop phase shift Φ such that NEMS oscillation at ωi is enforced roots in the
Fig. 1. Auto-calibrating mass balance. (a) System architecture. (b) Calibration process.
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that amplifies signals above ωL , differentiates till ωd and integrates above ωd. The choice of ζ¼0.65 stands for maximally accurate derivation at minimal noise bandwidth performance loss [10]. Saturation of ∂t2 ω yields a bipolar digital signal ( 71) with phasedependent average voltage V Δϕ ðϕÞ, depicted in Fig. 3(b), negative below, positive above and zero at ϕi, which is notably independent of the NEMS quality factor. The phase-to-voltage conversion gain K ϕ ¼ ∂ϕ V Δϕ depends on the modulation amplitude and is given by Fig. 3(c), in absolute terms for ideal differentiators (ωd =ω0 ¼ 1) and relatively to this ideal value for the limited bandwidth differentiator of Fig. 3(b). Fig. 2. Impact of phase shift on ω for ϕ A ϕi 7 8π ; ϕi and Δϕ ¼ 5π .
5. NEMS mass balance system nonlinearity of the NEMS phase–frequency relation 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω0 π > > ½1 þ 1 þ ð2Q tan ðϕÞÞ2 if ϕ A π; ½ > < 2Q tan ðϕÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω¼ ω0 π > 2 > > : 2Q tan ðϕÞ ½1 1 þ ð2Q tan ðϕÞÞ if ϕ A 2 ; 0½
ð4Þ
derived from Eq. (2). The reasonable hypothesis that the NEMS presents the highest Q among all loop elements makes a sinusoidal modulation of the loop phase Φ to cause an identical modulation of the NEMS phase ϕ, which results in a variation of its frequency ω, as Fig. 2 reveals. The key observation is that the frequency's inflection points, distinguishable as the extrema of its time derivative ∂t ω or zeros of ∂t2 ω, are spread over a period with an irregularity defined by the signed average distance between ϕ and ϕi. Is the NEMS oscillating below ωi, then the down-time of ∂t2 ω exceeds its up-time, while the reverse is true for frequencies above ωi and equilibrium is obtained at ωi.
4. Controllability and observability Practical implementation of the calibration-loop consequently requires Φ to be controllable and ∂t2 ω to be observable. 4.1. Controllability of Φ The phase modulation can be performed by inserting the circuit of Fig. 3(a) s 1 1 1 ωt ¼ with ωt ¼ TðsÞ ¼ s RC ðR þ ΔR sin ðωm tÞÞC 0 þ1 ωt
ð5Þ
into the NEMS signal path. A sinusoidal control of R at a frequency ωm ⪡ω0 introduces a highly linear delay 2 arctanðω0 =ωt Þ 2ω0 =ωt with variational amplitude Δϕ ¼ 2ω0 ΔRC and no impact on the signal amplitude jTðsÞj ¼ 1. 4.2. Observability of ∂t2 ω Information on the NEMS frequency is available through the control voltage of the VCO in the topology of Fig. 1(a), which consists of the ϕ-controllable technique of [8], completed by the calibration loop of Φ. If the phase modulation frequency ωm is chosen such that ωm ⪡ωPLL ⪡ω0 , then the impact of Φ on the NEMS frequency is retrievable from the low-pass filtered VCO control voltage, which unveils ∂t2 ω after two differentiations, performed by the second order bandpass filter of Fig. 3(b) DðsÞ ¼
ω2d
s ωL s2 þ 2ζωd s þ ω2d
ð6Þ
The proposed auto-calibrating mass balance of Fig. 1(a) omits frequency division, necessary to lock on the field effect or the piezoresistive NEMS signal, as will be shown in Section 6, for the sake of clarity. A periodic modulation of the delay Δϕ modulates the NEMS frequency nonlinearly. PFD-spikes are removed via a lowpass filter F spike ðsÞ and DðsÞ differentiates the nonlinearity twice, before saturating it. The calibration-loop is closed by a filter F Φ ðsÞ that adjusts the NEMS phase ϕ to ϕi. The loop acts as a PLL tracking ϕi, which is certainly unknown, but implicitly defined by the NEMS and made observable through the injected phase modulation. Given the involved frequencies, the PFD and F PLL ðsÞ are transparent to this loop and TðsÞ has unity gain. A low frequency (ωc ⪡ωm ) third order PLL is necessary to prevent interference between the injected modulation and the resulting compensation of Φ. The calibration loop filter F Φ ðsÞ may consist of a zero-order integrator (unity gain at K=K ϕ β), a first-order integrator (zero at α) and a lowpass filter (pole at β), forming the closed-loop transfer function: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 sin φ > K ¼ ð2πωc Þ2 11 þ > sin φ > < ϕ Kðs þ αÞ ; ¼ α ¼ 2πωc 1 cossinφ φ ϕi s3 þ βs2 þ Ks þ Kα > > > : β ¼ 2πωc cos φ 1 sin φ where ωc is the loop bandwidth and φ the phase margin. The graphs of Fig. 1(b) illustrate the settling of a critically damped third order (ωc ¼ 10 5 ω0 , φ ¼ 501) calibration loop for a NEMS embedded in a critically damped second order PLL (ωPLL ¼ 10 1 ω0 , Q PLL ¼ 12), presented in Section 7. The VCO centre frequency is 0.9ω0, the gain 0.1ω0, and the injected variation (Δϕ¼ π=20, ωm ¼10 3 ω0 ). Upon convergence, the calibration loop may be disconnected to remove the modulation-induced residual ripple.
6. CNT-NEMS resonator Now that it has been shown that the system of Fig. 1(a) allows for high-speed and high-resolution mass sensing, we take a closer look at suspended carbon nanotube NEMS, as they were identified as promising candidates for sensing. First we demonstrate how motional information is converted into an electrical signal, then we show how inertial sensing and tube straining affect the dynamics. 6.1. Transduction mechanism The sensor resonator is a CNT of radius r suspended at a height h over a trench of length L. Its extremes are denoted as source and drain in analogy to a MOS transistor. The gate is formed by a nearby electrode and exerts an electrostatic force onto the NEMS,
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Fig. 3. Phase modulation for controllability and signal differentiation for observability. (a) Generation of phase shift ϕ ¼ ϕ þ Δϕ sin ðωm tÞ. (b) Average of sgnð∂t2 ωÞ. (c) Phase– voltage gain K ϕ jϕi ðV=radÞ.
