Investigating electronic, optical, and structural properties of beryllium oxide zigzag nanotubes using DFT

In this study, we employ density functional theory and the Siesta code to investigate the electronic and optical properties of beryllium oxide (BeO) zigzag nanotubes (n,0) with n = 6, 8, 10, 12, 14, 16. Our research aims to elucidate the characteristics of BeO nanotubes and their potential applications. Notably, we found that the bandgap energy of BeO nanotubes increases with diameter, indicating superior conductivity in smaller-diameter nanotubes. Our findings align closely with experimental data, particularly when using the GGA-WC functional. Additionally, we calculated nanotube buckling decrease with diameter, revealing its negligible impact on these structures. The static refractive index of BeO nanotubes remains consistent at approximately 1.1, with an optical absorption peak around 9 eV. Our research offers valuable insights into the electronic and optical properties of BeO nanotubes, which have implications for various applications.


Introduction
Due to its exceptional characteristics, including high thermal conductivity [1], high melting point [2], high hardness [3] and high electrical resistivity [4], beryllium oxide (BeO) has attracted scientists' attention as an intriguing issue.BeO is frequently used in many high-performance semiconductor components due to its superior thermal conductivity and electrical resistance.Additionally, the bulk of beryllium oxide has a band gap of around 10.63 eV [5,6], which qualifies it for use in optoelectronic devices [7].Moreover, BeO is widely used in the nuclear industry for both fission and fusion power production because of its exceptional mechanical, thermal, and chemical qualities [8].
Many scientists have explored the characteristics of BeO experimentally and theoretically during the last few decades [9][10][11][12][13][14][15].The thermal characteristics of BeO, such as enthalpy and specific heat, were simulated theoretically by Wei et al. in the DFT framework [16].Mashhadzade et al. [17] theoretically investigated the mechanical and electronic characteristics of BeO graphene-like structures to study the defects that can occur in the synthesis of BeO.They have shown that the Young's modulus for the pristine BeO graphene-like sheet to be equal to 1.110 TPa.The Mechanical, electronic and stability properties of multi-walled BeO nanotubes and nanopeapods have been investigated by Y. Rostamiyan et al. [18].According to their calculations, the band gap of zigzag BeO nanotubes ((8,0) to (24,0)) is between 6 and 7 eV, and the Young's modulus of the BeO nanotube (14,0) is equivalent to 700.12 Gpa.Fathalian et al. [19] also performed an ab initio study of the optical properties of BeO nanotubes with different diameters and chiralities using the full-potential linearized augmented plane wave method implemented in the WIEN2k code.They found that the optical gap of BeO nanotubes decreases with increasing diameter and that the optical spectra are sensitive to the chirality of the nanotubes.However, their study did not cover all the possible zigzag configurations of BeO nanotubes, which may have different optical behaviors.Despite the wealth of research on BeO, there remain unanswered questions and unexplored areas, particularly in relation to its behavior at the nanoscale.Previous studies have investigated the properties of various BeO nanostructures, including nanosheets, nanorods, and nanopeapods.However, the study of BeO nanotubes has been largely overlooked.Nanotubes, with their one-dimensional geometry and quantum confinement effects, are expected to exhibit novel electronic and optical properties that could open up new avenues for application.
In this study, we aim to fill this gap in the literature by exploring the electronic and optical properties of BeO zigzag nanotubes using density functional theory (DFT).We compare our findings with both experimental and theoretical data for monolayer BeO and other available data.Through this investigation, we hope to not only advance our understanding of BeO nanotubes but also highlight their potential for novel applications.

Computational method
We have investigated and calculated the electronic, optical, and structural properties of zigzag BeO nanotubes (n,0) with n = 6, 8, 10, 12, 14 and 16 by applying Siesta code in the framework of DFT and using functionals such as GGA-BLYP, GGA-PBEJsJrLo, GGA-PBE, GGA-PBEsol, GGA-WC and LDA-CA.In order to do this, we have taken the cutoff energy and k-mesh into consideration as 600 Rydberg and 1 × 1 × 11, respectively.Herein, we have used TZP instead of DZP in the calculations for structural optimization.The average of the Be-O bond length is calculated to be 1.6 Å and the maximum force exerted on each atom is considered to be less than 0.02 eV Å −1 .

