# Graphene Nanoribbon Bending (Nanotubes): Interaction Force between QDs and Graphene

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}/Vs for a single layer of mechanically exfoliated graphene. In addition, they specifically minimized the substrate-induced scattering by etching under the channel to produce graphene suspended between gold contacts in their experiments [28]. At room temperature, and with such high carrier mobility, charge transport is essentially ballistic on the micrometer scale. Because this enables the fabrication of all-ballistic devices even at current integrated circuit (IC) channel lengths (as low as 45 nm), it has great significance for the semiconductor industry. Graphene possesses considerable optical properties, and it can be optically imagined, despite being only a single atom thick. The linear dispersion of the Dirac electrons makes broadband applications possible. Storable absorption is observed as a consequence of Pauli blocking, and the absence of equilibrium carriers results in hot luminescence. Chemical and physical treatments can also lead to luminescence. All of these properties make graphene an ideal photonic and optoelectronic material [27]. Chemical doping and defects in graphene-based materials are currently being actively explored as a potential source of innovation to tailor the electronic properties of these nanostructures. Recently, experiments on graphene have been extended to the fabrication and study of QD–graphene nanostructures [29,30]. The author of [30] observed that when an array of quantum dots placed on the modified Graphene, the graphene plate bends. In this work, we analyzed the effects of the polarization of the quantum dots’ electric field on the graphene nanoribbon, which caused this bend. Moreover, we converted a graphene nanoribbon into a graphene nanotube by applying electromagnetic waves. This technique helps to use one material in two distinct structural states in electro-optical devices.

## 2. Geometry and Crystallographic Structure of SWCNTs

_{h}in the graphene sheet can be indicated as a combination of primitive vectors. C

_{h}is defined as follows:

_{h}touches its tail when the graphene sheet is rolled into a tube, we call C

_{h}the chiral vector or roll-up vector of the nanotube [16]. Zigzag and armchair tubes are chiral nanotubes because of their high geometric symmetry. The angle θ is used for determining the electrical properties of the SWCNT, and is referred to as the chiral angle (see Figure 1). However, SWCNTs with a chiral angle of 0 < θ < π/6 are chiral nanotubes. The chiral angle is given as follows:

- Armchair (A) SWCNT n = m, C
_{h}= (n, n), θ = π/6. - Zigzag (Z) SWCNT m = 0, C
_{h}= (n, 0), θ = 0. - Chiral SWCNT n ≠ m ≠ 0, 0 < θ < π/6.

_{h}(see Figure 1). The diameter of SWCNTs can be determined from the following relation:

_{c−c}displays the bond length between adjacent carbon atoms of a cell (a

_{c−c}≈ 1.42 A) [11].

_{h}= (n, n), and θ = π/6, so that the vector T is on an axis that is in line with the length and will pass through the center of the sheet of graphene nanoribbon. We deem it important that the presence of hanging bonding on the edge of the graphene nanoribbon sheet creates the final bond and cross-sectional formation. The cross-sectional radius is for the SWCNT. According to Formula (3), n must be a non-integer value to create a hanging bond at the edge of the sheet. Thus, in our work, a width of 40 nanometers was determined to be sufficient for the graphene nanoribbon.

## 3. Mathematical Formalism

#### 3.1. Interaction between Quantum Dots and the Charged Graphene Sheet

_{2}.

_{f}), also known as chemical potential (µ

_{c}), has a finite value between 1 and −1 eV. For an electrostatic gate, whenever a DC voltage is connected to it, the charge stored inside the electrostatic capacitor is equivalent to the doped charge state in the graphene gate, which is proportional to the density of the charge carriers at the Fermi level. The density of carriers at the Fermi level can be reduced to the following form through the Fermi–Dirac distribution and the density of two-dimensional material states [38,39]:

_{d}is the Fermi–Dirac distribution, which is defined by the Fermi level (E

_{f}) and the Boltzmann constant (KB = 8.617 × 10

^{−5}ev·K

^{−1}). An electron moves at a velocity equal to 106 m·s

^{−1}in a graphene structure, known as Fermi velocity (V

_{f}). ħ is a symbol of Planck’s decline (ħ = 6.5875 × 10

^{−16}eV). Thus, the graphene sheet can be charged as much as Q = q·n

_{s}, where q is the amount of electric charge (q = 1.6 × 10

^{−19}C).

