Abstract
Transistor biosensors are massfabricationcompatible devices of interest for point of care diagnosis as well as molecular interaction studies. While the actual transistor gates in processors reach the sub10 nm range for optimum integration and power consumption, studies on design rules for the signaltonoise ratio (S/N) optimization in transistorbased biosensors have been so far restricted to 1 µm^{2} device gate area, a range where the discrete nature of the defects can be neglected. In this study, which combines experiments and theoretical analysis at both numerical and analytical levels, we extend such investigation to the nanometer range and highlight the effect of doping type as well as the noise suppression opportunities offered at this scale. In particular, we show that, when a single trap is active near the conductive channel, the noise can be suppressed even beyond the thermal limit by monitoring the trap occupancy probability in an approach analog to the stochastic resonance effect used in biological systems.
Introduction
Transistorbased biosensors are now widely used as integrated semiconductor devices for genome sequencing^{1}. They are still being further integrated for statistical study^{1,2}, pico or nanoliter volume analysis^{3,4,5} or singlemolecule sensing^{6,7}. Smallest nanotransistorbased biosensors are very similar to massproduction stateoftheart semiconductor transistors^{8,9}, have dimensions close to small biological objects (see Fig. 1), and tend to have a very low charge noise \({S}_{q}\). In particular, the subelementary charge or singlecharge sensitivity ability in liquid^{10} is promising for the development of the nonoptical version of singlemolecule digital nanoarrays^{11}, or nanoelectrochemistry^{12}. In contrast, larger devices, whose dimensions are typically similar to a biological cell (see Fig. 1), have a larger charge noise but tend to have a very low inputreferred voltage noise \({S}_{{V}_{G}}\). Several studies have recently investigated quantitatively the role of gate area \(A\) for \({S}_{{V}_{G}}\) noise^{13,14} in the range \(A>1\) µm^{2}. In principle, \({S}_{{V}_{G}}\) reflects the smallest change in analyte concentration that can be detected with such biosensors. The simplest and effective model for noise (see Eqs. (1a) and (1b)) is based on the fluctuation of the number of active defects (typically gate oxide traps)^{15}.
where \({S}_{I}\) is the power spectral density of current noise, \({g}_{m}\) is the transconductance, \(q\) is the elementary charge, \({N}_{ot}\) is the density of oxide traps (or other charge trapping source), \({C}_{G}\) is the gate oxide capacitance per surface unit, and \(f\) is the frequency.
It appears from Eqs. (1a) and (1b) that \({S}_{q}\) and \({S}_{{V}_{G}}\) scale as \(A\) and \({A}^{1}\), respectively. Equations (1a) and (1b) assume that \({N}_{ot} A\gg 1\), and that the interaction of chemical species with the sensor itself is the signal, therefore it does not affect \({S}_{{V}_{G}}\)_{.} This latter assumption seems to be valid in the majority of cases^{5,10,16} if we neglect drift effects affecting the very lowfrequency signal^{17,18}. Considering a typical lower range of \({N}_{ot}=1{0}^{8} c{m}^{2}\) for the stateoftheart devices, we see that \({N}_{ot} A=1\) corresponds to \(A=1 {\mu m}^{2}\) as a lower boundary^{13,14,15}, and therefore, one can wonder what is the optimal design rule when \(A<1 {\mu m}^{2}\). Even more interestingly, nanoscale devices can offer unique opportunities for noise suppression due to the absence of traps^{10}, correlation effects^{19,20}, and singletrap phenomena^{21,22,23}. Here, we address both questions and show in particular that, by analogy with the stochastic resonance (SR) noise suppression approach found in biological systems, the exploitation of singletrap phenomena in nanoscale devices can be quantitatively described and used for noise suppression, even beyond the thermal noise limit^{2,24}.