expressed as FðtÞ ¼
VðtÞ2 πϵ0 L 2 h h arccosh r
ð7Þ
where ϵ0 is the vacuum permittivity. This force accelerates the NEMS as a driven harmonic oscillator, balancing the driving, restoring and frictional forces: ∂t2 x^ þ2ζω0 ∂t x^ þω20 x^ ¼
FðtÞ meff
ð8Þ
where the damping ratio ζ ¼ 1=2Q is inversely proportional to the NEMS quality factor Q. Supposing a sinusoidal gate voltage VðtÞ, the driving force at twice that frequency (see Eq. (7)) causes an oscillation half-amplitude at the middle of the beam equal to F0 ^ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðtÞ sin ðωt þ ϕÞ meff ð2ωω0 ζÞ2 þ ðω20 ω2 Þ2
ð9Þ
ð11Þ
and the ubiquitous piezoresistive property of CNTs renders it detectable due to its effect on the current through the tube δI piezo ðtÞ ¼ I 0 βϵðtÞ
δI piezo V 3 5 δI cap ω0
ð12Þ
where I0 is the bias current and a linear dependence of the CNT resistance on the gauge factor β is hypothesized. This
ð13Þ
indicates that higher driving voltages are necessary to compensate for the parasitical gate-drain coupling of high-frequency resonators.
6.2. Sensitivity and frequency tuning The hypothesized mode shape of Eq. (10) entails the expression of the effective mass: meff ¼
delayed with respect to the drive by an angle within ½ π; 0 ϕ ¼ arctanð2ωω0 ζ=ðω2 ω20 ÞÞ. The CNT's shape may be approximated by [11] ^ xðtÞ 2πl 1 cos ð10Þ xðl; tÞ ¼ 2 L This motion of the tube induces a strain π 2 ^ xðtÞ ϵðtÞ ¼ 2L
piezoresistive current has a fourfold frequency with respect to the gate voltage (see Eqs. (11) and (7)), allowing a bandpass filter to spectrally separate the motional information from any parasitical capacitive coupling between the gate and drain electrodes. The piezoresistive to capacitive signal ratio scaling as
3 2 πr Lρ 8
ð14Þ
with ρ being the true density of a single CNT. This effective mass, along with the effective spring constant, defines the NEMS eigenfrequency, derived from [11] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ffi ω0 E π r ΔL ¼ f0 ¼ þ ð15Þ L 2π 3ρL2 L2 where E is the CNT Young modulus and ΔL an elongation of the tube induced by stretching. From Eq. (15) it becomes apparent that the eigenfrequency is tunable, a feature that allows us to extend the proposed circuitry to a wider range of CNTs (Section 8). Adsorption of a particle of mass Δm onto the CNT surface with uniform position probability distribution results in an expected change in effective mass Δmeff ¼ 38Δm and translates into a shift of resonance frequency. Using Eq. (14), the smallest detectable
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particle mass expresses as Δm ¼
16 Δf 0 Δf m ¼ 2πr 2 Lρ 0 3 f 0 eff f0
ð16Þ
From Eq. (16) it is apparent that short and thin tubes foster sensitivity. An upper bound on sensitivity is set by the NEMS dynamics and defined by thermomechanical and adsorption–desorption noise, temperature fluctuations and momentum exchange [1]. A practical limit is set by the sensor circuitry noise and will be assessed here after. Fig. 4 presents the resulting model of the CNT-NEMS for Simulink, whose output current (Id) is composed of the piezoresistive and capacitive contributions. It is actuated via the gate voltage (Vg) and tunable by changing the source position (Ps).
7. System topology and implementation Fig. 5 illustrates how the NEMS eigenfrequency, necessary to infer changes in sensed mass, can be extracted and tracked automatically. The CNT-NEMS can be stretched by a MEMS actuator [12] at its source. A low-noise amplifier (LNA) [9] enhances the signal strength out of the NEMS. A bandpass filter (BPF) then allows us to isolate the motional piezoresistive information at four times the actuation frequency, from the parasitical capacitive feedthrough. A phase–frequency detector (PFD) generates pulses of polarity and duration proportional to the relative phase- and frequency-advance of the NEMS with respect to a voltage controlled oscillator (VCO). This VCO drives the NEMS via the gate electrode and resonance is obtained if the VCO frequency corresponds to twice the NEMS eigenfrequency. The electrostatic force's proportionality to the square of the voltage (Eq. (7)) and symmetry of the piezoresistive effect with respect to position (Eq. (11)) generate
indeed a current with double (for square wave actuation) or fourfold (for sinewave actuation) frequency out of the NEMS. The VCO forms, together with a charge pump (CP) and a loop filter (LF), a phase locked loop (PLL), which locks on a frequency and phase as imposed by the NEMS. Only a frequency close to the NEMS' eigenfrequency produces an overall closed-loop phase congruent modulo 1801, implying a phase-lock at this frequency, once the loop phase Φ has been calibrated. Any particle binding to the CNT, or a change in the ambient atmosphere, would translate into a shift of the eigenfrequency, which is detected by a change of the LF output voltage and of the VCO's frequency. This basic functionality is completed by frequency dividers, which enable the use of higher frequency VCOs. These have the twofold advantage of occupying less chip-area and provide a reduced loop noise. An on-chip emulator of the CNT-NEMS allows for loop self-testing in the absence of an operational sensor. A start-up ramp sweeps through the frequencies and a lock-detector block closes the PLL once NEMS oscillation has been detected. The closed-loop sensor behaviour is illustrated by Fig. 6 for the parameter set reported in Table 1. Values are extracted from the literature for the CNT-NEMS and correspond to measured parameters of the printed circuit board (PCB) and integrated circuit (IC) implementations. The 250 nm long CNT has an eigenfrequency of 317 MHz and a quality factor of 50. The loop filter has been designed to obtain a critically damped loop (Q ¼ 12) with a natural frequency of 10 MHz. For a driving amplitude of 300 mV, the CNT's oscillation builds up within hundreds of nanoseconds and reaches a steady-state half-amplitude of 7 nm, corresponding to a piezoresistive current modulation of 200 nA at 634 MHz. The implementation of the system is presented in Fig. 7. The PCB hosts all blocks as they were presented in Fig. 5. The enframed
Fig. 4. Tunable piezoresistive and capacitive CNT-NEMS model for Simulink.
Fig. 5. NEMS sensor in closed loop.
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Fig. 6. Simulated PLL locking on CNT-NEMS.
