J. Nig. Soc. Phys. Sci. 5 (2023) 1083 Journal of the Nigerian Society of Physical Sciences Theoretical Investigation of Diameter Effects and Edge Configuration on the Optical Properties of Graphdiyne Nanotubes in the Presence of Electric Field T. M. J. Abdulkadhima,∗, S. A. A. Alsaatia, M. H. Shinenb aCollege of Basic Education, University of Babylon, Babylon 51002, Iraq bCollege of Science, University of Babylon, Babylon 51002, Iraq Abstract In this research, the structural, electronic and optical properties of the armchair (ant) and zigzag (znt) Graphdiyne nanotubes (GDY-NT) with different diameters were studied based on density functional theory (DFT). The computations were done using SIESTA code, based on linear combination of localized atomic orbitals (LCAO) method and the generalized gradient approximation (GGA). The results from the band structure analysis show that all these nanotubes are semiconductors with direct band gap at gamma point. The band gap of the nanotubes is clearly dependent on the nanotube diameter, and by increasing the nanotube diameter, the band gap is decreased. Optical properties such as dielectric function; absorption coefficient, optical conductivity and refractive index were examined and calculated for all samples. The results show that all these functions have an inverse relationship with the nanotube diameter and a direct relationship with the band gap. The effect of applying the external electric field with intensity of 0.1 V/Å, 0.2 V/Å in the direction of x-axis (perpendicular to the nanotube axis) on the structural and electronic features of these nanotubes has been studied and calculated. DOI:10.46481/jnsps.2023.1083 Keywords: Graphdiyne nanotubes, Electric field, Band gap, Optical properties, SIESTA code, Density functional theory Article History : Received: 22 September 2022 Received in revised form: 15 November 2022 Accepted for publication: 26 November 2022 Published: 02 March 2023 © 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: K. Sakthipandi 1. Introduction Unlike natural structures of carbon that often have two sta- ble allotropes of diamond and graphite, and sp3 and sp2 hy- bridized carbon atoms, respectively, a wide range of synthetic carbon allotropes have been synthesized or theoretically pre- dicted. This has made the present day become known as the ∗Corresponding author tel. no: +964 7817977099, +964 7816022376 Email address: taqi.mohammed@uobabylon.edu.iq (T. M. J. Abdulkadhim) era of carbon allotropes [1]. As the most prominent and most notable achievements, the discovery of Fullerenes [2], carbon nanotubes [3], and Graphene [4-6] represents the prominent and novel types of zero-, one- and two-dimensional carbon struc- tures that have the carbon structures with sp2 hybridization. In addition, the tendency of carbon to form the three types of sp1, sp2 and sp3 hybridization states allows for the creation of mul- tiple combinations of carbon allotropes by linking the carbon atoms with different hybridizations [7-9]. One of the most interesting families of carbon allotropes introduced in recent years is the Graphyne family [8]. This car- 1 Abdulkadhim et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1083 2 bon allotrope has a flat structure with a one atom thick single- plate lattice that is made by replacing some = C = C = bonds in Graphene with -C≡C- acetylene bonds. These structures have two non-equivalent types of carbon atoms: sp2 hybridized car- bon atoms with three identical bonds and sp hybridized carbon atoms with two identical bonds [10]. In Graphdiyne, a member of the Graphyne family, there are also two sp and sp2 hybrids of carbon atom [11]. The sp2 hybridized carbon atoms form a hexagonal ring, which are connected together by di-acetylene sp connections. Compared to Graphene, Graphdiyne has many interesting properties such as uniformly distributed holes, strong π bonding, controllable electronic features, lower density, high electrical conductivity excellent hardness and high temperature resistance that these properties are due to the sp and sp2 hybrids and the natural holes in its structure [11]. These plane carbon lattices with high number of π bond, density lower than that of Graphene, and modifiable electronic properties are the promising materials in the nano-electronics industry. It is also believed that various forms of Graphyne with multiple functionalities can compete with the older carbon structures such as Fullerenes, carbon nanotubes or Graphene and respond positively to the increasing need for carbon nanos- tructures. 