Buckling calculation
Following structural optimization simulations, it was discovered that oxygen and beryllium atoms are attracted by the exterior and interior rings of the nanotube, respectively.This procedure explains how the structure has been buckled.We have determined the radius of anions (oxygen atoms) and subsequently the radius of cations (beryllium atoms) in order to compute the buckling of BeO nanotube structures.Then, we defined the difference between these two radii as buckling, denoted by = r a − r k ( r a = D a ∕2 , r k = D k ∕2 ) which is shown in Fig. 1.
The obtained buckling values for zigzag nanotubes (n,0) with n = 6, 8, 10, 12, 14 and 16 are given in Table 1.According to the findings, buckling is decreased when nanotube diameter is increased.The calculations are in good agreement with earlier works [29,30].Moreover, there is no buckling in the experimental findings for monolayer beryllium oxide [31].As a result, when selecting a nanotube with a sufficiently large radius, we expect that our calculations will demonstrate zero buckling in good agreement with experiment findings.The obtained buckling for the nanotubes with (n,0), n = 6, 8, 10, 12 and 16 shows that the buckling is smaller than 0.063 Å for all these nanotubes.This finding is in agreement with earlier studies by Baima [30] and Gorbunova [29].To determine the buckling tendency, we also created a buckling diagram in accordance with nanotubes radius.As seen in Fig. 2, the decreasing slope is steeper in bigger nanotubes.This trend is consistent with Ref. [31,33] and indicates that the buckling computed by these functionals (GGA-BLYP, GGA-PBEJsJrLo, GGA-PBE, GGA-PBEsol, GGA-WC and LDA-CA) in large nanotubes would be near to zero, resulting in buckling-free nanosheets.Additionally, the graph data shows that the WC functional exhibits less buckling than the others.Furthermore, all functionals yield findings that are nearly equivalent for large nanotubes.
Table 2 reveals that the higher energy gaps result in superior semiconducting characteristics, and the band gap increased gradually from the lowest magnitude of Eg = 5.04 eV for BeO (6,0) to the maximum magnitude of Eg = 5.62 eV for BeO (16,0).In other words, the fact that BeO nanotubes' bandgap energy was raised from 5.04 to 5.62 eV by increasing their diameters clearly shows that nanotubes with smaller diameters had better conductivity characteristics [18].The band gaps were calculated using a variety of functionals, and the results agree with both the conclusions of theoretical research as well as the results of the monolayer experiments.
The slope of rising band gap energy is less steep for nanotubes with larger radii, as seen in Table 2 and Fig. 5. Looking at this graph, we anticipate that for large-diameter nanotubes, the bandgap will approach the value for a nanosheet, which is 6.0 eV.Their band gap values are also shown in Fig. 4 and 5.In contrast to CNTs [35,36], larger BeO nanotubes exhibit wider band gaps.The primary cause is that π + σ hybridization is weaker in BeO tubes with lower radii [37,19].

