Flexure-FET biosensor to break the fundamental sensitivity limits of nanobiosensors using nonlinear electromechanical coupling

Abstract

In this article, we propose a Flexure-FET (flexure sensitive field effect transistor) ultrasensitive biosensor that utilizes the nonlinear electromechanical coupling to overcome the fundamental sensitivity limits of classical electrical or mechanical nanoscale biosensors. The stiffness of the suspended gate of Flexure-FET changes with the capture of the target biomolecules, and the corresponding change in the gate shape or deflection is reflected in the drain current of FET. The Flexure-FET is configured to operate such that the gate is biased near pull-in instability, and the FET-channel is biased in the subthreshold regime. In this coupled nonlinear operating mode, the sensitivity (S) of Flexure-FET with respect to the captured molecule density (N_s) is shown to be exponentially higher than that of any other electrical or mechanical biosensor. In other words, while , classical electrical or mechanical biosensors are limited to S_classical ∼ γ₃N_S or γ₄ ln(N_S), where γ_i are sensor-specific constants. In addition, the proposed sensor can detect both charged and charge-neutral biomolecules, without requiring a reference electrode or any sophisticated instrumentation, making it a potential candidate for various low-cost, point-of-care applications.

Nanoscale biosensors are widely regarded as a potential candidate for ultrasensitive, label-free detection of biochemical molecules. Among the various technologies, significant research have focused on developing ultrasensitive nanoscale electrical (1) and mechanical (2) biosensors. Despite remarkable progress over the last decade, these technologies have fundamental challenges that limit opportunities for further improvement in their sensitivity (Fig. 1A) (3–6). For example, the sensitivity of electrical nanobiosensors such as Si-Nanowire (NW) FET (field effect transistor) (Fig. 1B) is severely suppressed by the electrostatic screening due to the presence of other ions/charged biomolecules in the solution (7), which limits its sensitivity to vary linearly (in subthreshold regime) (3, 7) or logarithmically (in accumulation regime) (4, 7, 8, 9) with respect to the captured molecule density N_s. Moreover, the miniaturization and stability of the reference electrode have been a persistent problem, especially for lab-on-chip applications (1). Finally, it is difficult to detect charge-neutral biological entities such as viruses or proteins using charge-based electrical nanobiosensor schemes.

Fig. 1.

(A) Sensitivity S of different types of biosensors; e.g., (B) electrical (Si-NW FET) in which transduction is achieved by modulation of channel conductivity (G) when charged biomolecules are captured by the gate. (C) Transduction in cantilever-based nanomechanical biosensors is achieved by change in its mass, stiffness, or surface stress. Nanocantilever can be operated in dynamic mode (mass change-based detection using shift in resonance frequency) or in static mode (surface stress change based detection using piezoresistive material). (D) Proposed Flexure-FET biosensor in which transduction is achieved due to change in the stiffness of the suspended gate. (E) Operation of Flexure-FET below pull-in. Displacement of the suspended gate (y) and drain current (I_DS) as a function of applied gate bias V_G. The y changes rapidly near pull-in (V_G ≈ V_PI) and I_DS increases exponentially with V_G in the subthreshold regime (V_G < V_T).

In contrast, nanomechanical biosensors like nanocantilevers (10, 11) (Fig. 1C) do not require biomolecules to be charged for detection. Here, the capture of target molecules on the cantilever surface modulates its mass, stiffness, and/or surface stress (5, 11, 12). This change in the mechanical properties of the cantilever can then be observed as a change in its resonance frequency (dynamic mode), mechanical deflection, or change in the resistance of a piezoresistive material (static mode) attached to the cantilever (6, 13). Unfortunately, typical optical detection schemes (10) require complex instrumentation which may preclude them from many low-cost point-of-care applications. Further, the response of nanomechanical biosensors varies only linearly (5) or logarithmically (6, 14, 15) with the change in the mass or surface stress of the cantilever, and therefore these sensors may not be sufficiently sensitive to detect target molecules at very low analyte concentrations, unless sophisticated, low-noise setup is used.