2. Computational Method The atomic structure of nanotubes has been investigated by density functional theory (DFT) approach and using SIESTA computational code based on LCAO method and the GGA-PBE approximation for the exchange-correlation functional of e-e in- teraction [12-13]. Brillouin zone (BZ) division was done by Monkhorst–Pack method. The optimum number of k points selected for the armchair Graphdiyne nanotubes was 1 × 1 × 7 and for the zigzag Graphdiyne nanotubes, it was 1 × 1 × 11. In the computations, the optimal cutoff energy used for the armchair Graphdiyne nanotubes was 300 Ry and for the zigzag Graphdiyne nanotubes, it was 400 Ry. A large enough vacuum layer (10 Å and 12 Å for the armchair and zigzag nanotubes, re- spectively) is considered for all nanotubes to avoid interacting with adjacent structures. All these nanotubes are relaxed until the total atomic force value reaches less than 0.01eV/Å. The optimized lattice constant obtained for the armchair and zigzag nanotubes were 16.71Å and 9.46Å, The dimeter of znt2, znt3, znt4, ant2, ant3 and ant4 after relaxation is obtained equal to 10.25, 15.56, 20.81, 6.42, 9.08 and 12.04, respectively. In order to perform the optical computations, the energy range in which the optical properties are investigated was se- lected from zero to 40eV. In the optical computations, the spin- orbit interaction is neglected because it will not have much im- pact on the results. In this section, one of the important pa- rameters is the kind of the optical polarization. In all structures studied, the polarized light and its propagation along the x di- rection were considered. The optical properties such as the real part of the dielectric function, absorption coefficient, refractive Table 1. The Optimized Bond Lengths of the znt and ant Graphdiyne Nanotubes after Relaxation Optimized bonding length (Å) Bonding type 1.40 =C − C ≡ 1.43 =C = C = 1.35 = C − C ≡ 1.24 − C ≡ C −− index and reflection coefficient were calculated using the imag- inary part of the dielectric function and the Kramers–Kronig re- lations [14]. In the final step, the efficacy of applying the electric field in the direction of x-axis, with the sizes of 0.1 V/Å, 0.2 V/Å on the structural and electronic features of the zigzag and armchair Graphdiyne nanotubes was studied. 3. Results and discussion 3.1. Investigating the Structural Properties Similar to carbon nanotubes, Graphdiyne nanotubes are pro- duced by rolling the Graphdiyne sheet (plate). Similar to carbon nanotubes, the denomination (n, m) can be used for Graphdiyne nanotubes. However, unlike carbon nanotubes, (n, n) repre- sents the zigzag type and (n, 0) represents the armchair type. Nanotube Maker software was used to obtain the coordinates of points forming nanotubes [14]. In the study, the zigzag (2, 2), (3, 3), (4, 4) and armchair (2, 0), (3, 0), (4, 0) Graphdiyne nanotubes were investigated. Figures 1 and 2 show the struc- ture of the znt and ant nanotubes examined after optimization, respectively. The bond lengths of nanotubes after the structural relaxation are presented in Table 1. The values obtained are in agreement with the results of other researchers [15]. The length of carbon-carbon bonds in the hexagonal ring and the di-acetylene bonds are not the same, indicating the existence of different hybrids in the carbon bonds of Graphdiyne nan- otubes. This difference leads to more structural flexibility for Graphdiyne nanotubes compared to carbon nanotubes [16-17]. Another important difference between Graphdiyne nanotubes and carbon nanotubes is that GDY-NT have the uniformly dis- tributed holes in their walls, which facilitates the electron trans- port in the nanotube wall and it is important for the applications such as hydrogen storage. 