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Elseddawy Ahmed M., Phillips Adel H., Bayoumi Ahmed S. Carbon nanotube-based nanoelectromechanical resonator as mass biosensor. Chinese Physics B, 2020, 29(7): 078501  
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Carbon nanotube-based nanoelectromechanical resonator as mass biosensor
Elseddawy Ahmed M.1, †, Phillips Adel H.2, ‡, Bayoumi Ahmed S3, §
Faculty of Graduate Studies, University of Windsor, Windsor, Ontario, Canada
Faculty of Engineering, Ain-Shams University, Cairo, Egypt
Faculty of Engineering, Kafr-Elsheikh Unversity, Kafr-Elsheikh, Egypt

 

† Corresponding author. E-mail: elsedda@uwindsor.ca adel_phillips@eng.asu.edu.eg ahmed.bayoumi@eng.kfs.edu.eg

Abstract

The use of single walled carbon nanotube-based nanoelectromechanical system (NEMS) resonator to sense the biomolecules’ mass is investigated under the influence of an external ac-field. A single walled carbon nanotube (SWCNT) cantilever has been proposed and studied if the mass is attached at the tip or various intermediate positions. The shift of the resonant frequency and the quality factor have been investigated and show high sensitivity to the attached mass of biomolecule and its position. The proposed SWCNT-based NEMS resonator is a good candidate for sensing and tracing biomolecules’ mass as concentration of acetone in human exhale, resulting in a painless, correct, and simple diabetics’ diagnosis.

PACS: ;85.30.De;;72.80.Vp;;85.40.Qx;
Keyword:;single walled carbon nanotube;;nanoelectromechanical system resonator;;ac-field;;biomolecule;
1. Introduction

Strides throughout nanotechnology resulted in novel tools which enhance fields of sensors and electronics. New engineered materials offer novel properties that enhance state-of-the-art components in cost, speed, power consumption, sensitivity, and density on chip.[1–7] single walled carbon nanotubes (SWCNTs) have attracted considerable attention for the last two decades.[8,9] Their unique mechanical, electrical, and optical properties have made them an attractive candidate for many applications.[10,12,13] Recently, mechanical properties of carbon nanotubes (CNTs) have inspired research.[14,15] Although applications focus primarily on bulk tensile strength, the low mass combined with the record Young’s modulus about 1.2 TPa also results in extremely large increase in resonance frequency of SWCNT resonators.[16–20] CNTs are, also, expected to have major impact on several nanoelectromechanical system (NEMS) applications.[21–25]

NEMS technologies made it possible to detect physical quantities sensitively, such as spin,[21,26,27] molecular mass,[28–35] quantum states,[36] thermal fluctuation,[37] coupled resonance,[38,39] and biochemical reactions.[40,42] Also, NEMS technologies have allowed the rapid, reliable, and label-free detection of specific disease-related molecules, suggesting their potential in specific diseases early diagnosis such as cancer.[43–46] These impact values of NEMS offer motivations to proceed the investigation on the use of single walled carbon nanotube based NEMS resonator to sense trace acetone concentration in human exhale, resulting in a simple, reliable, and painless diabetics breath diagnosis.

The previous literature results[47] emphasized that the quality factor and resonant frequency shift of the proposed NEMS device are highly sensitive to the dimensions (length and diameter) of the SWCNT. The authors of Ref. [47] investigated the case when the biomolecule is only at the tip of the cantilever. Now, it is interesting to investigate the case when the biomolecule is at various intermediate positions along the SWCNT cantilever’s length. There are four sections in this paper: the introduction, the theoretical model and its formalism, the numerical results for the proposed SWCNT based NEMS, and conclusion.

