# An Analytical Conductance Model for Gas Detection Based on a Zigzag Carbon Nanotube Sensor

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## Abstract

**:**

_{2}, H

_{2}O, and CH

_{4}). Therefore, we start with the modeling of the energy band structure by acquiring the new energy dispersion relation for the ZCNT and introducing the gas adsorption effects to the band structure model. Then, the electrical conductance of the ZCNT is modeled and formulated while the gas adsorption effect is considered in the conductance model. The band structure analysis indicates that, the semiconducting ZCNT experiences band gap variation after the adsorption of the gases. Furthermore, the bandgap variation influences the conductance of the ZCNT and the results exhibit increments of the ZCNT conductance in the presence of target gases while the minimum conductance shifted upward around the neutrality point. Besides, the I-V characteristics of the sensor are extracted from the conductance model and its variations after adsorption of different gas molecules are monitored and investigated. To verify the accuracy of the proposed models, the conductance model is compared with previous experimental and modeling data and a good consensus is observed. It can be concluded that the proposed analytical models can successfully be applied to predict sensor behavior against different gas molecules.

## 1. Introduction

_{3}, NO, NO

_{2}, etc. [25,26,27,28]. CNTs show superior performance because they have a high specific surface area that provides many sites for the gas molecules adsorption, and they have hollow geometry that aid to increase the sensitivity and decrease the operating temperature [29].

_{3}and CO

_{2}gas molecules by [30,31]. In the reported works, they considered gas concentration as a function of CNT and graphene and GNR conductance and tried to empirically formulate I-V variation after gas adsorption. Similar to this, another study was performed to detect NO

_{2}gas using graphene-based FET platforms by [32]. They reported that NO

_{2}concentration remains a factor that affects channel density of states, conductance and current-voltage characteristics. Furthermore, the gas concertation effect on the capacitance of the channel has been investigated by [33]. The capacitance has been considered as a function of CO

_{2}gas concertation and using the relationship between current and capacitance, I-V characteristics in the presence of gas were derived. These studies have used completely similar approaches in the modeling process as they introduced some fitting parameters to fit the suggested models to experimental data. They did not provide physical and mathematical justification for fitting parameters used in the model. Therefore, the proposed models are neither analytical nor physical. In addition, the tight-binding (TB) approach has been adapted by various researchers for modeling nanostructure and nanosensors. For example, the TB approach was adapted to model graphene nanoribbon gas sensor to investigate the average density of states and the semiconducting energy gap of the GNR affected by the gas adsorption [34]. In addition, this technique has been used for modeling of GNR band structure and conductance [35], Boron nitride nanoribbons and SiC band structures modeling [36,37], modeling of gas effects on the local density of states of the GNR [38] and etc. In the reported works using the TB approach, they did not model nonequilibrium conditions and only statistical states have been analyzed that have been applied on graphene and GNR mostly. Other modeling techniques such as neural network model, Lasso and genetic algorithms or Wolkenstein adsorption theory and model have been used for modeling gas sensors based on carbon nanostructures [39,40]. However, these works did not investigate device physics and the sensing mechanism and the phenomena that occur inside the device, and just tried to interpret the output and formulate the output curves using algorithms.

## 2. Materials and Methods

^{th}unit cell, there are four neighbor unit cells. In the modeling of ZCNT energy band structure, the interaction of the carbon atoms was taken into account by hopping integral parameter. On the other hand, as discussed earlier, the electrical properties of carbon nanotubes strongly depend on their energy band structure. Based on the above concepts, we introduced another varying integral parameter and onsite energy parameter of the adsorbed molecule to address the interaction between gas molecules and carbon atoms and apply molecular adsorption effect on the ZCNT band structure. Definitely, this phenomenon will affect electrical properties such as conductance and current-voltage properties. Therefore, first, we start with modeling of the energy band structure by developing an energy dispersion relation with considering the molecular adsorption effect. Afterward, the conductance and I-V characteristics of the ZCNT are modeled through the energy dispersion relation.

#### 2.1. Energy Band Structure Modeling

_{0}are the carbon-carbon hopping energy and onsite energy of the carbon atom, ${t}^{\prime}$ which is introduced as the hopping energy parameter between a carbon atom and an adsorbed gas molecule, a

_{1}and a

_{2}are the lattice vectors and ${E}_{0}^{\prime}$ is the on-site energy of the gas molecule. The achieved Schrödinger equation shows those neighbors (the nearest neighbors) that affect the nth unit cell and their energy.

