Carbon-Based Band Gap Engineering in the h-BN Analytical Modeling

The absence of a band gap in graphene is a hindrance to its application in electronic devices. Alternately, the complete replacement of carbon atoms with B and N atoms in graphene structures led to the formation of hexagonal boron nitride (h-BN) and caused the opening of its gap. Now, an exciting possibility is a partial substitution of C atoms with B and N atoms in the graphene structure, which caused the formation of a boron nitride composite with specified stoichiometry. BC2N nanotubes are more stable than other triple compounds due to the existence of a maximum number of B–N and C–C bonds. This paper focused on the nearest neighbor’s tight-binding method to explore the dispersion relation of BC2N, which has no chemical bond between its carbon atoms. More specifically, the band dispersion of this specific structure and the effects of energy hopping in boron–carbon and nitrogen–carbon atoms on the band gap are studied. Besides, the band structure is achieved from density functional theory (DFT) using the generalized gradient approximations (GGA) approximation method. This calculation shows that this specific structure is semimetal, and the band gap energy is 0.167 ev.


Introduction
Nowadays, frontier technologies, especially in the semiconductor and battery industries, are improving material scaling to enhance the efficiency of their products. In particular, foiling unwanted transport between anode and cathode in the batteries is engaged by material performance. One the other hand, from the semiconductor-based point of view, one of the available techniques to progress the artifact is hidden in the BN material performance, which can be controlled by supplementary carbon atoms. For example, in the form of BC 2 N, the band gap can be controlled effortlessly. On the other hand, the BC 2 N allotropes as negative electrodes and anchoring materials for Li-S has been reported [1,2]. Since an increase in the numbers and efficiency of transistors diminishes the cost and leads to computers with faster processing ability, the number of such transistors on a microchip is commonly on the rise. Given this, it seems very cost effective to reduce the size of transistors [3,4]. However, this minimizing process will be finally stopped due to the need for developments in electronic commerce; thus, alternative technologies are needed [3]. Hence, the search for a matter with good nanoelectronics

Modeling
The NNTB approximation method is used to model the band structure of BC2N, based on Figure 1a [28]. In this regard, one-way is used to investigate how the electron-electron interaction in the structure of crystalline solves a separable Schrödinger equation, and this model is achieved by means of a time-independent Schrödinger equation. The outcome of the time-independent equation (Schrödinger) is written in the form of Where ∅ 0 is the wave function, and h(k) is the matrix equation as In Equation (2), stands for the Hamiltonian matrix equation, k is the wave vector, and and are the displacements of the m th and n th unit cells from the origin, respectively. On the other hand, for this special model, no chemical bonds between carbon-carbon bonds are considered. Now, in this case, B assigns , , and to the onsite energies of carbon, boron, and nitrogen atoms, respectively, and the overlapping energy between the two atoms of B-C and N-C are designated by = and = , respectively. Thus, Equation (2) can be presented via: where h1 = (E0 + 2 × E0 × (cos(k.r1) + cos(k.r2)))E, h2 = t + t × (cos(k.r1)), h3 = tp + tp × (cos(k.r1)), h4 = t × (cos(k.r2)), k = kx i + ky j, r1 = 3 × a0 i, and r2 = 3a0i + √3 a0j. For interpretation purposes, it is assumed that Figure 1b is an ideal hexagonal shape and has an equal bond length of a0 = 1.446 Å between boron and nitrogen atoms. Therefore, to define the energy eigenvalues in the matrix form, the energy matrix is diagonalized by det[ The energy eigenvalues based on the Schrödinger equation solution lead to the following four energy dispersion relations for this particular BC2N structure due to the introduced unit cell, as shown in Figure 1a.

Modeling
The NNTB approximation method is used to model the band structure of BC 2 N, based on Figure 1a [28]. In this regard, one-way is used to investigate how the electron-electron interaction in the structure of crystalline solves a separable Schrödinger equation, and this model is achieved by means of a time-independent Schrödinger equation. The outcome of the time-independent equation (Schrödinger) is written in the form of where ∅ 0 is the wave function, and h(k) is the matrix equation as In Equation (2), H mn stands for the Hamiltonian matrix equation, k is the wave vector, and d m and d n are the displacements of the m th and n th unit cells from the origin, respectively. On the other hand, for this special model, no chemical bonds between carbon-carbon bonds are considered. Now, in this case, B assigns E oC , E oB , and E oN to the onsite energies of carbon, boron, and nitrogen atoms, respectively, and the overlapping energy between the two atoms of B-C and N-C are designated by t bc = t and t cn = t p , respectively. Thus, Equation (2) can be presented via: where h 1 = (E 0 + 2 × E 0 × (cos(k.r 1 ) + cos(k.r 2 )))E, h 2 = t + t × (cos(k.r 1 )), h 3 = t p + t p × (cos(k.r 1 )), h 4 = t × (cos(k.r 2 )), k = k x i + k y j, r 1 = 3 × a 0 i, and r 2 = 3a 0 i + √ 3 a 0 j. For interpretation purposes, it is assumed that Figure 1b is an ideal hexagonal shape and has an equal bond length of a 0 = 1.446 Å between boron and nitrogen atoms. Therefore, to define the energy eigenvalues in the matrix form, the energy matrix is diagonalized by The energy eigenvalues based on the Schrödinger equation solution lead to the following four energy dispersion relations for this particular BC2N structure due to the introduced unit cell, as shown in Figure 1a.
As shown in Figure 3, the possible band gaps based on the presented mode are analyzed and plotted in three-dimensional (right-hand side figures) and two-dimensional (left-hand side figures) forms in comparison with each other starting with E1 as shown in Figure 2a, followed by the accumulation of band structures E2 to E4.  (6), it is found that the fourth possible band gap E 4 is 1.3254 ev.