2. Theoretical model

The proposed model is a SWCNT cantilever acting as an NEMS resonator device which uses two leads in coupling with electronic transport. The influences of the induced ac-field and magnetic field are considered in the electron tunneling in the proposed device. Additional tunneling channels are induced when the carriers exchange energy with photons of energy ℏω from the induced ac-field of frequency ω, which represent photon-assisted tunneling.[48]

Figure 1 shows the SWCNT cantilever which is activated to vibrate in the y-direction. The shift of resonance frequency, Δυ, and quality factor Q are given by[49]

where Cg is the coupling gate capacitance with SWCNT, CCNT represents the carbon nanotube capacitance, the gate voltage is Vg, the probability of tunneling considering the photon-assisted tunneling (PAT) is Γ, and G represents the electronic conductance of SWCNT. The resonance frequency of the SWCNT cantilever in Eqs. (1) and (2) is represented as υ. This resonant frequency υ will take the following expression according to the following two cases:

The first case in which the biomolecule is attached at the tip of the SWCNT cantilever (see Fig. 1(a)). So, the effective spring constant kspring (see Eqs. (1) and (2) and the resonant frequency υ are given by[50,51]

where Y represents the Young’s modulus of SWCNT, I represents its moment of inertia, the length of the SWCNT cantilever is L, ρ represents the density of SWCNT, A is its cross section area, and mbio-mol is the mass of the attached biomolecule.

The second case in which the biomolecule is concentrated at any point along the SWCNT cantilever (see Fig. 1(b)). So, the effective spring constant kspring (see Eqs. (1) and (2)) and the resonant frequency υ will take the following form:[50–52]

where a represents the distance from the fixed end of the SWCNT cantilever at the moment when the biomolecule is positioned (see Fig. 1(b)).

Fig. 1. Sketch of single walled carbon nanotube cantilever.

In Fig. 1(b), F is the weight of the attached biomolecule. Landauer–Buttiker formula[53,54] is used to calculate the device conductance G (Eqs. (1) and (2)

where –∂ fFD/∂ E represents the first derivative of the Fermi–Dirac distribution function and is calculated as

Here, kB is the Boltzmann constant, T is the absolute temperature, E is the tunneling carriers’ energy, and EF is the Fermi-energy.

The expression of ΓwithPhoton has been derived by applying boundary conditions to the eigen-functions obtained through solving Dirac equation[55,56] as

where Jn is the nth order Bessel function of the first order appropriate to the nth different side bands, Vac represents the induced ac-field’s peak value, the electron charge is e, the source–drain voltage is represented as Vsd, and fFD represents the Fermi–Dirac distribution function. In Eq. (8), Γn (E + nℏω) is calculated as[55,56]

where the wave vector kn is given by

Here Vb represents the barrier height at the interface between the cantilever and leads, the applied magnetic field is represented by B, ℏ represents the reduced Planck’s constant, m* represents the charge carrier’s effective mass, and qn can be calculated using the length, L, of the SWCNT cantilever

3. Results and discussion

The proposed biomolecule sensor based on the SWCNT system will be analyzed numerically for two cases; the biomolecule is positioned at the tip of the SWCNT cantilever in the first case, and in the second case, the biomolecule is at various intermediate positions. Acetone molecule is the aim of this investigation as the diabetic patients’ exhale contains it.[51,57] And any biomolecule of the same mass rang would give consistent results.

The parameters in Eq. (7) for the conductance G[55,56] are EF = 0.125 eV, m* = 0.054 me (where me represents the free electron mass), Cg = 0.4 nF, and CCNT = 0.25 nF. This investigation focuses on constant diameter and so that constant Young’s modulus and density, and the effect of the SWCN diameter variation is not considered.

Figure 2 illustrates the behavior of conductance G with photon energy, E, of the induced ac-field at various magnitudes of the applied magnetic field, B. This figure shows an oscillatory behavior of conductance with different peak heights, small side resonant peaks surrounding the main resonant peaks. The cause of the small side-peaks might be the PAT procedure such as the photon absorption and emission.[48] Ac-field induces shifts in the energy levels, causing resonant peaks at positions of photonic sidebands characterized by n = 0, ±1, ±2, ± The observed peak height (see Fig. 2) is determined by a Bessel function (see Eq. (8)).[48,58] It is also noticed that with the increase of the photon energy, the peak height increases. This profile of the conductance with the photon energy could be due to the photon induced sideband resonances.[55,56,58] As the induced photons’ energy increases, the interaction between them and transport carriers strongly increases, and this interaction affects the sidebands and tunneling rates.[48,55,58]

Fig. 2. The behavior of conductance, G, with photon energy at various magnitudes of the applied magnetic field.

