pubs.acs.org/NanoLett
Signal-to-Noise Ratio in Carbon Nanotube Electromechanical Piezoresistive Sensors Thomas Helbling,* Cosmin Roman, and Christofer Hierold Micro and Nanosystems, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland ABSTRACT The signal-to-noise ratio (SNR) of piezoresistive transducers based on carbon nanotube field effect transistors (CNFETs) is an essential yet unexplored performance metric. Here, we show that the SNR of CNFET piezoresistors made of small band gap semiconducting SWNTs (SGS-SWNTs) depends strongly on the gate bias voltage. The SNR is found by combining low frequency 1/f noise with the piezoresistive signal. We find that SGS-CNFET piezoresistors are best operated at device off-state, where strain resolution is maximal. KEYWORDS Carbon nanotube transistors, noise, signal-to-noise ratio, electromechanical sensors, strain gauges
C
arbon nanotubes (CNTs) are known for a number of exceptional properties and have attracted great attention as functional material for next generation sensor devices including chemical1 and biochemical2 sensors, optical converters,3,4 resonators,5 and electromechanical transducers.6-10 Strain gauges are an important class of electromechanical transducers with broad application potential. Single-walled carbon nanotube based (SWNT) strain gauges offer several advantages including high gauge factors (GF), low power consumption, and small stiffness compared to the strain-applying structures. In terms of GFs, it was shown that SWNTs outperform state-of-the-art silicon strain gauges.6,8,9,11,12 However, although a high sensitivity is necessary and important, this criterion alone cannot be sufficient for sensors operated close to the limit of detection, where noise is expected to play an equally important role. Carbon nanotube-based field effect transistors (CNFETs) exhibit non-negligible low frequency 1/f noise due to their reduced dimensions and thus reduced number of carriers.13 In these conditions, a meaningful performance metric is the signal-to-noise ratio (SNR) that incorporates both sensitivity and noise. We have recently shown that GFs in CNFETs depend on the applied gate voltage Vg.12,14 On the other hand, the noise level is also known to depend on Vg in CNFETs.15,16 These two facts suggest that SNR in piezoresistive CNFET sensors might also depend on Vg, providing a way to maximize SNR by electrical sensor biasing. Recently, Heller et al.17 have discussed the SNR dependence on Vg in the context of biosensors, showing that the maximum of SNR does not coincide with the sensitivity maximum. The sensing mechanism in CNFET strain gauges is of a complete different physical nature motivating a separate SNR study, not performed thus far to our knowledge.
In this Letter, we investigate the SNR and its dependence on the gate voltage Vg for CNFET strain gauges employing small band gap single-walled carbon nanotubes (SGSSWNTs) as transistor channel. SGS-SWNTs are characterized by ambipolar conduction and small on-off ratios.18,19 These strain gauges consist of an individual SWNT channel encapsulated by Al2O3 and a top gate electrode as shown in Figure 1a. We utilize a membrane configuration as a test bench to strain the CNFETs in the low strain regime by applying differential pressure as shown in Figure 1b. The fabrication of pressure sensors, described in detail elsewhere,18 consists mainly in SWNT growth, deposition of CNFET source and drain electrodes, and encapsulation of transistors by atomic layer deposition (ALD) of Al2O3 followed by a backside DRIE-ICP process to release the membrane. Figure 1b shows a cross sectional schematic of a carbon nanotube strain gauge integrated into a test bench structure which is essentially a diaphragm-type pressure sensor. The diaphragms are circular double-layer membranes made of SiO2/Al2O3 (70 nm/70 nm thick) with a radius r ) 45 µm. A differential pressure, p, applied across the sensor membrane leads to a deflection of the membrane which induces strain in the CNFETs.6,9,18 The CNFET strain gauges are embedded in between the bottom SiO2 and the top Al2O3 layer and contacted by source and drain electrodes as shown schematically in Figure 1a. The gate terminal is on top of the ALD Al2O3 layer. Electrical CNFET gate characteristics measurements are conducted by applying a constant voltage, Vds, across the source-drain electrodes and sweeping the gate voltage Vg, from -Vgmax to +Vgmax. Typical Id-Vg characteristics obtained from two SGS-CNFET strain gauges encapsulated by ALD Al2O3 measured at zero differential pressure across the membrane are shown in Figure 1c. The locations of the minimal Id current, Imin, are indicated in Figure 1c and delimit the off-state of the transistors.