Scaling effect on charge and voltage noise and experimental results
Figure 2a,b show some of the stateoftheart experimental results of charge noise (\({S}_{q}^{ 0.5}\)) and inputreferred voltage noise \({S}_{{V}_{G}}\) as a function of \(A\), with \(A\) down to a few hundred \({nm}^{2}\), including some additional experimental data obtained for both N and Ptype devices in the range \(A<1 {\mu m}^{2}\) (see Supplementary Information (SI) for the details). Rather counterintuitively, we see that guidelines based on Eqs. (1a) and (1b) can be considered as reasonable approximations over the full range of \(A\), even in the absence of traps. A way to understand this is to introduce the “charge noise”, that simply considers \({S}_{q}\) as a constant in the whole gate bias range. It was initially introduced by the mesoscopic physics community to describe noise in elementary chargesensitive electrometers (whose origin was typically attributed to fluctuating charges in the substrate)^{25}, but also used by the biosensors community to explain noise in nanoscale silicon transistors^{10,13}, carbon nanotubes^{26}, graphene FETs^{27} or PEDOT:PSSbased organic electrochemical transistor devices^{16}.
A quantitative attempt to the charge noise came from the measurement of noise in trapfree devices, and the origin was attributed to the dielectric polarization (DP) noise related to the thermal fluctuation of dipoles in the gate oxide^{10}. The consideration of a typical dielectric loss tangent tg δ = 3.8 × 10^{–3} for the SiO_{2}^{10,29} gate dielectric could provide a quantitative description as follows:
where \(k\) is the Boltzmann constant and \(T\) is the temperature.
Interestingly, we stress that considering a doublelayer capacitance of 0.2 F m^{2} and tg δ = 5 × 10^{–3} Eq. (2a) could also provide a quantitative agreement to the charge noise measured for liquidgated carbon nanotube transistor sensors^{26} (see Fig. 2a).
Noise suppression in nanoscale devices
Nanoscale devices offer unique opportunities for noise suppression. In this section, we propose a critical review of the various approaches and present the results for both N and Ptype nanodevices.
Ptype and Ntype subµm devices
As it is predicted by Eq. (1a), \({S}_{{V}_{G}}\) noise is inversely proportional to the gate area for both Ntype and Ptype FETs. However, it should be noted that the authors in Ref.^{15} suggested that Ptype transistors might have a lower noise level than Ntype devices due to lower \({N}_{ot}\) for Ptype structures in relation to different tunneling parameters (e.g. carriers effective mass) for electrons and heavy holes. Our experiments performed with nanoscale devices fabricated in the same technological run show that this is not necessarily the case for subµm devices (see Fig. 2a,b). One reason could be that the energy distribution of the few traps in scaled devices is pretty similar for both N and Ptype structures performed with the same fabrication protocol. Another one would be that the Coulomb repulsion effect between traps could be more effective for Ptype devices. Such an effect is seen only when a transistor has multiple traps^{19} (e.g. typically for micrometric devices). In the case of a single trap, \({S}_{{V}_{G}}\) noise increases by up to two orders of magnitude when compared to no trap for both Ntype and Ptype devices due to a randomtelegraph signal (RTS) noise whose amplitude \(\Delta I={g}_{m}\times {q}^{*}/({C}_{G}\times A)\), where \({q}^{*}\) being an effective charge of about \(0.5 q\) for SiO_{2} that accounts for image charge effects^{19}. RTS noise has a Lorentzian power spectrum shape (see Fig. 1) that can be evaluated as^{19}:
where \(g\) denotes the trap occupancy probability (gfactor) given by:
where \({\tau }_{e}\) is the emission time of a charge from the trap and \({\tau }_{c}\) is the capture time in the trap. RTS noise is maximized relative to the background DP noise at the frequency equal to the Lorentzian corner frequency \({f}_{0}\). By considering the probability of the trap to be occupied equal to 50%, the maximum of RTS noise can be estimated as^{19}:
According to Eq. (5), RTS noise tends to increase with capacitance decrease showing a stronger dependence than DP noise. RTS noise is typically above DP noise (see Fig. 2b,c). Such behavior demonstrates the effect of the presence of a single trap on the nanotransistor biosensor performance. However, as we discuss below, RTS noise can be suppressed by considering the singlecarrier trappingdetrapping process as a signal rather than a parasitic effect. Moreover, better performance of nanobiosensors exploiting RTS is expected for devices covered with highk dielectrics. Typically, highk materials possess higher dielectric constants and lower values of dielectric loss tangent compared to the conventional SiO_{2}. This leads to larger RTS amplitude and lower DP noise and, therefore, the improved performance of singletrap phenomena is expected in nanotransistors with highk gate insulators.
