Electronic and optical properties of boron nitride nanoribbons exploiting DFT

We have used density functional theory and the SIESTA code to investigate the electronic bandgap and optical spectra of zigzag boron nitride nanoribbons n-ZBNNRs (n = 18, 22, 26, 30). The electronic structure of the simulated nanoribbons reveals that they are semiconductors with a bandgap of 4.87–4.95 eV. For energies greater than 5 eV, the real part of the dielectric function suggests that ZBNNRs are negative refractive index materials. n-ZBNNRs also have a lower static refractive index than armchair graphene nanoribbons. The fundamental reason is that the boron nitride nanoribbons have more localized electrons. The optical absorption of n-ZBNNRs (n = 18, 22, 26, 30) is anisotropic for the x, y, and z polarizations as well. Moreover, Eoptical gap18-BNNRs = 4.95 eV, Eoptical gap22-BNNRs = 4.94 eV, Eoptical gap26-BNNRs = 4.92 eV, and E optical gap30-BNNRs = 4.87 eV. The maximum optical extinction occurs at E ≈ 6–7 eV for the y polarization and E ≈ 5–6 eV for the z-direction. Besides, by around 6 eV for the y polarization (and 5 eV for the z polarization), σ is almost 1500 Ω−1 cm−1 (and 860–1500 Ω−1 cm−1). The imaginary part of the optical conductivity indicates that for the low energy range, n-ZBNNRs (n = 18, 22, 26, 30) supply the conditions of the transverse-electric mode existence. In addition, for Ey>6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{y} \, > \,6$$\end{document} eV and Ex>5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{x} \, > \,5$$\end{document}, n-ZBNNRs (n = 18, 22, 26, 30) support the transverse-magnetic plasmons.


Introduction
A boron nitride monolayer is a single layer of boron and nitride atoms arranged in a twodimensional honeycomb lattice nanostructure. Reports show that graphene and boron nitride monolayers have different physical properties and similar two-dimensional configurations (Peng et al. 2012). Two-dimensional boron nitride has a bandgap of 5.9 eV. It is a potential candidate for dielectric, coating, and some device applications (Bhimanapati et al. 2016), as well as a range of high-pressure and high-temperature applications (Cheng et al. 2017 Boron nitride nanoribbons (BNNRs) are a class of promising materials for applications such as photovoltaic, electronic, piezoelectric, and optoelectronic devices (Qi et al. 2012). Boron nitride nanoribbons, nanotubes, and nano-cocoons have been synthesized by Fathalizadeh et al. (2014) exploiting the direct feedstock injection into an extended pressure. Additionally, Sinitskii et al. (2014) have used the longitudinal splitting of boron nitride nanotubes by the potassium vapor technique to obtain bulk quantities of high-quality boron nitride nanoribbons. The results indicate that the BNNTs synthesized by different methods may demonstrate different reactivity in the potassium-splitting reaction. Boron nitride nanoribbons have been produced successfully via other techniques (Chen et al. 2008;Wei et al. 2011;Jiang et al. 2021). Wang et al. (2011a) have investigated the exciton binding energies and optical absorption of armchair and zigzag boron nitride nanoribbons (BNNRs) (w = 6, 7, 8) employing a GW-Bethe-Salpeter equation approach (Salpeter and Bethe 1951). The electronic properties of armchair boron nitride nanoribbons (ABNNRs) (N = 8-20) in an external electric field have been studied using the tight-binding method by Chegel and Behzad (2014). Furthermore, the electronic band structure and dielectric function of ABNNR (w = 25) in the absence and presence of an external electric field have been simulated (Hasani and Chegell 2020). The results suggest that the change in the electronic bandgap of BNNRs with a larger width is more sensitive to the external electric field. The structural and electronic properties of Fe-terminated armchair boron nitride nanoribbons have been studied using first-principles calculations (Wang et al. 2011b). The data indicate that the Fe-ABNNRs have antiferromagnetic metallic behavior.
Nanoribbons (Mousavi et al. 2022;Dazmiri and Badehian 2021) and nanotubes Gharbavi and Allah Badehian 2016;Gharbavi and Badehian 2015) show periodic behavior with an increasing number of atoms in the nanostructure. Wide bandgap semiconductors have attracted scientists' attention due to their applications in electronic and high-performance optoelectronic devices (Pearton et al. 2003;Tien and Ho 2019). In the present study, we investigate the electronic bandgap and optical characteristics of zigzag boron nitride nanoribbons (n-ZBNNRs) (n = 18, 22, 26, 30) using density functional theory (DFT), which to our knowledge, has not achieved yet.