* Corresponding author,
[email protected]. Received for review: 03/23/2010 Published on Web: 08/24/2010 © 2010 American Chemical Society
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FIGURE 2. Real time sensor outputs, Id(t) for CNFET #2 measured at different gate voltages, namely, (a) Imin for Vg ) -3 V, (b) Vg ) -2 V, and (c) Vg ) -1.5 V under the pressure stimuli p(t) applied to the membrane shown in (d).
studies have shown that the piezoresistive response is coming from strain modulating the electronic band gap of SWNTs. Accordingly, SGS-CNFET #1 exhibits a decrease of the electronic band gap under strain, whereas SGS-CNFET #2 exhibits a band gap increase under strain. Plots a-c of Figure 2 show three sensor output measurements Id(t) of SGS-CNFET #2 recorded at different Vg values and Vds ) 60 mV. For all three measurements the same pressure stimulus p(t) as shown in Figure 2d was applied. Figure 2a was recorded at Imin where the transistor is in the off-state. This is the location of maximum sensitivity, where GF ) 240 ( 18 is extracted, which is defined as GF ) (∆R/R0)ε-1.12 Figure 2b,c shows measurements near maximum transconductance, gm, of the transistor at Vg ) -2 V and Vg ) -1.5 V, respectively (see Id-Vg characteristic of CNFET #2 in Figure 1c). The extracted GF(Vg) values of SGS-CNFET #2 are GF(-2 V) ) 124.2 ( 16 and GF(-1.5 V) ) 68 ( 20. Plots a-c of Figure 2 suggest that the signal-to-noise ratio depends strongly on Vg, setting the goal of finding Vg that maximizes SNR. A rigorous definition for SNR will be given later, but roughly speaking SNR is directly proportional to the sensitivity GF and inversely proportional to the noise amplitude A. In the following, the GF and A dependence on Vg will be studied separately before combining them into an SNR(Vg) characteristic. Research on low frequency noise in SWNT has uncovered large 1/f low-frequency signal fluctuations that are described by the current noise power spectral density, SI,15,16,22-33 following the law,
FIGURE 1. Pressure sensor device schematic and electromechanical measurements on SGS-CNFETs. (a) Close up cross sectional schematic of CNFETs with source and drain electrodes contacting the SWNT channel and a top gate electrode. (b) Cross sectional schematic of a pressure sensor device made of a double layer circular membrane and CNFETs embedded in between two layers. (c) Ambipolar Id-Vg characteristics of two typical SGS-CNFETs and (d) electromechanical response of the same two CNFETs to pressure stimuli p(t) indicated in the inset. The CNFET Vg bias is set to operate the strain gauges close to Imin.
Electromechanical sensor measurements of the same SGSCNFETs are shown in Figure 1d. Rectangular pressure pulses with first increasing (ramp up) and then decreasing (ramp down) amplitudes are applied across the sensor membrane with seven different pressure amplitudes linearly distributed between 0 and 40 kPa. (See inset of Figure 1d.) The Id(t) responses of the two SGS-CNFETs to identical pressure stimuli in Figure 1d are different in both magnitude and sign. For SGSCNFET #1 Id increases while for SGS-CNFET #2 it decreases upon increasing pressure. Previous electrical7,9,10 and optical20,21 © 2010 American Chemical Society
SI(f ) )
AId2 fα
(1)
where A is the noise amplitude, f is the frequency, and R is a system-dependent exponent typically valued around 1. 3351
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The parameter encapsulating most information on 1/f noise is A. A sets the magnitude of SI, is independent of f, but depends on device type, geometry, and electrical bias, most notably Vg. For bulk semiconductor devices, A obeys Hooge’s law, depending on Vg through its inverse proportionality to the number of carriers in the channel. However, recently it was proposed by Tersoff that for semiconducting SWNTs, A depends mainly on the device transconductance,31 violating as such Hooge’s law. This was shown experimentally by Ma¨nnik et al. for CNFETs made from semiconducting SWNTs in the realm of biosensing.16 Although, low-frequency noise has been measured for both SGS-CNFETs, only the measurements for CNFET #1 are shown here while similar measurements for CNFET #2 are shown in Figure S1 in Supporting Information. Figure 3a shows the Id-Vg characteristic of the SGS-CNFET #1 for Vgmax ) 4.5 V and Vds ) 30 mV. The extracted frequency exponent R is constant throughout the whole Vg measurement range, with a value R ) 0.99 ( 0.06. Noise amplitude values, A, recorded for Vg ∈ [-4.5,4.5] V are plotted in Figure 3b. Individual 1/f noise spectra measurements corresponding to distinct symbols in plots a and b of Figure 3 (black square, blue triangle, red circle and green star) are shown in Figure S2 (Supporting Information) for completeness. We observe from Figure 3b that A changes by 1 order of magnitude when Vg is swept through the different CNFET operation regions, being locally minimum in the off-state (A ) 2.8 × 10-6 at Vg ) 2.1 V or minimum Id) and on-state (A(Vg > 1 V) ) (1.3 ( 0.14) × 10-6) and maximum in the subthreshold region (A ) 2.1 × 10-5 at Vg ) 1.6 V or maximum gm). The second quantity influencing SNR, besides the noise amplitude A, is sensitivity (GF). An analysis of GF and its scaling with Vg for the SGS-CNFET strain gauge shown in Figure 3a is being presented elsewhere.12 Figure 3c (reproduced from ref 12) plots (GF)2 against Vg, showing the strong dependence of (GF)2 on Vg. A maximum GF ) -463 ( 32 is measured at Imin of device #1 at Vg ) -2 V. Moving away from Imin, GF decreases rapidly toward almost zero in the on-state of the device. We now investigate the SNR in CNFET strain gauges by combining the results from Figure 3b and Figure 3c. SNR is defined as the sensor signal power divided by the integrated noise power. The sensor signal of a piezoresistive CNFET sensor is defined as the change of drain current, ∆Id, under strain. ∆Id can be expressed as a function of GF and the uniaxial strain ε by