Dualgate devices
The use of dualgates (gate coupling effect^{2,22}) for the nanotransistor biosensors in which a liquidgate remains fixed and a backgate is monitored has attracted substantial interest due to the possibility to capacitively amplify the signal by the ratio of the top gate to the bottom gate capacitances^{30}. This approach is, however, not necessarily providing a larger signaltonoise ratio (S/N) as the noise is amplified exactly in the same manner. Still, one can argue that this approach provides noisefree amplification, which can simplify the electronics acquisition setup.
Defectfree devices
The second noise suppression effect due to the “nanometer dimension” is the fact that there are statistically no oxide traps for devices of a few tens of nanometers (see Fig. 2b). The gain compared to devices with traps (at fixed capacitance) is about a factor 12 (1200%)^{19}. One could have expected a gain of several orders of magnitude in \({S}_{{V}_{G}}\), but it is restricted due to the presence of the thermal DP noise (see Fig. 2b,d). As Eqs. (1b) and (2b) have different dependence on \({C}_{G}\), \({S}_{{V}_{G}}\) is relatively lower for DP noise with thicker oxides when compared to the trapping/detrapping noise (see Fig. 2d).
Singletrap phenomena as a stochastic resonance effect
The third noise suppression effect, as introduced in^{21,22}, aims to exploit the presence of a single active trap in a gate dielectric layer of a nanotransistor, where RTS noise is observed (see Fig. 3a,b). Such an RTS effect is usually avoided as it increases the noise level (see Fig. 2b,c), but if RTS parameters (i.e. trap occupancy probability, time constants) are monitored (see Fig. 3b), then RTS noise becomes a signal. Intuitively, one could expect that the use of RTS as a signal would provide a gain corresponding to the difference between a singletrap and a trapfree device, e.g. between one and two orders of magnitude (see Fig. 2a,b). Below, we show that the potential of singletrap phenomena for the noise suppression is even larger and that it is similar to the SR effect observed in biology^{31}, enabling here to overpass the thermal DP noise limit. The idea beyond this is that the addition of white noise to a signal that is nonmeasurable below a given threshold can become measurable (see Fig. 3c). As RTS is nothing but a white noise below a cutoff frequency that is added to the signal of interest, there are obviously some similarities (Fig. 3d). However, a technical difference comes from the fact that RTS time constants are related to the signal of interest (surface potential), which is usually not the case for the white noise. Below, we combine theory and experiments to push the limits of noise suppression with singletrap phenomena.
Noise suppression beyond the thermal limit
In this section, the aim is to propose a theoretical framework for the signaltonoise ratio in the case of the singletrap phenomena approach. We consider the trap occupancy probability as the signal and evaluate the noise of \(g\) to determine the S/N ratio (see Fig. 3e,f). We demonstrate experimentally, numerically, and analytically that under optimized conditions, the S/N ratio can be beyond that of the thermal noise in trapfree devices.
Trap occupancy probability \({\varvec{g}}\) and numerical simulation
The usual signal in transistorbased biosensors is a shift of drain current, and current fluctuations are noise. In contrast, we define the signal in singletrapbased biosensors^{21,22,32} as trap occupancy probability \(g\). To calculate the gfactor noise (fluctuations in time) considering twolevel RTS time trace, one can extract \(g\left(t\right)\) over a given window \(\Theta \) directly from the distribution of the voltage fluctuations (see Fig. 3f). Then, by sliding the window along with the RTS time trace one can obtain a new time trace with the trap occupancy factor fluctuations in time. The timedomain gfactor data can be then translated into frequency spectrum resulting in the power spectral density \({S}_{g}\).