To overcome the respective limitations of classical electrical and mechanical nanoscale biosensors, we propose the concept of a Flexure-FET biosensor that integrates the key advantages of both technologies but does not suffer from the limitations of either approach. The Flexure-FET consists of a nanoplate channel biased through a thin-film suspended gate (Fig. 1D). Although the structure is similar to that of a suspended-gate FET (16), nano-electromechanical (NEM) FET (17), or resonant gate transistor (18), we call the device Flexure-FET to emphasize its distinctive nonlinear operation specifically optimized for ultrasensitive detection of biomolecules. As shown in Fig. 1E, the ultra high sensitivity arises from the coupling of two electromechanical nonlinear responses, namely (i) spring-softening (19) in which stiffness decreases nonlinearly with the applied gate bias V_G and vanishes at the pull-in point (for detailed discussions on pull-in instability, see refs. 20, 21), and (ii) subthreshold electrical conduction (22) in which current depends exponentially on the surface potential (Fig. S1). Such nonlinear electromechanical coupling enables exponentially high sensitivity for Flexure-FET sensors (Fig. 1A), which is fundamentally unachievable by exclusive use of existing nanoscale electrical or mechanical biosensors. Moreover, the reliance of change in stiffness (23, 24) ensures screening-free detection of charged/neutral molecules, with no need for a reference electrode, and the measurement of drain current for detection requires no complex instrumentation. It should be noted that from a mechanical perspective, the Flexure-FET operates close to pull-in instability, a critical point. Similar critical point sensing has also been reported for vapor sensors (25) that operate close to bucking-instability (25) and for mass sensors that operate close to saddle-node bifurcation (26), and their higher sensitivity has been confirmed experimentally. However, beyond the critical point sensing, the integrated transistor-action in the subthreshold regime provides the Flexure-FET an additional exponential sensitivity (and simpler direct current readout) that could not be achieved by the classical nonlinear sensor schemes.

Theory of Flexure-FET

Sensor Configuration Before Target Capture.

The operating principle of Flexure-FET can be understood using the well established spring-mass model (Fig. 2) (17, 18). With the application of gate bias V_G, the gate moves downward toward the dielectric (y vs. V_G curve in Fig. 1E), and the corresponding increase in gate capacitance is reflected in the increased drain current I_DS, as shown in Fig. 1E. The static behavior of the device is dictated by the balance of spring and electrostatic forces; i.e.,

[1]

where is the stiffness, α is a geometrical factor, E is the Young’s modulus, W is the width, H is the thickness, L is the length of the gate electrode, y₀ is the air-gap, y is the position of the gate electrode, ϵ₀ is the permittivity of free space, E_air is the electric field in the air, and A = WL is the area of the gate electrode. The electric field below the membrane E_air is equal to ϵ_sE_s(ψ_s), where ϵ_s is the dielectric constant of the substrate, and

[2a]

where E_s(ψ_s) is the electric field at the substrate-dielectric interface (22, page 64, for a detailed derivation of Eq. 2a), ψ_s is the surface potential, q is the charge of an electron, N_A is the substrate doping, k_B is the Boltzmann constant, T is the absolute temperature, and n_i is the intrinsic carrier concentration in the substrate. The voltage drop in air (yϵ_sE_s(ψ_s)), dielectric (), and substrate (ψ_s) can be related to the applied gate bias V_G as follows-

[2b]

where, y_d is the dielectric thickness and ϵ_d is the dielectric constant. Eqs. 1 and 2 are solved self-consistently for y and ψ_s at each V_G. The corresponding inversion charge density (Q_i) in the channel and drain current (I_DS) are given by

[3]

[4]

where μ_n is the channel mobility for electrons, V_DS is the applied drain to source voltage. Fig. 1E shows the steady-state response of Flexure-FET as a function of biasing voltage V_G, obtained from the numerical simulations of Eqs. 1–4.

Fig. 2.

(A) and (B). Equivalent spring-mass model of Flexure-FET. Stiffness changes from k to k + Δk after the capture of biomolecules, and therefore position of the gate changes from y to y + Δy, which results in the modulation of drain current from I_DS1 to I_DS2.

Flexure-FET Response to Target Capture.