Computational approach
The first-principles calculation has been performed in the framework of the DFT (Hohenberg and Kohn 1964;Kohn and Sham 1965) as implemented in the SIESTA program package (Soler et al. 2002). The spin-polarized Perdew-Burke-Ernzerhof (PBE) parameterizations of the GGA approach (Perdew et al. 1997;Ernzerhof and Perdew 1998) have been exploited for the exchange-correlation energy ( E XC ). A convergence basis set of 1.0 × 10 −3 eV/Å within the Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization method (Byrd et al. 1992;Dai 2002) has been utilized to relax the quasi-crystals of ZBNNRs. The convergence criterion for the density matrix is 0.0001. The tetragonal unit cell has adopted a 1 × 1 × 100 structure, and the energy cut-off is set to 120Ry. Figure 1 represents the crystal lattice of 30-ZBNNRs. The edge boron and nitride atoms are passivated by hydrogen.

DOS and band structure
Reports suggest that wide bandgap semiconductors reveal many advantages in electronic and optoelectronic applications (Tien and Ho 2019). The total density of states (TDOS) and band structure of the zigzag boron nitride nanoribbons (n = 18, 22, 26, 30) have been calculated, and the results are represented in Fig. 2. Figure 2 confirms that the n-ZBNNRs (n = 18, 22, 26, 30) have semiconducting behavior with a direct bandgap. Table 1 compares the electronic bandgaps of different nanoribbons. As can be seen, the simulated nanoribbons are semiconductors with a 4.87-4.95 eV bandgap. In comparison, zigzag GNRs are either metal or small gap semiconductors. Compared to graphene nanoribbons, boron nitride nanoribbons have a larger bandgap. In-plane lattice constants of hexagonal boron nitride and graphite are 2.50 Å and 2.46 Å, respectively (Ishigami et al. 2003). However, the chemical bonding between boron and nitrogen is more challenging to break than that of carbon-carbon (Badehian and Vatankhah 2022;Gharbavi and Badehian 2014). This difference increases the localized electrons in BNNRs, which results in wider bandgaps through BNNRs. The bandgaps of ZBNNRs shrink with increasing width as well (Table 1). Furthermore, in comparison with monolayer boron nitride (E g = 6.04 eV) (Wickramaratne et al. 2018), the bandgap of the BNNRs is narrower. The reason behind this is because the nanoribbons DFT 1.69 ± 0.

Dielectric function and refractive index
The dielectric function, ε , as a function of the incoming light frequency ( ) is calculated by Ambrosch-Draxl and Sofo (2006): 1 and 2 are the real and imaginary parts of the dielectric function, respectively (Wooten 1973): m is the electronic mass and are the directional components, c k and v k are vacant conduction and filled valence states, and e is the electronic charge magnitude (Wooten 1973). Figure 3 illustrates the real and imaginary parts of the dielectric function of n-ZBNNRs (n = 18, 22, 26, 30). The Fermi level is set to zero.
The curves indicate that ( ) is anisotropic. Re ( ) confirms that ZBNNRs are negative refractive index materials for energies higher than 5 eV. The data also suggest that the optical spectra are constant in the x-direction.
Based on the x and y components of Im (inter) , the first optical absorption of ZBNNRs takes place at E ≈ 5-6 eV. Furthermore, the other optical absorption peaks are at energies below 7 eV.
The static refractive index, n( ) , as a function of frequency, is calculated by Byrd et al. (1992): The values are given in Table 2. Based on the data, n 0 (y) and n 0 (z) are higher than n 0 (x). The reason is that the mass density of the supercell for the yand z polarizations is higher than the x polarization. The static refractive index of the simulated nanoribbons is also lower than that of 25-AGNRs (Hasani and Chegell 2020) and ZGNRs (Mousavi et al. 2022). The fundamental cause, as previously stated, is the localized electrons in the boron nitride nanoribbons. Figure 3 shows that for the y and z polarizations, Re ( ) declines from 1.7-2.5 at E ≈ 5 eV to 0 at E ≈ 6 eV.
(1) The refractive index of the simulated BNNRs increases by diameter, and reaches its zenith in the BN thin film (Rah et al. 2019). The optical properties of zigzag and armchair BNNRs were calculated by Wang et al. (2011a). According to their findings, BNNRs with wide bandgaps have higher exciton binding energies. FL Shyu's tight-binding research (2014) also looks into the absorption frequencies of ABNNRs and ZBNNRs. Furthermore, the imaginary component of AGNR20BN40's dielectric function has been calculated using tight binding and first-principles calculations (Nematian et al. 2012).