∆Id )
-GFε I 1 + GFε d0
FIGURE 3. Low-frequency 1/f noise and electromechanical sensitivity (GF) of SGS-CNFET #1. (a) Gate characteristic measured from Vg ) -4.5 to +4.5 V. (b) Extracted low-frequency noise amplitudes A(Vg). A is minimum in the device on-state and at Imin and largest at maximum transconductance. Distinct symbols (black square, blue triangle, red circle and green star) belong to individual noise spectra shown in Figure S2 (Supporting Information). (c) GF2 of the same SGS-CNFET (reproduced from ref 12). It is maximum at Imin and reduced toward both on-state regions.
bandwidth, defined by two corner frequencies f1 and f2.The SNR is then
SNR )
∫f f SI df 2
1
(2)
)
∆Id2 AId02 ln f2 /f1
≈
(GF)2 1 ε2 A ln f2 /f1
(3)
Since f1 and f2 are defined only at the sensor design time, for the sake of definiteness we take ln(f2/f1) to be 1. The right side of eq 3 is valid only for small strains, which are in our sensors in a range of ε ∈[0, 5 × 10-4]as extracted by finite element analysis.12 The SNR analysis is carried out assuming a moderate strain value of ε ) 10-4.
where Id0 is the drain current at ε ) 0. The low frequency noise power in a CNFET is determined by integrating the current noise PSD defined in eq 1 over the measurement © 2010 American Chemical Society
∆Id2
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device, then SNR is reduced toward the n- and p-type onstate region by 2 orders of magnitude. For the SNR analysis it was assumed that A is independent of the small strain ε for the same applied Vg. This simplifying hypothesis is not true a priori and needs experimental verification which is carried out in the Supporting Information (Figure S3). The discussion of the results begins with an analysis of low-frequency noise measurements based on the previously mentioned charge noise model due to Tersoff.31 In this model, noise is seen to have two components, one of classical 1/f type, proportional to Id2, and the other due to channel potential fluctuations, proportional to (gm/Id)2, with gm the transistor transconductance slope. Tersoff’s model predicts for the noise amplitude A, a constant value in the on-state, a (possibly) maximum value in the transition on/ off region (if gm is large enough), and a local minimum value in the off region (at Imin.) Indeed, the behavior of the model fits the measured noise amplitude dependence on Vg throughout the entire measurement range. On the contrary, according to the classical Hooge model, A should have reached a global maximum at Imin, which is in contradiction to the measurements. This analysis thus confirms that low-frequency noise in SGS-CNFETs obeys the charge noise model of Tersoff rather than the classical Hooge model. A fit of the Tersoff model on our noise amplitude data is shown in the Supporting Information (Figure S4). We also studied lowfrequency noise on two different transistors fabricated on one long SWNT which shows that A is intrinsic to the SWNT and is not dominated by local and time varying effects (see also Figure S4 in Supporting Information). In addition to that we tested, the dependency of noise on the absolute value of Vg and on the dependency of Vg sweep direction (see Figure S5 in Supporting Information). Due to hysteresis Vg corresponding to Imin is changed in the order of volts when changing Vg sweep direction. All additional results clearly show that A(Vg) depends on gm(Vg) and not on the absolute Vg value. We now discuss the gate voltage dependence of SNR. Equation 3 expresses the relationship of the SNR of a CNFET to its strain sensitivity GF and its noise amplitude A. Measurements such as those plotted in Figure 3b,c show that both values, A and GF strongly depend on Vg. The piezoresistive GF is attributed to small changes in the electronic band gap of CNFETs under strain and has a maximum at Imin as mentioned before. On the other hand, A(Vg) in the on/off region around Imin is dominated by potential fluctuations at the CNFET channel (via the gm term). Hence, even though both properties, A and GF, depend on Vg, the positions of their minima and maxima are independent because band gap opening and potential fluctuations are uncorrelated physical mechanisms. Fortunately, it so happens that the maximum strain sensitivity of a SGS-CNFET falls at a position where the noise amplitude is minimal. Nevertheless, this is not so for all CNFET-based sensors. For example, in bio-
FIGURE 4. Signal-to-noise ratio for SGS-CNFET #1 (a) and #2 (b). The SNR peaks at Imin and is reduced toward the n- and p-type on-state conductance.