For transduction, the proposed Flexure-FET biosensor utilizes the change in suspended gate stiffness from k to k + Δk, (12, 24, 27–29) due to the capture of biomolecules. The change in stiffness due to the capture of biomolecules has been demonstrated by several recent experiments of mass sensing using nanocantilever-based resonators (12, 27, 28) (Fig. S2). This well known observation of stiffness change has been attributed to the change in the membrane thickness, Young’s modulus, and/or surface stress of the beam (12, 23, 24, 30). Indeed, Craighead (27) suggests its use as a basis of a new class of mechanical biosensor.

In the following analysis, we model change in k by change in the effective thickness H of the gate (ΔH), although it should be stressed that the conclusions do not depend on the particular hypothesis regarding Δk. For now, we ignore the details of the spatial distribution of molecules associated with random sequential adsorption (31) and assume a uniform distribution of adsorbed molecules on the sensor surface. Therefore, the conservation of volume suggests ΔH = N_sA_tH_t, where N_s is the area density, A_t is the effective cross-sectional area, and H_t is the effective thickness of the target molecule. Using the fact that , the change in stiffness Δk due to ΔH( ≪ H) can be related to adsorbed molecule density N_s as follows:

[5]

For simplicity, we have taken the Young’s modulus of captured molecules to be the same as that of the membrane, but this is obviously not necessary, and the theory can be generalized by the methods developed in Tamayo, Ramos, Mertens, and Calleja (23).

Combining Eqs. 1 and 2b, we get k(y₀ - y)y² ≈ ϵ₀A(V_G - ψ_s)²/2. Now, the change in gate position Δy for small change in stiffness Δk due to capture of biomolecules is given as

[6]

If Flexure-FET is biased close to pull-in (), the nonlinear Δy² term dominates the linear Δy term in Eq. 6. It is essential to bias the Flexure-FET in this nonlinear, close-to-pull-in regime for maximum sensitivity. Using Eqs. 5 and 6, we find

[7]

where is a bias and device dependent constant.

Since the electrostatic force in subthreshold regime is given by (Eq. 2a), the corresponding change in the surface potential Δψ_s is obtained by perturbation of Eq. 1; i.e.,

[8]

Using Eqs. 2a, 3, and 4, we can calculate the drain current I_DS in the subthreshold regime as follows,

[9]

Now, the ratio of the drain current before (I_DS1) and after (I_DS2) capture of biomolecules (in terms of the change in surface potential Δψ_s) is given by

[10]

Using Eqs. 8 and 10, the ratio I_DS1/I_DS2 is given by

[11]

Therefore, if Flexure-FET is operated close to pull-in and in subthreshold regime, sensitivity S (using Eqs. 5, 7, and 11) is given by

[12]

where and . The sensitivity S is defined as I_DS1/I_DS2, because I_DS decreases after capture (see next text section).

Eq. 12 is the key result of the paper and shows how nonlinear interaction between mechanical (spring-softening) and electrical (subthreshold) aspects of sensing leads to an exponential sensitivity to capture of biomolecules. Such gain in sensitivity is impossible to achieve exclusively by electrical or mechanical sensing mechanisms.

Numerical Confirmation of Flexure-FET Response.

The compact analytical expression of sensitivity of the Flexure-FET sensor can be validated by the self-consistent numerical solution of Eqs. 1–4. The results for the change in sensor characteristics due to the capture of biomolecules are summarized in Fig. 3. For example, Fig. 3A shows y vs. V_G before and after capture of target molecules. After the capture, the gate moves up (for a fixed V_G) due to increased restoring spring force (because of increase in the k; Fig. 3A). Interestingly, change in gate position Δy is maximum close to pull-in due to spring-softening effect, as shown in Fig. 3B (see Figs. S3, S4 and S5 in SI Text for experimental validation). The change in gate position Δy is directly reflected in change in I_DS. Fig. 3C shows I_DS vs. V_G before and after capture of biomolecules. Interestingly, I_DS decreases after capture due to increased separation between the gate and the dielectric (hence decreased capacitance). The corresponding ratio of the currents I_DS1 (before capture) and I_DS2 (after capture) increases exponentially with Δy (Fig. 3D), and becomes maximum near pull-in. Note that the results from detailed numerical simulations are accurately anticipated by Eq. 11, thus validating the analytical model described in the previous section. Therefore, by operating the Flexure-FET close to mechanical pull-in and in electrical subthreshold regime, orders of magnitude change in I_DS can be easily achieved for typical surface density of N_s = 5 × 10¹² cm^-2, projected area of the biomolecule, with R_t = 1 nm, and H_t = 5.1 nm. These parameters translate to just an equivalent Δk ∼ 6%. Note that to achieve the maximum sensitivity, it is important to bias the Flexure-FET in subthreshold regime below pull-in (i.e., V_T ≈ V_PI).