3.2. Electronic Properties of GDY-NT The band structure of the znt and ant Graphdiyne nanotubes is shown in Figures 3 and 4, respectively. The Fermi energy was selected at the zero point. All these GDY-NT are semicon- ductors with the direct band gap at gamma point of the first BZ. The values of band gap and diameter of the zigzag and arm- chair nanotubes are presented in Table 2. Similar to carbon nanotubes, the diameter of Graphdiyne nanotubes can be ob- tained from the following relationship: d = a p √ n2+nm+m2 2 Abdulkadhim et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1083 3 Figure 1. Structure of the Zigzag Graphdiyne Nanotubes after Relaxation that a is the lattice constant of GDY-NT [5]. As shown in these Figures, because of the quantum confine- ment effect, the band gap value of the znt and ant nanotubes is obviously dependent on the nanotube diameter, and decreases with increasing the nanotube diameter. In addition, the arm- chair nanotubes have a smaller diameter and the larger band gap than the zigzag nanotubes. Figures 5 and 6 show the density of the total states (DOS) of the znt and ant Graphdiyne nanotubes, respectively. The DOS at Fermi energy is zero, which confirms the semiconducting be- havior of GDY-NT. Examining the band structure and the DOS, we observe that as the diameter of the GDY-NT increases, the band gap decreases and the DOS increases. 3.3. Optical Properties of Nanotubes 3.3.1. Dielectric Function The complex dielectric function describes the optical prop- erties of solids. This function is used to describe the crys- tal response to electromagnetic fields, which depends on the band structure of crystal [18]. In Figures 7 and 8, the dia- grams of the real and imaginary parts of the dielectric function Figure 2. Structure of the Armchair Graphdiyne Nanotubes (Double wall) after Relaxation Table 2. The Value of Band Gap and Diameter of the znt and ant Graphdiyne Nanotubes Nano tube diameter (Å) Band gap (eV) Structure 10.25 0.65 znt2 15.56 0.55 znt3 20.81 0.50 znt4 6.42 0.95 ant2 9.08 0.65 ant3 12.04 0.55 ant4 Figure 3. Band Structure of the znt Graphdiyne Nanotubes of the Graphdiynenanotubes are plotted for the polarized inci- dent light in the direction of x-axis (perpendicular to the axis of Graphdiynenanotubes), respectively. The imaginary part of the dielectric function partly reflects the actual transfers between occupied and unoccupied states. We also know that the inter- band transition is due to excitation at the absorption edges, and the intra-band transition is the result of volumetric plasmons (absorption by free electrons). Therefore, in diagram 7, the relatively high increase in energy values less than 5eV in the 3 Abdulkadhim et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1083 4 Figure 4. Band Structure of the Armchair Graphdiyne Nanotubes Figure 5. DOS of the Zigzag GDY-NT imaginary part of the dielectric function, as well as the almost constant behavior at energies above 30eV confirm these absorp- tion approaches. Comparison among different nanotubes with various diame- ters shows that the highest dielectric function intensity is related to AGDYNT (2, 0) nanotube which has the smallest diameter and the largest gap, and the lowest dielectric function intensity is related to ZGDYNT (4, 4) nanotube that has the largest diam- eter and the smallest gap. The results show that the dielectric function intensity is inversely related to the diameter of the nan- otubes and there is a direct relationship between the dielectric function intensity and the band gap. Figure 6. DOS of the Armchair GDY-NT Figure 7. Real Part of the Dielectric Function of GraphdiyneNanotubes for the Polarized Light in the Direction of X-Axis 3.3.2. Optical Absorption The absorption spectrum of GDY-NT with different diame- ters is illustrated in Figure 9. The Figure shows the permitted optical transitions of the electron between the empty states of the conduction band and the full states of the valence band. The threshold energy for transition in the absorption spectrum cor- responds to the gap size in the band structure of Graphdiyne- nanotubes and it means that the electrons are excited and make a transient by receiving the least energy greater than the band gap value. The comparison between different nanotubes with various diameters also shows that the highest absorption is related to 4 Abdulkadhim et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1083 5 Figure 8. Imaginary Part of the Dielectric Function of GraphdiyneNanotubes for the Polarized Light in the Direction of X-Axis Figure 9. The Diagram of Absorption Spectra of GraphdiyneNanotubes with Different Diameters AGDYNT (2, 0) nanotube