Table 1 Sensor parameters. Parameter
Symbol
Value
Unit
Length Diameter Height Damping Bias Gauge factor Effective mass Eigenfrequency
L 2r h ζ I0 β m ω0
250 1 100 1/100 1 100 155 317
nm nm nm – μA – zg MHz
Loop frequency Loop quality factor
ωn Q PLL
10 1/2
MHz –
blocks have then been integrated in an 180 nm technology, forming a 2.085 2.165 mm2 chip [8]. This scale-down from spectrum analyzers, over a PCB towards an IC, illustrates the potential of small-size and low-cost autonomous sensor nodes. The integrated programmable VCOs are tunable from 600 MHz to 771 MHz, as can be observed from Fig. 8. The VCO gain peaks around 1.8 V for a maximum of 42 MHz=V at the high frequency end (728–771 MHz) and 27 MHz=V at the low frequency end (600–625 MHz). A value of K VCO equal to 40 MHz V has been used for the simulation of Fig. 6, where the VCO oscillates at twice the CNT eigenfrequency, that is 634 MHz. The PFD and the CP mirror a 2π phase interval onto the 1.8 V range, translating into a gain of K PFD ¼ 286 mV=rad. The loop dynamics are then set by a first order integrator (LF), sized for a critically damped Q PLL ¼ 12 closed loop time-constant ωn ¼10 MHz, to achieve steady-state as fast as possible without considerable overshoot: LFðjωÞ ¼
1 þ jðω=Q ωn Þ jðωK PFD K VCO =ω2n Þ
ð17Þ
8. Sensor performance In a next step the CNT parameter space, that is promising for sensor operation with the presented IC, is confined. Finally the mass sensor's resolution and the response time are evaluated. 8.1. Suitable CNT space The degree of universality of the presented circuitry may be expressed by its invariance with respect to CNT parameters. Given the ubiquity of piezoresistivity in CNTs and supposing its electrical
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effect to exceed the sensitivity of the front-end LNA, merely the NEMS eigenfrequency may hinder joint sensor operation of CNTs operated by the IC. To this purpose, the eigenfrequency (Eq. (15)) for common CNT dimensions has been evaluated and is compared to the IC's tuning range. General CNT diameters range from 1 to 10 nm. Given that mass sensitivity scales inversely to tube length (Eq. (16)), we limit our analysis to rather short tubes with lengths from 100 nm to 1 μm. Considering that the piezoresistive current modulation occurs at twice the motional frequency, the VCO's tuning range (600–771 MHz) is suitable for CNTs with eigenfrequencies in the 300–385 MHz range. On-chip frequency division by 2, 4 and 8 completes this interval by the 150–192 MHz, 75– 96 MHz and 38–48 MHz ranges respectively. Adequate CNT dimensions for those ranges are indicated in black in Fig. 9. The use of a MEMS actuator with an appropriate set of levers may be used to stretch the CNT by pulling on its source, thereby tuning its eigenfrequency (Eq. (15)). Such an approach extends the IC's compatibility to a large part of the CNT parameter space (grey zones in Fig. 9). The displacement resolution required to shift the CNT's eigenfrequency into the IC's range varies from 100 pm down to 1 pm. Though piezoelectric macro-actuators reach picometre resolution, their integrated MEMS versions achieve only a 100 pm resolution [12]. In combination with a simple lever ratio of 1:10, they could tune a substantial part of the CNT's to the IC's operating range. These piezoelectric MEMS are low-power, exhibit high forces and present large travel under robust operation. Their actuation voltage exceeds standard IC supply, and CMOS compatibility of the process may be an issue. Though magnetic MEMS trade resolution for lower actuation voltage, simple CMOS process compatibility is achieved by electrostatic and thermal actuators, at the cost of an insufficient 10 nm resolution, making piezoelectric MEMS the preferred candidate for precise frequency tuning.
8.2. Mass sensitivity and detection speed To assess how close a sensor, based on the presented IC, may get to the ultimate limits of inertial mass sensing [12], its root mean square jitter has been measured at 190 MHz (T~0 ¼ 50 263 ps) and was found to be σ~ ¼ 10:55 ps. Under the hypothesis that crossover fluctuations follow a normal distribution, the VCO output signal's period is a random variable following a Gaussian distribution: 0 sffiffiffiffiffiffiffiffiffiffiffi!2 1 T0 2 A T N @T 0 ; ð18Þ σ~ T~0 The estimation error on a CNT's eigenfrequency (f 0 ¼ 1=T 0 ), interfaced by the IC, decreases consequently with the square root of the number of observed periods n, resulting in a relative frequency error of Δf Δt ασ ¼ pffiffiffi f 0 T0 T0 n
ð19Þ
where α represents the desired certainty, which amounts to 99.7% for α¼3. Defining an observation time t meas , during which crossovers are counted, the number of observations writes n ¼ t meas =T 0 . Feeding Eq. (16) with these expressions leads to a 3σ confidence mass resolution σ~ 1 Δmcircuit ¼ 6π ρ r 2 L qffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi t meas T~0
ð20Þ
that improves with the square root of measurement time. This upper bound to sensitivity may be compared to the
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Fig. 7. PCB and IC sensor implementation.
Fig. 8. Measured VCO transfer characteristic.
Fig. 10. Sensor mass sensitivity and detection speed.
oscillation, while it takes a tenth of a second to detect the presence of a hydrogen atom. It shall be noted that the measurement converges towards the correct value of Δm only if the loop phase delay Φ has been calibrated previously.
9. Conclusion
Fig. 9. Acceptable CNT device space for sensor operation with presented IC.
thermomechanical noise intrinsic to the CNT-NEMS [1] pffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 kB Tmeff ΔmNEMS ¼ pffiffiffi 3 Q ω30 x^ 2 t meas
ð21Þ
and is found to be less restrictive than this thermal limit at room temperature, for the 300 mV actuation. Fig. 10 illustrates that the proposed sensor may detect binding proteins within a single
This contribution categorized state-of-the-art inertial mass balances into ω-controllable and ϕ-controllable methods. Highresolution weighing was hitherto the exclusive realm of the former, which unfortunately cannot perform instant sensing, while the latter may operate continuously, but are subject to offset. In this prospect, a novel nonlinear mass sensor calibration technique has been presented, and was shown to have the potential towards simultaneous high-speed and high-resolution mass sensing. A case study has been conducted, presenting an integrated circuit for mass balance applications of a suspended carbon nanotube resonator (CNT-NEMS). The concept is derived from a phase-locked loop that locks on and tracks the CNT-NEMS's eigenfrequency. The circuit's integration in a 180 nm technology shows sufficiently low jitter to detect single protons within the tenth of a second and proteins within nanoseconds. A CNT-NEMS model has been presented and demonstrates usability of the
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integrated circuitry with a wide range of CNTs, an important step towards small-size and low-cost sensor nodes for environmental and healthcare applications.