Experimental results
Figure 4a shows the twolevel drain current fluctuations measured for the 100 nm wide and 100 nm long liquidgated Si NW FET. The device was fabricated using a previously reported protocol^{32}. A brief description of the main fabrication steps is also presented in Supporting information (SI) of this work. All noise measurements were performed in a custombuilt Faraday cage using a fullyautomated ultralownoise measurement setup^{22,33}. The transistor demonstrating RTS noise behavior was biased in the linear operation regime and measured at room temperature. To extract drain current states for measured RTS time trace a method based on a hidden Markov model^{34,35} was applied (see Fig. 4a). Average capture and emission time constants characterizing measured RTS process are shown in Fig. 4b. The average emission characteristic time remains about constant, while the average capture characteristic time demonstrates a strong dependence on the liquidgate voltage applied. It should be noted that such behavior of RTS time constants is typical for liquidgated nanowirebased FET devices^{22,32,36}.
In order not to be limited statistically and to have long enough RTS time traces for calculation of gfactor noise, we also generated RTS noise numerically using master Equations^{37} (see Equations S1S3 in SI) with additional consideration of DP noise^{10}. Simulated RTS noise has characteristics similar to that obtained for experimentally measured RTS noise as it is shown in Fig. 4b. The trap occupancy factor noise taken at 10 Hz for both measured and simulated time traces is plotted in Fig. 4c. It should be noted that the data shown in Fig. 4c is obtained for RTS with \(g=0.5\), which corresponds to the case when the trap energy level coincides with the Fermi level of the system. At this condition the number of transition events between the states is maximized, so the noise introduced by the calculation of the trap occupancy factor (gfactor noise) is also maximized.
As can be seen from Fig. 4c, the gfactor noise decreases with increasing the time window \(\Theta \). The dependence of gfactor noise against the time window can be explained by considering the fact that the larger time window contains more transition events enabling gfactor to be estimated with higher accuracy, as illustrated in Fig. 3f.
gfactor noise analytical model
Let’s consider a twolevel RTS signal \({X}_{t}\) that jumps between states 0 and 1. The transition probabilities \(P\) for an RTS with states (0, 1) and rates \((\lambda ,\mu )\) to jump from states 0 to 1, and 1 to 0, respectively, are given by Kolmogorov´s forward equation:
Then, we consider the trap occupancy probability \(g\) (our signal), averaged over a time window \(\Theta \) and defined as:
where \({1}_{\left\{{X}_{S}=1\right\}}\) is the indicator function (equal to 1 if \({X}_{S}=1\)). To obtain the autocorrelation function \(C\left(s\right)\) of \(g\), we consider the expected value \(E\) and obtain:
where \(s\) is the time lag. Then, \(C\left(s\right)\) can be written as:
We see that the autocorrelation function follows two regimes that are related to the averaging filter and the stochastic charge transfer, respectively (see Fig.S5). After some simplifications, an analytical model can be obtained for the power spectral density of \(g\), in the case where \(\lambda =\mu =\gamma \) (i.e. \(g=0.5\)) as:
where \(\omega =2\pi f\) and \(\Theta \) is a duration of a sliding time window (see Fig. 3(f)).
Inputreferred gfactor noise \({{\varvec{S}}}_{{\varvec{g}}{\varvec{g}}}\)
To compare the performance and efficiency of the nanotransistor sensors exploiting singletrap phenomena, one should first introduce and calculate an equivalent inputreferred noise caused by the variation of the gfactor. This can be done similarly as for the voltage noise (see Eq. (1b)) defining the inputreferred trap occupancy factor noise as:
where \({S}_{g}\) is the gfactor power spectral density and \({g}_{g}\) is the gfactor derivative calculated as \(\frac{\partial g}{{\partial V}_{G}}\).