which has the smallest diameter and largest gap, and the lowest absorption is related to ZGDYNT (4, 4) nanotube which has the largest diameter and the smallest gap. The results show that the optical absorption coefficient is inversely related to the diameter of the nanotubes and is directly related to the band gap. 3.3.3. Optical Conductivity Figure 10 shows the changes of optical conductivity of Graphdiynenanotubes with different diameters compared to the incident photon energy. As can be seen, the diagram of optical conductivity is proportional to the imaginary part of the dielec- tric function. Generally, at the energies that present in the imag- inary part of the peak dielectric function, these peaks are also seen in the real part of the optical conductivity. In this spec- trum, the conductivity increases up to 20eV and then decreases, and at energies with the highest absorption, we have the optical conductivity maximum. The results show that the optical con- ductivity has an inverse relationship with the diameter of the nanotubes and a direct relationship with the band gap. 3.3.4. Reflection The reflection diagram of GDY-NT with different diameters is plotted in Figure 11. According to this diagram, we find that the magnitude of the reflection is generally very small at the en- ergies above 6eV. We also have the maximum reflection at the Figure 10. The Diagram of Optical Conductivity of GDY-NT with Different Diameters for the Polarized Light in the Direction of X-Axis Figure 11. Reflection Diagram of GraphdiyneNanotubes with Different Diam- eters energies where the highest absorption occurs. Here, a compar- ison of the reflection coefficient of the nanotubes with different diameters shows that the reflection coefficient is inversely re- lated to the diameter of the nanotubes and directly related to the band gap. 3.3.5. Refractive Index The value of the refractive index at zero energy is called the static refractive index, which is equal to the static dielectric con- stant. The static refractive index of the znt and ant Graphdiyne- nanotubes with different diameters is reported in Table 3. If, by increasing the frequency, the refractive index increases, this be- havior is called the normal dispersion, which is the usual behav- ior of all transparent materials. In areas where the slope of the diagram of refractive index is negative, it is related to absorp- tion. In this area, when passing through the material, longer - wavelength light breaks more than short - wavelength light, which this behavior is called the anomalous dispersion. The diagram of refractive index and extinction coefficient of GDY- NT with different diameters are shown in Figures 12 and 13, respectively. As you can see in these diagrams, the refractive index reaches its lowest value at the energies between 4eV and 6eV, so the reflection increases, which is consistent with Figure 13. 5 Abdulkadhim et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1083 6 Table 3. Static Refractive Index of GraphdiyneNanotubes ant2 ant3 ant4 znt2 znt3 znt4 1.68 1.64 1.5 1.62 1.64 1.57 Figure 12. Diagram of Refractive Index of GDY-NT with Different Diameters Figure 13. The Diagram of Extinction Coefficient of GDY-NT with Different Diameters 3.4. Effect of Applying the External Electric Field In the final step, the electric field along the x-axis with the sizes of 0.1V/Å and 0. 2V/Å was applied to the znt and ant Graphdiynenanotubes with different diameters. The geometric structure and band structure of the Graphdiyne nanotubes were compared in the absence and presence of the external electric field. The results show that by applying the electric field in the direction of the x-axis and with different sizes to the ant2 nan- otube, its geometric structure and band structure is not changed (Figure 14). For the ant3 nanotubes, as shown in Figure 15, there is little change in the energy bands in the presence of the electric field of 0.1V/Å, and only slight shift (about 0.02 to 0.05eV) is seen in the energy bands. In addition, the size of the band gap is not changed. But by applying the electric field of 0.2V/Å, the band structure is completely changed and the size of band gap increased (Figure 15). In addition, by increas- ing the external electric field, the diameter of ant3 nanotube is also increased. Figure 16 shows the band