Acknowledgements This research has been funded by Nano-Tera.ch, a program of the Swiss Confederation, evaluated by SNSF. References [1] K.L. Ekinci, Y.T. Yang, M. Roukes, Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems, J. Appl. Phys. 95 (5) (2004) 2682–2689. http://dx.doi.org/10.1063/1.1642738. [2] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, A. Bachtold, A nanomechanical mass sensor with yoctogram resolution, Nat. Nanotechnol. 7 (2012) 301–304. http://dx.doi.org/10.1038/nnano.2012.42. [3] V. Gouttenoire, et al., Digital and fm demodulation of a doubly clamped singlewalled carbon-nanotube oscillator: towards a nanotube cell phone, Small (2010) 1060–1065 http://dx.doi.org/10.1002/smll.200901984.
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[4] Y.T. Yang, C. Callegari, X.L. Feng, K.L. Ekinci, M.L. Roukes, Zeptogram-scale nanomechanical mass sensing, Nano Lett. 6 (4) (2006) 583–586. http://dx.doi. org/10.1021/nl052134m. [5] V. Sazonova, Y. Yaish, H. Ustunel, D. Roundy, T.A. Arias, P.L. McEuen, A tunable carbon nanotube electromechanical oscillator, Nature 431 (7006) (2004) 284–287. http://dx.doi.org/10.1038/nature02905. [6] C. Kauth, M. Pastre, M. Kayal, On-chip mass sensing at the physical limits of nanoelectromechanical systems, in: Advances in Sensors and Interfaces Conference Proceedings, 2013, pp. 131–135 http://dx.doi.org/10.1109/IWASI. 2013.6576089. [7] H.-Y. Chiu, P. Hung, H.W.C. Postma, M. Bockrath, Atomic-scale mass sensing using carbon nanotube resonators, Nano Lett. 8 (12) (2008) 4342–4346. http: //dx.doi.org/10.1021/nl802181c. [8] C. Kauth, M. Pastre, M. Kayal, A self-regulating oscillator for sensor operation of nanoelectromechanical systems, in: New Circuits and Systems Conference Proceedings, 2013, pp. 1–4. http://dx.doi.org/10.1109/NEWCAS.2013.6573652. [9] C. Kauth, M. Pastre, M. Kayal, Low-noise wideband circuit for closed-loop RF CNT-NEMS sensors, Int. J. Microelectron. Comput. Sci. 3 (2012) 67–72. [10] D. Nordlund, High accuracy analog differentiator, Rev. Sci. Instrum. 43 (11) (1972) 1699. http://dx.doi.org/10.1063/1.1685527. [11] H. Postma, I. Kozinsky, A. Husain, M.L. Roukes, Dynamic range of nanotubeand nanowire-based electromechanical systems, Appl. Phys. Lett. 86 (22) (2005) 223105-1–223105-3. http://dx.doi.org/10.1063/1.1929098. [12] D. Bell, T. Lu, N. Fleck, S. Spearing, Mems actuators and sensors: observations on their performance and selection for purpose, J. Micromech. Microeng. 15 (2005) 153–164.
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Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
A novel approach to high-speed high-resolution on-chip mass sensing C. Kauth n, M. Pastre, M. Kayal STI-IEL Electronics Laboratory, Ecole Polytechnique Fédérale de Lausanne, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Received 26 December 2013 Accepted 14 July 2014 Available online 2 August 2014
The state-of-the-art mass sensing so far has been rather developed along the resolution axis, reaching atomic-scale detection, than into the direction of high-speed. This paper reports a novel self-calibrating technique, making high-speed inertial mass sensors capable of instant high-resolution particle detection and weighing. The sensing nanoelectromechanical resonator is embedded into a phase-locked loop and the sensor-inherent nonlinear phase–frequency relation is exploited for auto-calibration. A tunable onchip carbon nanotube based mass balance serves as a case study of small-size and low-cost environmental and healthcare applications. Tunability and a phase-locked loop topology make the system widely universal and invariant to nanotube characteristics. Operational for tube eigenfrequencies up to 385 MHz, the circuit integration in a 180 nm technology achieves instantaneous zeptogram resolution, while yoctogram precision is obtained within the tenth of a second. These figures of merit range at the physical limits of carbon nanotube resonators, in both mass- and time-resolution. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Analytical models Carbon nanotubes Closed loop systems Nanoelectromechanical systems Nonlinear systems Oscillators Phase locked loops System analysis and design
1. Introduction The state-of-the-art mass sensing is predicated on the detection of a shift in a resonator's eigenfrequency when an additional mass binds to its surface. Evaluation of the theoretical ultimate limits to such inertial mass sensing [1] suggests that single proton weighing is feasible at room temperature. This yoctogram mass resolution has been achieved [2] at liquid-helium temperature with an ultrasensitive carbon nanotube resonator, operated in a mixer setup [3]. The equivalent room-temperature record ranges at a hundredfold worse resolution [4]. These high-resolution sensors find applications in atomic physics, biology and life science as molecule identifiers, gas detectors and cell weighing to name but a few. In contrast to this impressive mass resolution, the temporal resolution of such sensors appears unspectacular. Frequency sweeps that last several seconds [5] and feedback loops of tens of milliseconds [2] set the benchmark hitherto. By comparing this sensing speed to the quasi-gigahertz NEMS eigenfrequency, it becomes apparent that there is immense potential to improve sensing speed by several orders of magnitude. A novel approach to simultaneous high-speed and high-resolution nanomechanical mass sensing is presented in this contribution. Today's state-of-the-art mass sensors do not combine these two desirable properties yet: high-resolution methods [2] rely on frequency sweeps or mixing, which makes them inherently slow, while high-speed methods [6] present non-quantifiable offsets, which lead
n
Corresponding author. E-mail address: christian.kauth@epfl.ch (C. Kauth).
http://dx.doi.org/10.1016/j.mejo.2014.07.004 0026-2692/& 2014 Elsevier Ltd. All rights reserved.
to erroneous mass inference. An auto-calibrating system is shown to compensate offsets and hence enables high-resolution high-speed sensing. Such precise and fast mass detection allows observation of phenomena that so far are not observable because they are either too subtle (insufficient sensor resolution) or happen too fast (insufficient sensor speed), like chemical and biochemical reactions. The concept condensates into an implementable topology and suspended carbon nanotubes (CNT) are the resonator of choice, favoured by their low effective mass and acceptable quality factor. Their high elasticity allows us to tune the CNT for maximal compatibility with the interface circuitry via mechanical strain. Feedback electronics, implemented in a 180 nm technology, form a phase-locked loop (PLL) around the CNT-NEMS resonator in order to drive and sustain the latter's motion at resonance. Metamorphosed into small-size and low-cost sensor nodes for large-scale healthcare or environmental applications, this on-chip sensor circuit presents unprecedented speed, and mass resolution down to the physical limits of nanoelectromechanical systems. Section 2 brings forward the key elements for high-speed and highresolution mass sensing, and presents a novel topology combining both features. The physical phenomenon at the basis of this self-calibrating sensor topology is investigated in Section 3. Section 4 demonstrates that this nonlinear phenomenon can indeed be exploited in practice for calibration and the operation principle of the resulting high-resolution high-speed mass sensor is explained in Section 5. Section 6 starts a case study illustrating how suspended carbon nanotubes can fulfill the role of the sensing NEMS. The motional information processing system and its integration are the topics of Section 7. Section 8 identifies which CNTs are adequate for the proposed chip and translates the latter's jitter into sensor performance in terms of resolution and speed.