Fig. 3.

Change in the sensor characteristics due to capture of target molecules on the surface of the gate, (A)y vs. V_G before and after capture, and (B) corresponding change in the position of gate electrode Δy vs. V_G. The Δy increases rapidly near pull-in due to spring-softening effect. The capture of target molecules is directly mirrored in the change in I_DS. (C) I_DS vs. V_G before and after capture, and (D) corresponding ratio of the two currents I_DS1 (before capture) and I_DS2 (after capture) as a function of Δy. Symbols denote the numerical simulation and solid line analytical formula (Eq. 11). The device considered has the following typical parameters: L = 4 μm, W = 1 μm, H = 40 nm, E = 200 GPa, y₀ = 100 nm, y_d = 5 nm, ϵ_s = 11.7, ϵ_d = 3.9, N_A = 6e16 cm^-3.

Comparison with Classical Sensors

Next we compare the sensitivity of the proposed Flexure-FET sensor with the current nanoscale electrical/mechanical biosensors. Fig. 4A indicates that the Flexure-FET sensors are exponentially sensitive to change in stiffness or captured molecule density N_s (symbols: numerical simulation, solid line: analytical result; Eq. 12). In the following, we explain the origin of linear (or logarithmic) sensitivity for electrical and mechanical nanoscale biosensors.

Fig. 4.

Comparison of the sensitivity of different biosensors. Sensitivity S (A) Flexure-FET (symbols denote the numerical simulation). (B) Si-NW biosensors in subthreshold and accumulation regime. (C) Resonance mode nanomechanical biosensors. (D) Surface stress change-based piezoresistive nanomechanical biosensors, as a function of N_s or ρ. In (B–D), symbols are the experimental data and the line is the guide to the eye.

Electrical Nanobiosensors.

For Si-NW FET biosensors, which also have the optimal sensitivity in subthreshold regime (3), sensitivity S is defined to be the ratio of conductance G (after) and G₀ (before) capture of target molecules (assuming conductance increases after the capture). Therefore, using Eq. 9, S can be approximated as

[13]

Unfortunately, the detection of biomolecules in a fluidic environment involves electrostatic screening by other ions in the solution. Consequently, the surface potential scales logarithmically with biomolecule density; i.e., (q/k_BT)Δψ_s ∝ ln(δN_s) (7) , where δ is a constant that depends on ionic strength and properties of dielectric/fluid interface. Therefore, optimal sensitivity of Si-NW biosensors is given by

[14]

In Fig. 4B, S is plotted against volume concentration ρ, as the captured molecule density N_s ∝ ρ [linear regime of Langmuir isotherm (7)]. Therefore, all the conclusions regarding the dependence of sensitivity on N_s also hold for ρ. It should be noted that the reported sensitivity in the subthreshold regime (3) is actually sublinear (Fig. 4B), below the maximum sensitivity limit defined by Eq. 14 that can be achieved in this sensing regime. In the accumulation or the inversion regimes, S_SiNW ∝ Δψ_s (7), and therefore, S_SiNW ∝ ln(N_s), as shown in Fig. 4B (4, 7). Similar logarithmic dependence of sensitivity was reported in other references (8, 9) as well.

Mechanical Nanobiosensors.