^{th}unit cell and four neighbor unit cells and calculating the determinant of the h(k) matrix, and applying the periodic boundary condition to the propagation parameter k

_{y}, the energy dispersion relation for the ZCNT is achieved:

_{x}is the wave propagation along the x-direction. Applying Taylor expansion to Equation (7), the final form of the energy dispersion relation is modified as:

_{0}[35,42,43]. Thus, in most calculations, the effect of E

_{0}has been neglected. Therefore, here the values of the E

_{0}and ${E}_{0}^{\prime}$ are set to zero and we rely on ${t}^{\prime}$ to apply gas effects. However, this is important that all parameters be considered and appeared in our calculations. To apply the gas adsorption effect, the value of the ${t}^{\prime}$ for the adsorption of a specific gas on ZCNT can be calculated through [44]:

_{R}stands for the distance between two adjacent carbon atoms of ZCNT, t

_{R}is the hopping integral between carbon atoms and ${d}_{\alpha \beta}$ stands for the bond length between ZCNT surface and adsorbed gas molecule. The distance and orientation of each gas are assumed according to the reported study by [45]. Thus, the value of the ${t}^{\prime}$ for the adsorption of each gas is presented in Table 1.

#### 2.2. Conductance Model By Considering Gas Adsorption Effect

_{g}is the gate voltage and V

_{T}is the thermal voltage. Figure 5b illustrates the variation of the ZCNT conductance after the adsorption of various gas molecules. It can be seen that after gas adsorption conductance of the ZCNT rises. This is because of the increase of the charge carrier density in the ZCNT surface that leads to an increase in the conductivity of the channel. In other words, in the adsorption process of the gas molecules as a result of the covalent bond created between target molecules and ZCNT, electrons be released and injected to the ZCNT surface that increases the carrier concentration. The conductance variation of the ZCNT channel directly modulates the I-V characteristics of the sensor. Assuming that the source and substrate potentials are grounded, the ZCNT channel exhibits resistor characteristics in small voltages of the Source-Drain (V

_{DS}). In addition, the general conductance model of the ZCNT can be implemented to describe the relationship between current and conductance. Therefore, based on the I-V characteristics of the ZCNT-FET based devices, the performance of the gas sensor can be evaluated through Equation (16):

_{2}O and CO

_{2}adsorption. This means that in the process of H

_{2}O adsorption more electrons be injected to the channel and cause the highest electron concentration, thus the highest conductivity compared to the other two gases as depicted in Figure 6.

_{0}and I are the currents before and after adsorption respectively. It can be seen that the sensor has almost high sensitivity especially at low voltages while it has a very sharp slope and become constant for the higher voltages. Furthermore, the sensitivity of the sensor against different gases can be understood and compared easily.

## 3. Conclusions

_{2}, H

_{2}O, and CH

_{4}gases. To validate our calculations and mathematical models, a comparison study between our proposed conductance model and previously published experimental data was performed and an acceptable agreement was achieved. It can be concluded that our proposed models can be applied to study various properties of the carbon nanotubes to fabricate modern sensors with a strong performance.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**(

**a**) Comparison of the proposed conductance model with experimental data and other modeling work. Acceptable consensus between our model and previous data can be seen; (

**b**) Variation of the conductance in the presence of different gas molecules.

Adsorption Type | Distance from ZCNT Surface (Å) | Hopping Parameter |
---|---|---|

CH_{4} | ${d}_{\alpha \beta}=3.19$ | t_{C-CH4} = 0.445t_{R} |

CO_{2} | ${d}_{\alpha \beta}=3.23$ | t_{C-CO2} = 0.43t_{R} |

H_{2}O | ${d}_{\alpha \beta}=2.69$ | t_{C-H2O} = 0.528t_{R} |

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**MDPI and ACS Style**

Hosseingholipourasl, A.; Hafizah Syed Ariffin, S.; Ahmadi, M.T.; Rahimian Koloor, S.S.; Petrů, M.; Hamzah, A.
An Analytical Conductance Model for Gas Detection Based on a Zigzag Carbon Nanotube Sensor. *Sensors* **2020**, *20*, 357.
https://doi.org/10.3390/s20020357

**AMA Style**

Hosseingholipourasl A, Hafizah Syed Ariffin S, Ahmadi MT, Rahimian Koloor SS, Petrů M, Hamzah A.
An Analytical Conductance Model for Gas Detection Based on a Zigzag Carbon Nanotube Sensor. *Sensors*. 2020; 20(2):357.
https://doi.org/10.3390/s20020357

**Chicago/Turabian Style**

Hosseingholipourasl, Ali, Sharifah Hafizah Syed Ariffin, Mohammad Taghi Ahmadi, Seyed Saeid Rahimian Koloor, Michal Petrů, and Afiq Hamzah.
2020. "An Analytical Conductance Model for Gas Detection Based on a Zigzag Carbon Nanotube Sensor" *Sensors* 20, no. 2: 357.
https://doi.org/10.3390/s20020357