_{Q}, z

_{Q}). If we consider that the charged sheet of graphene nanoribbon with dimensions of 775 nm × 40 nm is on the XZ plane, we can calculate this field using Gauss’law [40]. Therefore, the electric field emitted by a charged sheet of graphene nanoribbon at the center of the quantum dots on it, which polarizes the quantum dots, is equal to:

_{r}permittivity. The polarization measure of quantum dots is given as follows:

^{2}− P]/4πε

_{0}R

^{3}). Thus, the total electric field induced by the polarized particles to the desired point ‘P’ with coordinate position (x

_{p}, y

_{p}, z

_{p}) is given as follows:

_{i}is the distance from the polarization center of each quantum dot to the desired point ‘P’. The electric field induced by quantum polarized particles creates an interaction between the quantum array and the charged sheet of graphene nanoribbon. The induced electric field exerts a force on the charged graphene sheet. Therefore, this force is proportional to the induced electric field.

#### 3.2. Analysis and Evaluation of Torque and Bending Force to Determine the Number and Arrangement of Quantum Dots on the Charged Sheet of Graphene Nanoribbon

_{p}), so according to Formula (7), ${R}_{i}^{}=\sqrt{{r}_{QD}{}^{2}+{({z}_{g}-{z}_{{Q}_{i}})}^{2}},\mathrm{cos}\phi =0,\mathrm{sin}\phi =-1$. In this case, according to Equation (8) and Formula (7), we have the following:

_{g}is at the center point of the distance between the two quantum dots (${z}_{g}=\frac{{z}_{Qi}+{z}_{Qi-1}}{2}$), Equation (12) is correct. Moreover, the force in the r-direction is zero.

_{g}from the center point of the distance between the two quantum dots to the edge of one of the quantum dots (${z}_{g}=\frac{{z}_{Qi}+{z}_{Qi-1}}{2}\pm {\Delta}_{\mathrm{max}}$). The maximum distance that two quantum dots can have from one another, from edge to edge, is denoted by 2∆

_{max}. Similarly, as shown in Figure 2, we define ∆ as two quantum dots’ distance from edge to edge—not from center to center—which is given by $\Delta =\left|\frac{{z}_{Qi}-{z}_{Qi-1}}{2}\right|-{r}_{QD}$. Thus, ∆

_{max}is $\left|\frac{{z}_{Qi}-{z}_{Qi-1}}{2}\right|-{r}_{QD}$ as well. To be able to solve Equation (12), we use variable change ($u=\frac{{z}_{g}-{z}_{Qi-1}}{{r}_{QD}},g=\frac{{z}_{Qi}-{z}_{g}}{{r}_{QD}}$). Because Z

_{g}is on the edge of one of the quantum dots $({z}_{g}=\frac{{z}_{Qi}+{z}_{Qi-1}}{2}+{\Delta}_{\mathrm{max}})$, ‘u’ and ‘g’ are $\frac{{z}_{Qi}-{z}_{Qi-1}}{{r}_{QD}}+1$ and 1, respectively. Therefore, according to this trend, Equation (12) becomes:

_{min}) of consecutive dots is equal to 2r

_{QD}. All in all, the distances that consecutive quantum dots can have are equal to $2{r}_{QD}\le \left|{z}_{Qi}-{z}_{Qi-1}\right|\le (1+\sqrt{5}){r}_{QD}$.

_{z}/r

_{QD}>> 1); as a result, the inductive force on the charged sheet of graphene nanoribbon is given as follows:

**D**is a coefficient proportional to the length and width of the graphene nanoribbon as well as the permeability of the quantum dots. Therefore, by changing

**D**, their physical characteristics will change. The matrix

**H**shows the force applied to any point of the charged graphene nanoribbon sheet in three directions x, y, and z.

#### 3.3. Application of Electromagnetic Waves to Control the Bending Rate of the Charged Sheet of Graphene Nanoribbon

_{gi}, Y

_{gi}, Z

_{gi}). According to Figure 5, to achieve the SWCNT structure, it is necessary to design a force that can cause the sheet to bend in the path of the vector ∆L, which indicates the length of the sheet displacement.