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2. Motivation and novelty Fast and precise identification of the NEMS eigenfrequency ω0 may be modelled as a search problem. A questioner interrogates an oracle 8 > < 1 if ω oω0 if ω ¼ ω0 ð1Þ O : ω↦ 0 > : þ 1 if ω 4ω 0 on whether the unknown eigenfrequency ω0 equals an arbitrary guess ω. The oracle answers with 1, 0 or þ 1 as a function of the relative position of ω and ω0. Two key parameters to high-speed high-resolution mass sensing crystallize from this formal description of the eigenfrequency search. First must the oracle provide fast and trustworthy responses, and second shall the questioner interpret the answer quickly and rapidly to improve the quality of the guess. The fastest way to determine a periodic signal's frequency is to measure exactly one period. Hence high NEMS eigenfrequencies are beneficial for the first criterion and suspended carbon nanotubes are a neat choice, as shown in Section 6. Noise may distort the oracle's answer, but can be countervailed by averaging over several periods. The resulting speed-precision trade-off is assessed in Section 8. The current section's emphasis lies on the inspection of techniques to rapidly observe the oracle's answer and formulate a precise guess on ω0. The following observations are independent of the exact NEMS nature and averaging. A generic dynamic NEMS may be modelled as a normalized damped harmonic oscillator ω0 1 ( 1 2 jHðsÞj ¼ A Q 4Q HðsÞ ¼ ; ð2Þ ω0 argðHðsÞÞ ¼ ϕ s2 þ s þ ω20 Q with eigenfrequency ω0 and quality factor Q. The eigenfrequency is observable via detection of the amplitude A peak at ωp or the phase ϕ inflection at ωi: vs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 1 t ð3Þ 4 2 1 ω p ¼ ω0 1 ; ωi ¼ ω 0 2Q 2 Q ωp ωi ω0 for sufficiently large Q. Adsorption of a particle onto the oscillating NEMS causes a relative change Δmeff =meff in the oscillator's effective mass, defined by the particle's weight and the binding position [7], and shifts the resonance characteristics as pffiffiffi pffiffiffi ðω0 ; Q Þ-ðω0 = α, αQ ), with α ¼ 1 þ ðΔm=mÞ. This shift in eigenfrequency is to be detected as fast and precise as possible. Today's techniques of doing so may be systematically categorized into
ω-controllable techniques drive the NEMS at a given frequency ω and observe its amplitude A or phase ϕ response. Such approaches benefit from a reduced noise bandwidth when
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operating the NEMS as a mixer [5]. The mixed signal out of the NEMS, however, has a much lower frequency than ω0. This makes mixing incompatible with the high-speed detection exigence, and feedback loops take tens of milliseconds [2] to improve the estimation of ω0. High-resolution weighing is enabled by the simultaneous extraction of ωp or ωi and Q from a response plot around the resonance peak [2], which takes seconds. It is concluded that ω-controllable methods are precise, but slow. ϕ-controllable techniques embed the NEMS into a phase locked loop (PLL) and loop its output onto its input [4,8], with a controllable phase shift Φ, which forces the NEMS to oscillate at a unique frequency ω. An adsorption-related shift in ω is directly observable, enabling high-speed weighing, but the error in the observed Δω0 scales as 1=Q tan ðϕÞ. If ϕ ¼ π=2, the error vanishes, but this ideal hypothesis is hard to satisfy in practice, where component delays play a role and are a priori hardly entirely corrigible. It is concluded that ϕ-controllable methods are fast, but have limited resolution.
As a matter of fact could the high-resolution and high-speed sensing hitherto be combined yet. An novel approach is shown in Fig. 1(a), where in situ calibration of the NEMS phase ϕ via a nonlinear technique adds precision to ϕ-controllable methods and opens the gate to high-speed high-resolution mass sensing. The NEMS is embedded into a phase locked loop, formed by a phase–frequency detector (PFD), a loop filter (F PLL ) and a voltage controlled oscillator (VCO). At steady state, the VCO oscillates at a frequency for which both feedback paths, one formed by the NEMS and the readout electronics [9], the other by a simple phase shifter, have exactly the same delay. Hence the NEMS is forced to oscillate at a precise frequency, at which ϕ satisfies the above constraint. This frequency depends consequently on the loop delay Φ. The additional feedback adjusts Φ so as to centre resonance at ωi, quasi-equal to ω0 for NEMS with decent Q. Fig. 1 (b) illustrates how the auto-calibration circuit adjusts the loop phase Φ so as to bring the NEMS phase ϕ to π=2, which reflects oscillation at ω0 and error-free sensing. Once the NEMS has been calibrated, Φ may simply be memorized and sensing happens precisely at the speed of ϕ-controllable methods. The underlying hypothesis for this memorization to work out is that the NEMS phase is far more sensitive to the oscillation frequency than any other circuit component. This condition is tacitly met in ϕ-controllable methods, as the PLL locks onto a frequency imposed by the circuit component with the highest quality factor, speak phase sensitivity, which must be the NEMS, if sensor operation is envisioned.
3. Physical phenomenon The technique to enable calibration of the feedback loop phase shift Φ such that NEMS oscillation at ωi is enforced roots in the
Fig. 1. Auto-calibrating mass balance. (a) System architecture. (b) Calibration process.
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that amplifies signals above ωL , differentiates till ωd and integrates above ωd. The choice of ζ¼0.65 stands for maximally accurate derivation at minimal noise bandwidth performance loss [10]. Saturation of ∂t2 ω yields a bipolar digital signal ( 71) with phasedependent average voltage V Δϕ ðϕÞ, depicted in Fig. 3(b), negative below, positive above and zero at ϕi, which is notably independent of the NEMS quality factor. The phase-to-voltage conversion gain K ϕ ¼ ∂ϕ V Δϕ depends on the modulation amplitude and is given by Fig. 3(c), in absolute terms for ideal differentiators (ωd =ω0 ¼ 1) and relatively to this ideal value for the limited bandwidth differentiator of Fig. 3(b). Fig. 2. Impact of phase shift on ω for ϕ A ϕi 7 8π ; ϕi and Δϕ ¼ 5π .