Experimental SNR results on the two SGS-CNFETs are shown in Figure 4. The SNR values calculated with eq 3 from individually measured A and GF are plotted against Vg in Figure 4a for SGS-CNFET #1 and in Figure 4b for SGS-CNFET #2. Red symbols in Figure 4 correspond to SNR data points that were calculated at Vg values at which discrete GF were measured (cf. Figure 3c for (GF)2 of CNFET #1). The corresponding noise amplitudes, A, were found by spline interpolation of measured A (cf. Figure 3b for measured A of CNFET #1) that in turn correspond to the dashed lines. Conversely, GFs corresponding to the dashed line are spline interpolated from measured GFs (corresponding to red symbols). The maximum SNR in Figure 4a is observed at Imin because at this position (GF)2 is maximum and simultaneously the low frequency noise amplitude exhibits a local minimum. Moving on the Vg axis from the location of Imin toward the p- and n-type on/off region of the transistor, (GF)2 decreases quickly while A rapidly increases by up to a decade. This is the explanation for the decrease of SNR outside of the transistor off-region. Toward the n-type onstate, for Vg > 0.5 V, A decreases by more than a decade compared to the n-type region of maximum gm at Vg ) -1.5 V (cf. Figure 3b). The significant decrease of (GF)2 at the n-type on-state region (Vg > 0.5 V) cannot be compensated by the reduced A compared to A at Imin. Hence, the SNR in the on-state is small despite significant lower noise amplitudes in this region. (GF)2 measured in the p-type region scales similarly to the n-type region. Quantitatively however SNR remains larger compared to the n-type onstate region. The SNR versus Vg of SGS-CNFET #2 (positive GFs) is shown in Figure 4b and is qualitatively similar to CNFET #1. The maximum SNR is reached at Imin of the © 2010 American Chemical Society
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(5)
chemical and chemical sensors both, the signal sensitivity (here associated with a gate threshold shift) and the noise partly share the same physical mechanism, namely, a change of the potential at the CNFET channel. Hence, in these types of sensors the SNR is truly a trade-off between sensitivity and noise, both reaching maximum at gm and thus SNR in this case is not maximum at Imin as shown by Heller et al.17 In summary, we have investigated the signal-to-noise ratio in carbon nanotube piezoresistive transducer elements that are embedded in membrane-based pressure sensors as test benches. We show on small band gap semiconducting CNFETs that the SNR is gate voltage dependent with a maximum at the transistor minimum current, Imin, and lowest in the transistor p-type and n-type on-state region. This effect is explained by the gate voltage dependency of both, the low-frequency noise amplitude, A(Vg), which scales as described by a charge-noise model, and the piezoresistive response GF(Vg) due to a change in the SWNT electronic band gap. These findings are verified in real time sensing experiments where maximum resolution is achieved at Imin. The calculated strain resolution of our devices is as small as 6.2 µε. These results suggest operating SGS-CNFET strain gauges at Imin to maximize sensor resolution.
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Acknowledgment. The authors thank S. Drittenbass and M. Putzi for help with the noise measurements, L. Durrer for the growth of SWNTs, and M. Mattmann, M. Muoth, S.W. Lee, R. Grundbacher, C. Stampfer, P. Studerus, D. Poulikakos, and B. Burg for helpful discussions. Support by ETH FIRST Lab and financial support by ETH Zurich (TH 13/053) are gratefully acknowledged.
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Supporting Information Available. Figures showing noise measurements of SGS-CNFET. This material is available free of charge via the Internet at http://pubs.acs.org.
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