Optical properties
In our study, we used density functional theory (DFT) methods to investigate the optical properties of nanotubes.However, it is important to note that DFT methods have limitations in accurately predicting optical properties.This is a well-known issue in the field and we have taken it into consideration when interpreting our results [38].Actually, a crystal's optical properties are a result of how it reacts to an electric field.Equation P = E , where is the Dielec- tric tensor, can be utilized to connect the polarization to the applied field when an external electric field affects the crystal.
The imaginary part of the dielectric function (in the range of linear optics and non-spin polarization) can be calculated using the following equation if the dielectric tensor in terms of frequency is taken to be ( ) = � ( ) + i �� ( ) [35, 39,  40]     where m is electron mass, and p is the momentum matrix element between α and β bands with the same crystal momentum k and , are dielectric tensor components.Also c k ,v k ,E c k , E k stand for empty states of conduction, occupied states of valence, the energies of conduction and valence, respectively.The real part of the dielectric function can be obtained using the Kramers-Kronig relation [37,40] as where P is the principal value of the integration.The refractive index, which is an important parameter to identify the optical properties of a material, can be obtained as below [41].
(2) where the static refractive index of the structure can be written by We have determined the absorption spectra for BeO zigzag nanotubes with (n,0) for n = 6, 8, 10, 12 and 16 as well as the real and imaginary components of the dielectric function.In order to do this, we used a number of exchange-correlation approximations (GGA-PBE, GGA-PBEsol, GGA-PBEJsJrLo and GGA-WC).The calculations related to the BeO nanotube (10,0) are given in Figs. 6, 7, 8 and 9.In Fig. 6, it can be seen that the GGA-PBE calculation reveals that the threshold for the onset of the energy absorption spectrum results in a lower energy than other functionals and that its fluctuations are bigger than those of the other functionals in the 8.5 to 10 eV range.Moreover, GGA-WC and GGA-PBEsol behave similarly, with fewer fluctuations than others in the 8.5 to 10 eV range.Figures 10 and 11 illustrate, with the use of GGA-WC, the real and imaginary ( 4)   3. The static refractive indices of zigzag nanotubes do not significantly differ from one another (Table 4).More specifically, the larger the radius of the BeO zigzag nanotube (BeO (16, 0)), the higher the static refractive index coefficients (Fig. 15).According to diagrams 12, 14, 15 and 16, when the electric field is parallel to the length of the nanotube, optical characteristics such as static refractive index and optical absorption are greater than those when the electric field is perpendicular to the nanotube axis.The data reported in Table 4 and plotted in Fig. 16 show that the static refractive index increases with the radius of the nanotube.
The optical band gap of a material can be calculated from its absorption spectrum using the Tauc relation [42].
This method involves plotting the absorption coefficient (α) as a function of photon energy (hν) and extrapolating the linear region of the plot to the x-axis to determine the band gap energy.As seen in Fig. 14, optical absorption in BeO zigzag nanotubes starts at energies around 6 eV.The electronic band gap refers to the energy difference between the top of the valence band and the bottom of the conduction band in a material.This determines the material's electrical conductivity, with larger band gaps corresponding to insulators and smaller band gaps corresponding to conductors or semiconductors.On the other hand, the optical band gap refers to the energy required for an electron to transition from the valence band to the conduction band by absorbing a photon.This determines the material's optical properties, such as its color and transparency.The optical band gap is typically larger than the electronic band gap due to excitonic effects.At energies around 9 to 10 eV, optical absorption reaches its maximum intensity.As the nanotube grows larger, the width of the absorption will decrease and the intensity of the absorption will increase.Additionally, the absorption intensity is higher when the field is applied perpendicularly to the nanotube's axis than when it is parallel.

Conclusions
In conclusion, our study utilized Density Functional Theory to explore the electronic and optical properties of BeO zigzag nanotubes (n,0) with n = 6 to n = 16.We discovered that the bandgap energy of these nanotubes increases with diameter, indicating that smaller-diameter nanotubes exhibit superior conductivity properties.This observation was further corroborated by the GGA-WC functional results, which closely align with experimental data.Moreover, we calculated the buckling of these nanotubes to be less than 0.063 Å, a negligible value that enhances their  potential for various applications.From an optical perspective, we found that the static refractive index remained consistent across different nanotube sizes, approximately 1.1, and the absorption peak occurred around 9 eV.These properties were not significantly affected by the increasing nanotube radius.These findings provide valuable insights into the characteristics of BeO nanotubes and their potential applications in optoelectronic devices and other fields.

Fig. 1
Fig. 1 Schematic view of zigzag BeO nanotube (8, 0). a The distance between two beryllium atoms facing each other ( D a ) b The distance between two oxygen atoms facing each other ( D k )