5. NEMS mass balance system nonlinearity of the NEMS phase–frequency relation 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω0 π > > ½1 þ 1 þ ð2Q tan ðϕÞÞ2 if ϕ A π; ½ > < 2Q tan ðϕÞ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω¼ ω0 π > 2 > > : 2Q tan ðϕÞ ½1 1 þ ð2Q tan ðϕÞÞ if ϕ A 2 ; 0½
ð4Þ
derived from Eq. (2). The reasonable hypothesis that the NEMS presents the highest Q among all loop elements makes a sinusoidal modulation of the loop phase Φ to cause an identical modulation of the NEMS phase ϕ, which results in a variation of its frequency ω, as Fig. 2 reveals. The key observation is that the frequency's inflection points, distinguishable as the extrema of its time derivative ∂t ω or zeros of ∂t2 ω, are spread over a period with an irregularity defined by the signed average distance between ϕ and ϕi. Is the NEMS oscillating below ωi, then the down-time of ∂t2 ω exceeds its up-time, while the reverse is true for frequencies above ωi and equilibrium is obtained at ωi.
4. Controllability and observability Practical implementation of the calibration-loop consequently requires Φ to be controllable and ∂t2 ω to be observable. 4.1. Controllability of Φ The phase modulation can be performed by inserting the circuit of Fig. 3(a) s 1 1 1 ωt ¼ with ωt ¼ TðsÞ ¼ s RC ðR þ ΔR sin ðωm tÞÞC 0 þ1 ωt
ð5Þ
into the NEMS signal path. A sinusoidal control of R at a frequency ωm ⪡ω0 introduces a highly linear delay 2 arctanðω0 =ωt Þ 2ω0 =ωt with variational amplitude Δϕ ¼ 2ω0 ΔRC and no impact on the signal amplitude jTðsÞj ¼ 1. 4.2. Observability of ∂t2 ω Information on the NEMS frequency is available through the control voltage of the VCO in the topology of Fig. 1(a), which consists of the ϕ-controllable technique of [8], completed by the calibration loop of Φ. If the phase modulation frequency ωm is chosen such that ωm ⪡ωPLL ⪡ω0 , then the impact of Φ on the NEMS frequency is retrievable from the low-pass filtered VCO control voltage, which unveils ∂t2 ω after two differentiations, performed by the second order bandpass filter of Fig. 3(b) DðsÞ ¼
ω2d
s ωL s2 þ 2ζωd s þ ω2d
ð6Þ
The proposed auto-calibrating mass balance of Fig. 1(a) omits frequency division, necessary to lock on the field effect or the piezoresistive NEMS signal, as will be shown in Section 6, for the sake of clarity. A periodic modulation of the delay Δϕ modulates the NEMS frequency nonlinearly. PFD-spikes are removed via a lowpass filter F spike ðsÞ and DðsÞ differentiates the nonlinearity twice, before saturating it. The calibration-loop is closed by a filter F Φ ðsÞ that adjusts the NEMS phase ϕ to ϕi. The loop acts as a PLL tracking ϕi, which is certainly unknown, but implicitly defined by the NEMS and made observable through the injected phase modulation. Given the involved frequencies, the PFD and F PLL ðsÞ are transparent to this loop and TðsÞ has unity gain. A low frequency (ωc ⪡ωm ) third order PLL is necessary to prevent interference between the injected modulation and the resulting compensation of Φ. The calibration loop filter F Φ ðsÞ may consist of a zero-order integrator (unity gain at K=K ϕ β), a first-order integrator (zero at α) and a lowpass filter (pole at β), forming the closed-loop transfer function: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 sin φ > K ¼ ð2πωc Þ2 11 þ > sin φ > < ϕ Kðs þ αÞ ; ¼ α ¼ 2πωc 1 cossinφ φ ϕi s3 þ βs2 þ Ks þ Kα > > > : β ¼ 2πωc cos φ 1 sin φ where ωc is the loop bandwidth and φ the phase margin. The graphs of Fig. 1(b) illustrate the settling of a critically damped third order (ωc ¼ 10 5 ω0 , φ ¼ 501) calibration loop for a NEMS embedded in a critically damped second order PLL (ωPLL ¼ 10 1 ω0 , Q PLL ¼ 12), presented in Section 7. The VCO centre frequency is 0.9ω0, the gain 0.1ω0, and the injected variation (Δϕ¼ π=20, ωm ¼10 3 ω0 ). Upon convergence, the calibration loop may be disconnected to remove the modulation-induced residual ripple.
6. CNT-NEMS resonator Now that it has been shown that the system of Fig. 1(a) allows for high-speed and high-resolution mass sensing, we take a closer look at suspended carbon nanotube NEMS, as they were identified as promising candidates for sensing. First we demonstrate how motional information is converted into an electrical signal, then we show how inertial sensing and tube straining affect the dynamics. 6.1. Transduction mechanism The sensor resonator is a CNT of radius r suspended at a height h over a trench of length L. Its extremes are denoted as source and drain in analogy to a MOS transistor. The gate is formed by a nearby electrode and exerts an electrostatic force onto the NEMS,
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Fig. 3. Phase modulation for controllability and signal differentiation for observability. (a) Generation of phase shift ϕ ¼ ϕ þ Δϕ sin ðωm tÞ. (b) Average of sgnð∂t2 ωÞ. (c) Phase– voltage gain K ϕ jϕi ðV=radÞ.
expressed as FðtÞ ¼
VðtÞ2 πϵ0 L 2 h h arccosh r
ð7Þ
where ϵ0 is the vacuum permittivity. This force accelerates the NEMS as a driven harmonic oscillator, balancing the driving, restoring and frictional forces: ∂t2 x^ þ2ζω0 ∂t x^ þω20 x^ ¼
FðtÞ meff
ð8Þ
where the damping ratio ζ ¼ 1=2Q is inversely proportional to the NEMS quality factor Q. Supposing a sinusoidal gate voltage VðtÞ, the driving force at twice that frequency (see Eq. (7)) causes an oscillation half-amplitude at the middle of the beam equal to F0 ^ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xðtÞ sin ðωt þ ϕÞ meff ð2ωω0 ζÞ2 þ ðω20 ω2 Þ2
ð9Þ
ð11Þ
and the ubiquitous piezoresistive property of CNTs renders it detectable due to its effect on the current through the tube δI piezo ðtÞ ¼ I 0 βϵðtÞ
δI piezo V 3 5 δI cap ω0
ð12Þ
where I0 is the bias current and a linear dependence of the CNT resistance on the gauge factor β is hypothesized. This
ð13Þ
indicates that higher driving voltages are necessary to compensate for the parasitical gate-drain coupling of high-frequency resonators.