Figure 5a shows the inputreferred voltage noise power spectral densities with Lorentzian fittings measured for the same 100 nm wide and 100 nm long liquidgated Si NW FET demonstrating pronounced twolevel RTS noise (see Fig. 4a). The dark blue dashed line represents the 1/f DP noise dependence calculated for the device using Eq. (2b) and considering tg δ = 3.8 × 10^{–3}. As can be seen, the measured voltage noise is larger than the inputreferred gfactor noise \({S}_{gg}\) calculated using Eqs. (10) and (11) with \(g=0.5\), \(\gamma =488 s^{1}\) (as for the experimental data), and \(\Theta =20 s\). Therefore, noise in the sensors exploiting the RTS phenomenon, in fact, can be suppressed when considering optimized conditions for calculation of gfactor. For this purpose, mainly three parameters need to be carefully considered: a time window \(\Theta \), RTS frequency \({f}_{0}\), and a slope of gfactor dependence on the gate voltage applied.
The importance of the time window \(\Theta \) on gfactor noise \({S}_{g}\) is shown in Fig. 4c. A large enough window that contains enough number of transition events (> 200)^{38,39} is needed for the meaningful statistical evaluation of \(g\). However, the number of switching events between two levels within a given period of time also strongly depends on the RTS corner frequency \({f}_{0}\). The number of transitions over time \(\Theta \) is higher for the highfrequency RTS than for the lowfrequency RTS in the case of the same trap occupancy probability. Therefore, the gfactor can be evaluated with more accuracy for the fast RTS considering the same amount of time as for the slow (lowfrequency) RTS process (see Fig. 5b).
The slope (steepness) of gfactor dependence on the gate voltage applied is another important parameter defining the efficiency of the single trapphenomena for biosensing. The gfactor curves with different slopes are shown in Fig. 5c. For the sensors exploiting the RTS effect, the signal is the pronounced changes in RTS parameters (i.e. gfactor, capture time, etc.) induced by the depletion or accumulation of charge carriers in the silicon nanowire when the charged biomolecules are attached to its surface. Therefore, the sensitivity for the RTSbased sensors can be written as:
The sensitivity dependence on the gfactor slope (steepness) is proven also experimentally^{22}. Moreover, according to Eq. (11), the inputreferred gfactor noise also strongly depends on the slope of the gfactor curve. As can be seen in Fig. 5d, \({S}_{gg}\) noise can be, in fact, decreased by up to an order of magnitude due to the effect of the gfactor slope.
Analysis of the signaltonoise ratio for trapbased nanobiosensors
The signaltonoise ratio is an important parameter for any sensor demonstrating its sensing capability. Therefore, this parameter needs to be carefully investigated for the nanotransistor sensors exploiting singletrap phenomena to optimize experimental conditions. Traditionally, for transistorbased biosensors monitoring threshold voltage shift as a signal, the signaltonoise ratio can be defined as follows:
where \({S}_{{V}_{G}}\) is the equivalent inputreferred voltage noise, and \(\delta {V}_{Th}\) is a threshold voltage shift caused by the interaction of the target biomolecule with the sensing surface of the biosensor. The signaltonoise ratio for nanobiosensors whose working principle is based on the singletrap phenomena can be determined similarly:
The S/N ratio calculated for RTS noise with different corner frequencies at \(g=0.5\) is shown in Fig. 6a. A larger number of transition events due to the higher RTS rate (\(\gamma ={\pi f}_{0}\)) results in smaller \({S}_{gg}\) noise (see Fig. 5b) which leads to the increase of the S/N ratio. Figure 6b demonstrates the S/N ratio calculated for RTS phenomena with different gfactor slopes (see Fig. 5c). The dashed line reflects the S/N level for the trapfree device with the same gate capacitance as for one with the single trap demonstrating DP noise only. As a signal, we used the threshold voltage shift of 5.9 mV caused by 0.1 pH change in the gating solution when considering ideal ionsensitive FETbased sensors. It can clearly be seen from Fig. 6a,b that under optimized conditions the S/N ratio can indeed substantially be increased even above the level expected for trapfree devices monitoring the threshold voltage shift as a signal.