Result and Discussion
It is found that there are four answers due to the variables of wave vector and energy. The first answer is E 1 (Figure 2a), which indicates a gap of 2.416 ev; the second answer is E 2 with a 1.212 ev band gap (Figure 2b), the band gap due to the third answer, E 3 , is equal to 0.7182 ev (Figure 2c), and finally, the fourth answer E 4 with a 1.3254 ev band gap (Figure 2d). The band structure that is plotted using E 1 , E 2 , E 3 , and E 4 , resulted from Equations (5) and (6), which demonstrates the suggested model, as shown in Figure 3. In the other words, the dispersion relation (Equation (6)) leads to the formation of the band structure, as shown in Figures 2 and 3.  BC2N; b) The band gap resulting from the Equation (3) is due to E1 is 2.416 ev; c) The band gap corresponding to Equation (4), namely E2, is about 1.212 ev; d) On the other hand, the band gap resulting from Equation (5) E3 is 0.7182 ev. Finally, as shown, through solving Equation (6), it is found that the fourth possible band gap E4 is 1.3254 ev.
As shown in Figure 3, the possible band gaps based on the presented mode are analyzed and plotted in three-dimensional (right-hand side figures) and two-dimensional (left-hand side figures) forms in comparison with each other starting with E1 as shown in Figure 2a, followed by the accumulation of band structures E2 to E4.  In the technology application, the ultimate lattice structure is under unwanted strain and stresses that cause band energy variation. To analyze this annoying effect, one of the imaginable methods is the overlap energy variation investigation, which can be realized in the form of lattice parameter variation. Therefore, the effect of overlap energy between carbon-nitrogen and carbonboron atoms as a disparity between the highest and lowest energy levels is considered as shown in Figure 4. In the other words, the reduction in the lattice parameter could cause an increase in the spatial overlap of the orbitals. On the other hand, the increased anti-bonding appears to be larger As shown in Figure 3, the possible band gaps based on the presented mode are analyzed and plotted in three-dimensional (right-hand side figures) and two-dimensional (left-hand side figures) forms in comparison with each other starting with E 1 as shown in Figure 2a, followed by the accumulation of band structures E 2 to E 4. In the technology application, the ultimate lattice structure is under unwanted strain and stresses that cause band energy variation. To analyze this annoying effect, one of the imaginable methods is the overlap energy variation investigation, which can be realized in the form of lattice parameter variation. Therefore, the effect of overlap energy between carbon-nitrogen and carbon-boron atoms as a disparity between the highest and lowest energy levels is considered as shown in Figure 4. In the other words, the reduction in the lattice parameter could cause an increase in the spatial overlap of the orbitals. On the other hand, the increased anti-bonding appears to be larger than the increased bonding. For simplification purposes, the hopping energy of boron-carbon and nitrogen-carbon is shown in terms of t and t p , respectively. As indicated in Figure 3, the energy overlaps between B-C, namely, t p , is fixed at 2.59 ev, and the energy overlaps between C-N, namely t, is changed. Therefore, band gap energies of about a) E 1 = 2.594 ev, b) E 1 = 4.266 ev, c) E 1 = 5.11 ev, and d) E 1 = 5.6 ev are reported. These results show that in accordance with the increase in t p , the band gap increases.  On the other hand, via changing the overlapping energy between nitrogen and carbon, namely t, an overlap between energy bands is reported. Besides, the effect of overlap energy between carbon and boron is plotted, as shown in Figure 5. On the other hand, via changing the overlapping energy between nitrogen and carbon, namely t, an overlap between energy bands is reported. Besides, the effect of overlap energy between carbon and boron is plotted, as shown in Figure 5.
It is concluded that any variation in the overlap energy indicates a direct effect on the BC 2 N band structure, which can be explained by an applied quantum confinement effect in the energy matrix. In the presented model, it can be assumed that the quantum confinement effect is buried in the overlap energy variation. On the other hand, density functional theory (DFT) and tight binding (TB) methods have been employed intensively in the material property investigation [29][30][31][32]. The DFT illustrates good agreement with experimental results; however, it is computationally very expensive. Therefore, its application has been limited to a small amount of atom calculations [32]. In contrary, the TB method in the band structure calculation without self-consistent progressions needs smaller amounts of computational possessions. Therefore, TB models have been applied in the large structure (up to millions of atoms) investigations [29]. In addition, the TB model often leads to analytical terminologies that improve the logical investigation of material properties; consequently, in this research, a TB model is being implemented. On the other hand, to simulate the DFT, we use an OPENMX3.8.5-open source computer code with a lattice constant of about a = b = 5.0503 Å and c = 15.2997 Å. In addition, for the linear combination of the pseudo-atomic localized basis set, 150 Ry cutoff energy and 10 × 10 × 1 k-point are used. The basis set function is B7.0-s2p2d1, N6.0-s2p2d1, and C6.0-s2p2d1 for boron, nitrogen, and carbon, respectively. The first symbol shows the chemical name together with the cutoff radius and the initial orbitals, which is shown via the last set. All data are achieved using generalized gradient approximations (GGA) and the PBE exchange-correlation functional. On the other hand, via changing the overlapping energy between nitrogen and carbon, namely t, an overlap between energy bands is reported. Besides, the effect of overlap energy between carbon and boron is plotted, as shown in Figure 5. It is concluded that any variation in the overlap energy indicates a direct effect on the BC2N band structure, which can be explained by an applied quantum confinement effect in the energy matrix. In the presented model, it can be assumed that the quantum confinement effect is buried in the overlap energy variation. On the other hand, density functional theory (DFT) and tight binding (TB) methods have been employed intensively in the material property investigation [29][30][31][32]. The DFT illustrates The result of the simulated structure is closer to the results of Figure 5 in response to the analytical model (comparison between Figures 5 and 6), which indicates that the band structure is about 0.167 ev. The comparison study between the presented model and DFT simulation result is carried out as shown in Figure 7. good agreement with experimental results; however, it is computationally very expensive. Therefore, its application has been limited to a small amount of atom calculations [32]. In contrary, the TB method in the band structure calculation without self-consistent progressions needs smaller amounts of computational possessions. Therefore, TB models have been applied in the large structure (up to millions of atoms) investigations [29]. In addition, the TB model often leads to analytical terminologies that improve the logical investigation of material properties; consequently, in this research, a TB model is being implemented. On the other hand, to simulate the DFT, we use an OPENMX3.8.5-open source computer code with a lattice constant of about a = b = 5.0503 Å and c = 15.2997 Å . In addition, for the linear combination of the pseudo-atomic localized basis set, 150 Ry cutoff energy and 10 × 10 × 1 k-point are used. The basis set function is B7.0-s2p2d1, N6.0-s2p2d1, and C6.0-s2p2d1 for boron, nitrogen, and carbon, respectively. The first symbol shows the chemical name together with the cutoff radius and the initial orbitals, which is shown via the last set. All data are achieved using generalized gradient approximations (GGA) and the PBE exchange-correlation functional. The result of the simulated structure is closer to the results of Figure 5 in response to the analytical model (comparison between Figures 5 and 6), which indicates that the band structure is about 0.167 ev. The comparison study between the presented model and DFT simulation result is carried out as shown in Figure 7.  The result of the simulated structure is closer to the results of Figure 5 in response to the analytical model (comparison between Figures 5 and 6), which indicates that the band structure is about 0.167 ev. The comparison study between the presented model and DFT simulation result is carried out as shown in Figure 7.  An acceptable agreement-especially on the k point-is detected, which confirms the accuracy of the proposed model. Therefore, the proposed analytical calculation could represent the prediction of electrical performance of BC 2 N, which also provides very fast modeling and simulation tools for band-gap investigation. On the other hand, the lattice parameter effect on the BC 2 N gap performance can be explored by the overlap energy gradient due to the stress and strain associated with the temperature or device fabrication limits.