6.2. Sensitivity and frequency tuning The hypothesized mode shape of Eq. (10) entails the expression of the effective mass: meff ¼
delayed with respect to the drive by an angle within ½ π; 0 ϕ ¼ arctanð2ωω0 ζ=ðω2 ω20 ÞÞ. The CNT's shape may be approximated by [11] ^ xðtÞ 2πl 1 cos ð10Þ xðl; tÞ ¼ 2 L This motion of the tube induces a strain π 2 ^ xðtÞ ϵðtÞ ¼ 2L
piezoresistive current has a fourfold frequency with respect to the gate voltage (see Eqs. (11) and (7)), allowing a bandpass filter to spectrally separate the motional information from any parasitical capacitive coupling between the gate and drain electrodes. The piezoresistive to capacitive signal ratio scaling as
3 2 πr Lρ 8
ð14Þ
with ρ being the true density of a single CNT. This effective mass, along with the effective spring constant, defines the NEMS eigenfrequency, derived from [11] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ffi ω0 E π r ΔL ¼ f0 ¼ þ ð15Þ L 2π 3ρL2 L2 where E is the CNT Young modulus and ΔL an elongation of the tube induced by stretching. From Eq. (15) it becomes apparent that the eigenfrequency is tunable, a feature that allows us to extend the proposed circuitry to a wider range of CNTs (Section 8). Adsorption of a particle of mass Δm onto the CNT surface with uniform position probability distribution results in an expected change in effective mass Δmeff ¼ 38Δm and translates into a shift of resonance frequency. Using Eq. (14), the smallest detectable
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particle mass expresses as Δm ¼
16 Δf 0 Δf m ¼ 2πr 2 Lρ 0 3 f 0 eff f0
ð16Þ
From Eq. (16) it is apparent that short and thin tubes foster sensitivity. An upper bound on sensitivity is set by the NEMS dynamics and defined by thermomechanical and adsorption–desorption noise, temperature fluctuations and momentum exchange [1]. A practical limit is set by the sensor circuitry noise and will be assessed here after. Fig. 4 presents the resulting model of the CNT-NEMS for Simulink, whose output current (Id) is composed of the piezoresistive and capacitive contributions. It is actuated via the gate voltage (Vg) and tunable by changing the source position (Ps).
7. System topology and implementation Fig. 5 illustrates how the NEMS eigenfrequency, necessary to infer changes in sensed mass, can be extracted and tracked automatically. The CNT-NEMS can be stretched by a MEMS actuator [12] at its source. A low-noise amplifier (LNA) [9] enhances the signal strength out of the NEMS. A bandpass filter (BPF) then allows us to isolate the motional piezoresistive information at four times the actuation frequency, from the parasitical capacitive feedthrough. A phase–frequency detector (PFD) generates pulses of polarity and duration proportional to the relative phase- and frequency-advance of the NEMS with respect to a voltage controlled oscillator (VCO). This VCO drives the NEMS via the gate electrode and resonance is obtained if the VCO frequency corresponds to twice the NEMS eigenfrequency. The electrostatic force's proportionality to the square of the voltage (Eq. (7)) and symmetry of the piezoresistive effect with respect to position (Eq. (11)) generate
indeed a current with double (for square wave actuation) or fourfold (for sinewave actuation) frequency out of the NEMS. The VCO forms, together with a charge pump (CP) and a loop filter (LF), a phase locked loop (PLL), which locks on a frequency and phase as imposed by the NEMS. Only a frequency close to the NEMS' eigenfrequency produces an overall closed-loop phase congruent modulo 1801, implying a phase-lock at this frequency, once the loop phase Φ has been calibrated. Any particle binding to the CNT, or a change in the ambient atmosphere, would translate into a shift of the eigenfrequency, which is detected by a change of the LF output voltage and of the VCO's frequency. This basic functionality is completed by frequency dividers, which enable the use of higher frequency VCOs. These have the twofold advantage of occupying less chip-area and provide a reduced loop noise. An on-chip emulator of the CNT-NEMS allows for loop self-testing in the absence of an operational sensor. A start-up ramp sweeps through the frequencies and a lock-detector block closes the PLL once NEMS oscillation has been detected. The closed-loop sensor behaviour is illustrated by Fig. 6 for the parameter set reported in Table 1. Values are extracted from the literature for the CNT-NEMS and correspond to measured parameters of the printed circuit board (PCB) and integrated circuit (IC) implementations. The 250 nm long CNT has an eigenfrequency of 317 MHz and a quality factor of 50. The loop filter has been designed to obtain a critically damped loop (Q ¼ 12) with a natural frequency of 10 MHz. For a driving amplitude of 300 mV, the CNT's oscillation builds up within hundreds of nanoseconds and reaches a steady-state half-amplitude of 7 nm, corresponding to a piezoresistive current modulation of 200 nA at 634 MHz. The implementation of the system is presented in Fig. 7. The PCB hosts all blocks as they were presented in Fig. 5. The enframed
Fig. 4. Tunable piezoresistive and capacitive CNT-NEMS model for Simulink.
Fig. 5. NEMS sensor in closed loop.
C. Kauth et al. / Microelectronics Journal 45 (2014) 1648–1655
Fig. 6. Simulated PLL locking on CNT-NEMS.