For nanomechanical biosensors such as resonance mode nanocantilever, the sensitivity S is defined as ω₀/ω, where ω is the resonance frequency after the capture of target biomolecules, and ω₀ is the resonance frequency before capture. Using the well known fact that , where k is the stiffness and m is the initial mass of the cantilever, S is given by

[15]

where m^∗ is the mass of individual biomolecule and Δm = N_sWLm^∗ is the added mass of the biomolecules. Therefore, the sensitivity of mechanical biosensor can only vary linearly with N_s. This theoretical prediction is confirmed by experimental data (5) in Fig. 4C. We emphasize that the nanomechanical biosensors—with careful design and appropriate instrumentation—can be extraordinarily sensitive; indeed, zeptogram mass detection (32) has been reported. Eq. 15 simply suggests that the sensitivity of such sensor still varies linearly with respect to N_s.

It is also important to realize that the linear sensitivity with N_s is achieved only if the change in stiffness due to capture of molecules is negligible. In general, however, capture of target molecules increases stiffness of the membrane. If this increase in stiffness compensates the corresponding increase in the mass, there might be no change in resonance frequency at all (12, 28), and the sensitivity could be vanishingly small. One must independently measure the change in the stiffness (29, 30) to decouple the mass effect from stiffness effect so that the mass of the adsorbed molecule can be correctly estimated. In contrast, the Flexure-FET relies only on the change in the stiffness and works in the static mode, and therefore requires no more than a simple measurement of the drain current.

Another class of nanocantilever sensor involves operation in the static mode, where the capture of the target molecules introduces a surface stress, which in turn bends the cantilever. The displacement Δy of the tip can in principle be measured using sophisticated optical readout methods, but a simpler approach can be used instead: One can measure the change in surface stress by measuring the change in the resistance of a piezoresistor attached to the cantilever. For these piezoresistive-based cantilever biosensors, the sensitivity is defined as the ratio of resistance before (R₀) and after (R) the capture of biomolecules. Fig. 4D shows a logarithmic dependence of S on ρ. Similar logarithmic dependence for surface stress change has also been reported (14, 15). We therefore conclude that these static mode sensors do not exceed linear sensitivity limit of classical sensors.

We summarize the results discussed in this section in Fig. 1A, where the sensitivity of various types of nanobiosensors has been plotted against normalized N_s, defined as the ratio of the measured quantity (either ρ or N_s) to the minimum measured ρ or N_s of the available data. Fig. 1A allows us to conclude that the Flexure-FET biosensor will be exponentially more sensitive compared to existing nanoscale electrical or mechanical biosensors.

Finally, we emphasize that each of the three sequential physical phenomena associated with the operation of Flexure-FET (stiffness change due to capture of biomolecules, pull-in instability, subthreshold conduction) (Fig. S1) has been individually confirmed by numerous experiments based on electromechanical resonators (18, 33) and suspended-gate FET (34). We provide a summary of these experiments in the SI Text (Figs. S2, S4, S5, and S6). In the SI Text, we also suggest that a simple reconfiguration of existing electromechanical resonators or suspended-gate FET in Flexure-FET mode can give rise to exponential sensitivity (Fig. S7).

Conclusion

In this paper, we have demonstrated how the Flexure-FET nanobiosensor achieves exponentially high sensitivity by combining two nonlinear characteristics of spring-softening and subthreshold conduction. This extreme high sensitivity of Flexure-FET, therefore, breaks the fundamental limits of linear or logarithmic sensitivity of classical nanoscale electrical or mechanical biosensors. There are broad ranges of applications that can benefit from this sensitivity gain. For example, the current genome sequencing schemes require PCR (polymerase chain reaction) amplification of DNA strands because of the lower sensitivity of existing biosensors. The high sensitivity of Flexure-FET can eliminate the requirement of multiplication step and hence reduce the cost of sequencing. In addition, we recall that the proposed sensing scheme (i) can detect both charged and charge-neutral molecules, (ii) does not rely on reference electrode (the fundamental roadblock of Si-NW type biosensors), and (iii) obviates the need for any sophisticated and difficult-to-integrate instrumentation. The sensitivity of Flexure-FET can be further enhanced by choosing a softer membrane (having low stiffness) such as some polymer with low Young’s modulus or an ultrathin membrane like graphene. Finally, let us emphasize that the sensing scheme is very general, which converts any change in the mechanical property of the gate electrode or change in the air-gap to the change in the drain current of the FET channel. Therefore, the proposed idea is not necessarily restricted to biomolecules detection but should find broader applications in gas/chemical/pressure sensing as well.