^{2}+ y

^{2}= (L

_{x}/2π)

^{2}. Accordingly, the displacement vector (ΔL) in the bend is given as follows:

^{g}

_{yi}) of the graphene nanoribbon sheet—which varies from the initial state to the final—using Gauss’law [41]. In the initial stages, the angle θ

_{1}could be varied between −$\frac{\pi}{4}$ and $\frac{\pi}{2}$. A bending angle (θ

_{s}) similar to the angle θ

_{1}(see Figure 5) was defined for the changing states of the charged graphene nanoribbon sheet. Ultimately, the electric field emitted from different states of graphene nanoribbons exists in the quantum dots, and is given as follows [41]:

_{r}= E

_{i}cos(ωt − kz), the polarization of the quantum array changes to the following form:

#### 3.4. Application of DC Voltage to Control the Bending Rate by Adjusting the Charge Rate of the Graphene Nanoribbon Sheet

_{1}equals cos

^{−1}($\frac{\pi}{\sqrt{1+{\pi}^{2}}}$) when the sheet of graphene nanoribbon is one plate. This changes with an increase in DC voltage. Therefore, under these assumptions, the field emitted from the quantum dots, as expressed in Equation (7), can be obtained as follows:

_{DC}[40, 41]. Capacitance is determined based on the introduced structure (C = ε

_{0}ε

_{sio2}L

_{x}L

_{z}/d). Moreover, it is sometimes expressed in surface units, which indicate the amount of charge stored per unit of area (C = ε

_{0}ε

_{sio}

_{2}L

_{x}L

_{z}/d) (C = ε

_{0}ε

_{sio}

_{2}/d). Therefore, if we want to calculate the potential energy in the graphene sheet, we can use the capacitance per unit of area, because we divide the graphene sheet into small integrated plates with an area of 1 nm

^{2}to calculate its displacement relative to the applied energy from the quantum array. In this case, the surface charge of graphene is proportional to its surface conductivity (Q

_{s}= σ

_{s}). The reason for this importance is that, when the charged test object is moved in the field by some external agent, the work done by the field on the charge is equal to the negative of the work done by the external agent causing the displacement. This work depends only on the particle’s initial and final coordinates. Hence, we can derive Equation (24):

## 4. Results and Discussion

_{gi}) of the graphene nanoribbon sheet through the array of quantum dots in three directions, as shown in Figure 6. The magnitude of this force is expressed in Equation (16).

_{s}changes value, the graphene sheet becomes more curved and induces more electric fields at the quantum dots. As a result, to counteract this effect on the quantum dots, it is necessary to increase the amplitude of the applied waves in the opposite direction of the electric field induced by the bent graphene sheet. The physical references that report the electric field emitted from the charged curved plates in their surroundings have values similar to those shown in Figure 7 at the specified angle. For example, the location of the pivot (bent on the charged sheet) when the bending angle is 90 degrees (a cylinder with a radius $\frac{{L}_{x}}{2\pi}$) is equal.

^{2}).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Illustration of the electric field emitted from the charged sheet of graphene nanoribbon and the polarization of QDs.

**Figure 4.**Schematic of the pivot in two positions: (

**a**) in the center of the sheet, and (

**b**) on the axis passing through the center along the length. The pivot is represented in green.

**Figure 5.**Introduction of the vector ∆L. In addition, schematic representation of the cross-section of the SWCNT, and visual expression of the bending management of the charged graphene nanoribbon sheet with the help of the bending vector ∆L.

**Figure 6.**The force applied to each section (z = z

_{gi}) of the graphene nanoribbon sheet through the quantum array in the (

**a**) r-direction, (

**b**) ϕ-direction, and (

**c**) z-direction.

**Figure 7.**Dependence of the amplitude of the electromagnetic waves applied by the electric field on the bending angle.

**Figure 8.**The relationship between the applied voltage and the force applied in the (

**a**) x-direction and (

**b**) y-direction to a graphene nanoribbon sheet (d = 25 nm).

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Armaghani, S.; Rostami, A.; Mirtaheri, P.
Graphene Nanoribbon Bending (Nanotubes): Interaction Force between QDs and Graphene. *Coatings* **2022**, *12*, 1341.
https://doi.org/10.3390/coatings12091341

**AMA Style**

Armaghani S, Rostami A, Mirtaheri P.
Graphene Nanoribbon Bending (Nanotubes): Interaction Force between QDs and Graphene. *Coatings*. 2022; 12(9):1341.
https://doi.org/10.3390/coatings12091341

**Chicago/Turabian Style**

Armaghani, Sahar, Ali Rostami, and Peyman Mirtaheri.
2022. "Graphene Nanoribbon Bending (Nanotubes): Interaction Force between QDs and Graphene" *Coatings* 12, no. 9: 1341.
https://doi.org/10.3390/coatings12091341