Table 1 Sensor parameters. Parameter
Symbol
Value
Unit
Length Diameter Height Damping Bias Gauge factor Effective mass Eigenfrequency
L 2r h ζ I0 β m ω0
250 1 100 1/100 1 100 155 317
nm nm nm – μA – zg MHz
Loop frequency Loop quality factor
ωn Q PLL
10 1/2
MHz –
blocks have then been integrated in an 180 nm technology, forming a 2.085 2.165 mm2 chip [8]. This scale-down from spectrum analyzers, over a PCB towards an IC, illustrates the potential of small-size and low-cost autonomous sensor nodes. The integrated programmable VCOs are tunable from 600 MHz to 771 MHz, as can be observed from Fig. 8. The VCO gain peaks around 1.8 V for a maximum of 42 MHz=V at the high frequency end (728–771 MHz) and 27 MHz=V at the low frequency end (600–625 MHz). A value of K VCO equal to 40 MHz V has been used for the simulation of Fig. 6, where the VCO oscillates at twice the CNT eigenfrequency, that is 634 MHz. The PFD and the CP mirror a 2π phase interval onto the 1.8 V range, translating into a gain of K PFD ¼ 286 mV=rad. The loop dynamics are then set by a first order integrator (LF), sized for a critically damped Q PLL ¼ 12 closed loop time-constant ωn ¼10 MHz, to achieve steady-state as fast as possible without considerable overshoot: LFðjωÞ ¼
1 þ jðω=Q ωn Þ jðωK PFD K VCO =ω2n Þ
ð17Þ
8. Sensor performance In a next step the CNT parameter space, that is promising for sensor operation with the presented IC, is confined. Finally the mass sensor's resolution and the response time are evaluated. 8.1. Suitable CNT space The degree of universality of the presented circuitry may be expressed by its invariance with respect to CNT parameters. Given the ubiquity of piezoresistivity in CNTs and supposing its electrical
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effect to exceed the sensitivity of the front-end LNA, merely the NEMS eigenfrequency may hinder joint sensor operation of CNTs operated by the IC. To this purpose, the eigenfrequency (Eq. (15)) for common CNT dimensions has been evaluated and is compared to the IC's tuning range. General CNT diameters range from 1 to 10 nm. Given that mass sensitivity scales inversely to tube length (Eq. (16)), we limit our analysis to rather short tubes with lengths from 100 nm to 1 μm. Considering that the piezoresistive current modulation occurs at twice the motional frequency, the VCO's tuning range (600–771 MHz) is suitable for CNTs with eigenfrequencies in the 300–385 MHz range. On-chip frequency division by 2, 4 and 8 completes this interval by the 150–192 MHz, 75– 96 MHz and 38–48 MHz ranges respectively. Adequate CNT dimensions for those ranges are indicated in black in Fig. 9. The use of a MEMS actuator with an appropriate set of levers may be used to stretch the CNT by pulling on its source, thereby tuning its eigenfrequency (Eq. (15)). Such an approach extends the IC's compatibility to a large part of the CNT parameter space (grey zones in Fig. 9). The displacement resolution required to shift the CNT's eigenfrequency into the IC's range varies from 100 pm down to 1 pm. Though piezoelectric macro-actuators reach picometre resolution, their integrated MEMS versions achieve only a 100 pm resolution [12]. In combination with a simple lever ratio of 1:10, they could tune a substantial part of the CNT's to the IC's operating range. These piezoelectric MEMS are low-power, exhibit high forces and present large travel under robust operation. Their actuation voltage exceeds standard IC supply, and CMOS compatibility of the process may be an issue. Though magnetic MEMS trade resolution for lower actuation voltage, simple CMOS process compatibility is achieved by electrostatic and thermal actuators, at the cost of an insufficient 10 nm resolution, making piezoelectric MEMS the preferred candidate for precise frequency tuning.
8.2. Mass sensitivity and detection speed To assess how close a sensor, based on the presented IC, may get to the ultimate limits of inertial mass sensing [12], its root mean square jitter has been measured at 190 MHz (T~0 ¼ 50 263 ps) and was found to be σ~ ¼ 10:55 ps. Under the hypothesis that crossover fluctuations follow a normal distribution, the VCO output signal's period is a random variable following a Gaussian distribution: 0 sffiffiffiffiffiffiffiffiffiffiffi!2 1 T0 2 A T N @T 0 ; ð18Þ σ~ T~0 The estimation error on a CNT's eigenfrequency (f 0 ¼ 1=T 0 ), interfaced by the IC, decreases consequently with the square root of the number of observed periods n, resulting in a relative frequency error of Δf Δt ασ ¼ pffiffiffi f 0 T0 T0 n
ð19Þ
where α represents the desired certainty, which amounts to 99.7% for α¼3. Defining an observation time t meas , during which crossovers are counted, the number of observations writes n ¼ t meas =T 0 . Feeding Eq. (16) with these expressions leads to a 3σ confidence mass resolution σ~ 1 Δmcircuit ¼ 6π ρ r 2 L qffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi t meas T~0
ð20Þ
that improves with the square root of measurement time. This upper bound to sensitivity may be compared to the
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Fig. 7. PCB and IC sensor implementation.
Fig. 8. Measured VCO transfer characteristic.
Fig. 10. Sensor mass sensitivity and detection speed.
oscillation, while it takes a tenth of a second to detect the presence of a hydrogen atom. It shall be noted that the measurement converges towards the correct value of Δm only if the loop phase delay Φ has been calibrated previously.
9. Conclusion
Fig. 9. Acceptable CNT device space for sensor operation with presented IC.
thermomechanical noise intrinsic to the CNT-NEMS [1] pffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 kB Tmeff ΔmNEMS ¼ pffiffiffi 3 Q ω30 x^ 2 t meas
ð21Þ
and is found to be less restrictive than this thermal limit at room temperature, for the 300 mV actuation. Fig. 10 illustrates that the proposed sensor may detect binding proteins within a single
This contribution categorized state-of-the-art inertial mass balances into ω-controllable and ϕ-controllable methods. Highresolution weighing was hitherto the exclusive realm of the former, which unfortunately cannot perform instant sensing, while the latter may operate continuously, but are subject to offset. In this prospect, a novel nonlinear mass sensor calibration technique has been presented, and was shown to have the potential towards simultaneous high-speed and high-resolution mass sensing. A case study has been conducted, presenting an integrated circuit for mass balance applications of a suspended carbon nanotube resonator (CNT-NEMS). The concept is derived from a phase-locked loop that locks on and tracks the CNT-NEMS's eigenfrequency. The circuit's integration in a 180 nm technology shows sufficiently low jitter to detect single protons within the tenth of a second and proteins within nanoseconds. A CNT-NEMS model has been presented and demonstrates usability of the
C. Kauth et al. / Microelectronics Journal 45 (2014) 1648–1655
integrated circuitry with a wide range of CNTs, an important step towards small-size and low-cost sensor nodes for environmental and healthcare applications.
Acknowledgements This research has been funded by Nano-Tera.ch, a program of the Swiss Confederation, evaluated by SNSF. References [1] K.L. Ekinci, Y.T. Yang, M. Roukes, Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems, J. Appl. Phys. 95 (5) (2004) 2682–2689. http://dx.doi.org/10.1063/1.1642738. [2] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, A. Bachtold, A nanomechanical mass sensor with yoctogram resolution, Nat. Nanotechnol. 7 (2012) 301–304. http://dx.doi.org/10.1038/nnano.2012.42. [3] V. Gouttenoire, et al., Digital and fm demodulation of a doubly clamped singlewalled carbon-nanotube oscillator: towards a nanotube cell phone, Small (2010) 1060–1065 http://dx.doi.org/10.1002/smll.200901984.
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