(i) Results for the first case when the biomolecule is positioned at the tip of the SWCNT cantilever: the Young’s modulus of CNT is equal to 1.0× 1015 ng/nm⋅s2 and its density ρ = 1.4× 10−12 ng/nm3.[47,51,56,57]

Figure 3(a) illustrates the behavior of resonant frequency shift Δυ versus the attached biomolecule mass mbio-mol at different photon energies. As shown in the figure, the resonant frequency shift decreases as the mass of the attached biomolecule increases. Also, we notice from this figure that the resonant frequency shift Δυ is very small, approximately 10−6 Hz. This small value of Δυ could be obtained by measuring the conductance, G, of the carbon nanotube device. It is known that the conductance of nanostructured devices could be measured with a high precision. It is well known that the SWCNT has been proposed as THz (infrared region) oscillators for novel applications as NEMS.[59,60] These ranges of frequencies are consistent with the induced photon energy of the ac-field (infrared region). This coupling between the mechanical vibration of the SWCNT and the applied photon energy affects prominently on the mass sensitivity.

Fig. 3. The behavior of (a) resonant frequency shift Δυ and (b) the inverse of the quality factor, (1/Q), versus the mass of the biomolecule at various photon energies.

Figure 3(b) illustrates the behavior of the quality factor inverse, (1/Q), versus the mass of the attached biomolecule, mbio-mol, at different photon energies. As shown in this figure, the inverse quality factor decreases as the mass of the attached biomolecule increases. Also the inverse quality factor increases as the photon energy of the induced ac-field increases. The higher quality factor (Q-factor) means higher sensitivity and more reliable performance of the present SWCNT based NEMS.[61] This sensitivity of Q-factor increases as the photon energy of the induced ac-field increases.

(ii) Results for the second case when the biomolecule is at various intermediate positions on the SWCNT cantilever.

Figure 4(a) illustrates the behavior of resonant frequency shift Δυ versus the attached biomolecule mass mbio-mol for the biomolecule at different positions a. We notice from this figure that Δυ is higher when a = 4 nm, that is, at a position of about 33 % of the SWCNT cantilever length, (L = 12 nm), compared to the other positions a = 6 nm and a = 8 nm.[32,52] For the three positions, Δυ decreases as mbio-mol increases.

Fig. 4. The behavior of (a) resonant frequency shift Δυ and (b) the variation of (1/Q) with mbio-mol at different a.

Figure 4(b) illustrates the behavior of the quality factor inverse (1/Q) versus the biomolecule mass for the biomolecule at different positions a. As shown in the figure, for all three intermediate positions, the inverse quality factor (1/Q) decreases as the mass mbio-mol increases. Also, (1/Q) is the highest for the case when a = 4 nm, which is 33 % of the length of SWCNT L = 12 nm.

In order to confirm the effect of the photon energy of the induced ac-field on both quality factor Q and resonant frequency shift Δυ, the calculations are performed for the case when a = 7 nm, that is, 58 % of the total length L = 12 nm. The results are shown in Figs. 5(a) and 5(b). As shown, Δυ and (1/Q) increase as the photon energy increases. This effect of photon energy shows that the interaction between photon and mechanical vibration of the SWCNT plays an important role in the mass sensitivity. The results are found to be consistent with those in the literature.[50,51,62–65]

Fig. 5. The behavior of (a) Δυ and (b) the variation of (1/Q) with mbio-mol at different photon energy.
4. Conclusion

SWCNT based sensor is investigated under effect of induced ac-field’s photons in the infrared region. The case of mass sensing studied is the biomolecule acetone which exists in the diabetic patients’ exhale. Results illustrate that the quality factor and change in resonant frequency are delicate to the attached biomolecule mass and its position. Also, the induced ac-field affects on both quality factor and change in resonant frequency. So, the SWCNT based nanoelectromechanical resonator could be used to sense acetone molecules and consequently for diabetes diagnosis.

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