Absorption
The absorption coefficient can be described by the following equation (Lashgari et al. 2016): ω , Re ij , and Im ij are the frequency, real and imaginary parts of the dielectric function, respectively. Figure 4 illustrates the optical absorption of n-ZBNNRs (n = 18, 22, 26, 30).
When light polarization is perpendicular or parallel to the layer, the optical absorption of n-ZBNNRs (n = 18, 22, 26, 30) is anisotropic (Fig. 4). The graph shows that the peaks in the y and z directions are higher than the x direction. From the optical transition between the lowest unoccupied molecular orbital (LUMO) and highest occupied molecular orbital (HOMO) sub-bands in Fig. 2, we can obtain the first optical bandgap and the first optical transitions. These optical transitions correspond to the energy of the optical absorption peaks in the z directions. See (Gharbavi and Badehian 2015;Badehian and Gharbavi 2021).
The nature of wide bandgap energy is appropriate for absorption or emitting ultraviolet (UV) light in optoelectronic devices(Tien and Ho 2019). The first optical bandgaps of BNNRs are as follows: E optical gap 18-BNNRs = 4.95 eV, E optical gap 22-BNNRs = 4.94 eV, E optical gap 26-BNNRs = 4.92 eV, and E optical gap 30-BNNRs = 4.87 eV. As can be seen, the first optical bandgap of ZBNNRs is higher than that of ZGNRs (Mousavi et al. 2022). Furthermore, the maximum optical bandgap in 18-ZBNNRs is 8.41 eV for the y polarization (y component of the highest peak). Schwab et al. (2012) suggest that the absorption peaks of the longest graphene nanoribbons are at 690 and 960 nm, although the numerical values of the optical absorption were not reported. Moreover, for the x direction, the absorption coefficient of the monolayer boron nitride(Jalilian and Safari 2016) and graphene (Ne et al. 2018) show their first peak at around 6.25 eV ( a~8 0 10 4 cm ) and 4.60 eV ( a~4 8 10 4 cm ), respectively.
The lower optical absorption of graphene is due to the lower mass density of carbon atoms in the supercell compared to boron and nitrogen in boron nitride (Failed XXXX).

Extinction
The extinction coefficient (k), as the imaginary part of the complex refractive index ( ∈ ii ), is described byAmbrosch-Draxl and Sofo (2006): Figure 5 suggests that the y and z polarizations of the extinction coefficient are above the z-direction. Additionally, the maximum optical extinction of n-ZBNNRs (n = 18, 22, 26, 30) occurs at E ≈ 6-7 eV for the y polarization and E ≈ 5-6 eV for the z-direction. This output corresponds to Im (inter) ( ) in Fig. 3.
The maxima of the optical extinction of a hexagonal boron nitride (h-BN) nanosheet are around 1.3 and 1.1 in the parallel and perpendicular to the plane polarizations, respectively (Beiranvand and Valedbagi 2015). These results are higher than those of the simulated nanoribbons. Therefore, the h-BN nanosheet has a lower transmission than n- 22,26,30). Previous experimental research also shows that the extinction coefficient (k) of the h-BN nanosheet containing octadecylamine (ODA) in tetrahydrofuran (THF) is equal to about 3 Lmol −1 cm −1 at 1000 nm (Lin et al. 2010).
The optical extinction of n-ZGNRs (n = 18, 22, 26, 30) (Mousavi et al. 2022) is higher than that of n-ZBNNRs (n = 18, 22, 26, 30) in this research. The result is in agreement with the imaginary part of the dielectric function. See Fig. 3 and (Mousavi et al. 2022). The peaks of the extinction index for the parallel and perpendicular to the graphene surface are about 0.7 and 1.3 respectively (Rani et al. 2014). Additionally, the optical extinction coefficients of the boron nitride nanosheet were predicted to be much lower than that of the graphene sheet, confirming the low-colored nature of the boron nitride nanosheet (Lin et al. 2010).