Discussion
The discrete number of traps in nanoscale devices offers a rich toolbox for optimizing the S/N ratio. In the case of the absence of traps, the dielectric loss of the gate oxide can be a tunable parameter in addition to the oxide thickness (see Fig. 2c,d). For singletrap phenomena, the RTS frequency is the main parameter as the best performance is obtained for a trap occupancy probability averaged over many events. The trap operation frequency is not easy to control, even though progress has been reported towards "ondemand" trap generation^{23}. Instead, a simple way to increase the trap operation frequency is to consider a larger gfactor by tuning the gate bias (see Fig. 6b). This comes from the fact that \({\tau }_{e}\) is almost constant (see Fig. 4b) and therefore, at relatively high \(g\), RTS corner frequency \({f}_{0}\approx 1/(2\pi {\tau }_{c})\). The alternative is to play with the slope of \(g\) (Fig. 5d). In principle, the slope is only determined by the temperature (Fermidistribution), and the trap depth (potential drop in the oxide), but in practice, punctual charges are very sensitive to correlation effects^{19} as often observed in electrochemical monolayers^{12}. A gain in the S/N ratio is obtained in the case of "attractive" interactions.
It should be noted that in order to improve the S/N ratio for the singletrap phenomena approach applied for the biomolecular detection, the time window for the analysis of biomolecular signal should be optimized taking into account the parameters of the designed transducers and definite type of biological object under study. For example, in Ref.^{32} we measured and analyzed 40 s long RTS time traces to detect very low concentrations of target biomolecules. As a result, enhanced sensitivity was achieved and demonstrated for the singletrap phenomena approach. Our estimations, based on the equations in the present paper, show that in the case of a 40 s long time window, the gfactor noise is substantially lower compared to the measured RTS noise. This results in a considerably improved S/N ratio and demonstrates the validity of the approach.
From a more general perspective, singletrap phenomena can be considered as an SR effect when a white noise added to a signal enables better sensitivity and performance. As in biological systems, the white noise source is embedded. However, the particularity of singletrap phenomena is that the "discrete nature" of this white noise source is exploited (as in other singleelectron devices^{40}) as well as the fact that it is related to the physical parameters. Finally, one could argue that the best way to exploit singletrap phenomena would be to keep the signal digital, as it is an energyefficient way of sensing and computing^{41}, i.e. without requiring analog to digital converters.
Summary and conclusion
The lowfrequency noise plays an important role in any type of sensors determining their capability to detect small signals coming from the analyte. In this work, we have proposed and discussed the noise suppression techniques for FETbased nanosensors including the exploitation of RTS noise as a signal. We demonstrated that the signaltonoise ratio can, in fact, considerably be increased for the singletrap phenomena approach. The results are very important for biosensing applications as well as for future nanotechnologies including the development of innovative chargetrap based memory devices and quantum computing systems.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors acknowledge the Seed Money funds for supporting a new international collaboration as part of the RTSBiosensor project. This work was partially supported by the JSPS CoretoCore Program (A. Advanced Research Networks). Y. Kutovyi greatly appreciates a research grant from the German Academic Exchange Service (DAAD). I. Madrid acknowledges the interdisciplinary research funds from CNRS for the BIOSTAT project. N. Clement acknowledges ANR SIBI and KAKENHI (26289238) scientific research projects. Y. Kutovyi and S. Vitusevich also gratefully acknowledge the Innovation Award of RWTH Aachen University as part of RWTH transparent 2016. Open access funding provided by Projekt DEAL.
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Kutovyi, Y., Madrid, I., Zadorozhnyi, I. et al. Noise suppression beyond the thermal limit with nanotransistor biosensors. Sci Rep 10, 12678 (2020). https://doi.org/10.1038/s4159802069493y
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DOI: https://doi.org/10.1038/s4159802069493y
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