Conclusions
BC 2 N is one of the most stable structures of C X (BN) y configuration. This structure is an in-between combination of hexagonal graphene and hexagonal boron nitride. The calculations showed that the band-gap energy of C X (BN) y is between pristine graphene with a band gap of zero and boron nitride with a band-gap energy of 5.5 eV. The band gap firmly depends on either the formation energy or binding energy. Conceding and doping in the graphene sheet with boron and nitrogen atoms firmly depends on its location and amount, which affects its stability. In the present study, we focused on the amazing and notable structures of the BC 2 N family. In this structure, no bonding is found between C-C atoms, which caused the inspiration of unique electronic and optical properties as well as its application in designing electronic devices. This study focused on the dispersion relation investigation using an analytical calculation based on the nearest neighbor's tight-binding method. This method supposes that the band gap of the main structure of BC 2 N alters from 2.41 in E 1 to 1.12 in E 2 and from 0.7182 eV to 1.3254 eV in E 4 . On the other hand, using figured alternating wave vector versus energy, it is found that the changes in overlap energy between boron-carbon and nitrogen-carbon can engender the band gap.
Author Contributions: The main idea are been suggested by M.T.A. which is been simulated by A.R. In the final form, the results are been discussed and supported by S.S.R.K. and M.P. All authors were in the administration team and contributed in the manuscript preparation. All authors have read and agreed to the published version of the manuscript.