Optical susceptibility
The linear optical susceptibility ( ) is calculated by the following formula (Lannoo 1977): P, ε 0 , and E are the macroscopic polarization, the optical dielectric constant, and the electric field, respectively. Lucking et al. (2018) have investigated the second-order susceptibility of hexagonal BN (h-BN) using the ABINIT code. Their calculations show that an h-BN monolayer exhibits a modest second harmonic response and a bandgap of 4.61 eV. Figure 6 illustrates the real and imaginary parts of the optical susceptibility of n-ZBNNRs (n = 18, 22, 26, 30). Corresponding to the real part of the Re ( ) function, Fig. 6 shows that the optical susceptibility fluctuates between around 4 eV and 8 eV. For the energy range of 4-8.49 eV, the y and z components of Im ( ) (Im ( χ) y and Im ( ) z ) are higher than the x component of Im ( χ ) (Im ( ) x ). Therefore, the interactions between the crystal lattice and the y and z polarizations of the incident beam are lower than that of the x-direction at E = 4-8.49 eV.
From Fig. 6b, one can see that the y component of χ Imaginary reaches its zenith at E ≈ 6 eV, which is proportional to Im (inter) ( ) . In comparison with ZBNNRs, the second order of the optical susceptibility in a single-layer boron nitride is = 4.2 × 10 −11 m/V (Kim et al. 2019). Moreover, Hendry et al. (2010) have reported that the third order of the optical susceptibility in graphene is high (10 -7 esu (electrostatic units). Figure 7 shows the optical conductivity of n-ZBNNRs (n = 18, 22, 26, 30). Analogous to the optical absorption function (Fig. 4), y and z polarizations of the real part of the optical conductivity reach their zenith at about 4-5 eV. At around 6 eV for the y polarization (and 5 eV for the z polarization), σ is almost 1500 Ω −1 cm −1 (and 860-1500 Ω −1 cm −1 ), which arises from the corresponding valence and conduction band transitions. Moreover, at the energy range ≈ 5-8 eV, the y component of the optical conductivity ( y ) is higher than z . The obtained results are higher than the optical conductivity of the electrical insulator h-BN nanosheets (Beiranvand and Valedbagi 2015;Xu and Ching 1991). The zeniths of optical conductivity in ZGNRs ( Re( )~1500 Ω −1 cm −1 ) (Mousavi et al. 2022) (Gonçalves et al. 2022). Additionally, the optical conductivity of a monolayer BN on different substrates such as quartz, PET, and sapphire at different temperatures has been reported by Bilal et al.(Bilal et al. 2020). To the best of our knowledge, there is no research on the optical conductivity of monolayer BN or BNNRs similar to the physical conditions in our simulation.

Conclusion
The electronic and optical behavior of n-ZBNNRs (n = 18, 22, 26, 30) have been calculated utilizing the DFT framework. The results show that the zigzag boron nitride nanoribbons have semiconducting behavior with a direct bandgap. In addition, compared to graphene nanoribbons, boron nitride nanoribbons have a larger bandgap. ZBNNRs are negative refractive index materials for energies higher than 5 eV as well. Moreover, the static refractive index of the simulated nanoribbons is almost lower than that of graphene nanoribbons. At around 6 eV for the y polarization (and 5 eV for the z polarization), the optical conductivity of n-ZBNNRs (n = 18, 22, 26, 30) is about 1500 Ω −1 cm −1 (and 860-1500 The calculated optical conductivity suggests that for the low energy range, n-ZBNNRs (n = 18, 22, 26, 30) supply the conditions of the transverse-electric (TE) mode existence. In addition, for E y > 6 eV and E x > 5 , n-ZBNNRs (n = 18, 22, 26, 30) support the transverse-magnetic (TM) plasmons.
Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.