structure of ant4 nanotubes in the presence and in the absence of the external electric field. In the presence of the electric field of 0.1V/Å, the size of band gap is not changed. But greater shift is seen in the energy bands com- Figure 14. Comparison of the Band Structure of ant2 Nanotubes in the Absence (E=0) and Presence of Electric Field (E=0.1 and 0.2V/Å) along the X-Axis Figure 15. Comparison of the Band Structure of ant3 Nanotubes in the Absence (E=0) and Presence of Electric Field (E=0.1 and 0.2V/Å) along the X-Axis pared to the ant3 nanotube (about 0.1eV). By applying the elec- tric field of 0.2V/Å, the band structure is completely changed and the size of band gap increased. The diameter of ant4 nan- otube is also increased by applying the electric field. The results are summarized in Table 4. The band structure of znt2 nanotubes in the absence and presence of the external electric field is compared in Figure 17. As can be seen in the Figure, by applying the electric field of 0.1V/Å, there is little change in the energy bands. The size of the band gap and the nanotube diameter also show a slight 6 Abdulkadhim et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1083 7 Table 4. The Size of Band Gap (Eg), Shift of Energy Bands and Diameter (d) of the Armchair Graphdiyne Nanotubes with Different Diameters in the Presence of the External Electric Field in the Direction of X-Axis and with the Sizes of 0.1V/Å and 0.2V/Å Shift (eV) d (Å) Eg (eV) structure E=0.2 E=0.1 E=0.2 E=0.1 E=0 E=0.2 E=0.1 E=0 0 0 5.8 5.8 5.8 6.42[3,4] 0.78 0.78 0.78 0.95[3,4] ant2 Structure changed 0.02- 0.05 9.2 9.07 9.04 9.08[3,4] 0.87 0.66 0.66 0.65[3,4] ant3 Structure changed 0.1 12.94 11.94 11.65 12.04[3,4] 0.65 0.57 0.57 0.55[3,4] ant4 Figure 16. Comparison of the Band Structure of ant4 Nanotubes in the Absence (E=0) and Presence of Electric Field (E=0.1 and 0.2V/Å) along the X-Axis decrease. As the external electric field intensity increases, the band structure is changed and the band gap is also decreased, but the nanotube diameter is slightly increased. The band structure of znt3 nanotube in the presence and in the absence of the external electric field is plotted in Figure 18. Applying the electric field of 0.1V/Å, there is little change in the energy bands. The size of the band gap and the nanotube diameter also show a slight decrease. As the intensity of the external electric field increases, there is a slight shift in the band structure (0.04eV). The size of the band gap and the nanotube diameter is also slightly decreased. In Figure 19, the band structure of znt4 nanotube is plotted in the absence and presence of the external electric field. By applying the electric field of 0.1V/Å, a shift of about 0.02eV is seen in the energy bands. The band gap value and the nanotube diameter also show a slight decrease. Applying the electric field of 0.2V/Å, the geometric structure and band structure of the nanotubes are completely changed. As shown in Figure 20, the shape of zn4 nanotube has become oval and the size of its band gap has decreased in the presence of E= 0.2V/Å. The results are summarized in Table 5. Given that the electric field is applied perpendicular to the axis of nanotubes, the atoms in the wall of the nanotubes have Figure 17. Comparison of the Band Structure of znt2 Nanotubes in the absence (E=0) and presence (E=0.1 and 0.2V/Å) of the Electric Field along the X-Axis the different potentials. The larger the nanotube diameter, the potential difference is also greater. Therefore, we expect that the geometric structure and band structure of nanotubes that are larger in diameter undergo more changes compared to nan- otubes that are smaller in diameter, which this is consistent with the results obtained. The znt4 nanotube, which has the largest diameter among all nanotubes investigated, exhibits the most changes in the geometric structure and band structure [19-20]. No research was conducted to study the effect of applying the external electric field on GDY-NT for comparison. The re- sults on other carbon nanostructures show that the effect of ap- plying the external electric field on the armchair Graphdiynenano- ribbons with various widths is different. Small width does not change the size of band gap. In larger widths (N> 13), by in- creasing the electric field intensity, the size of band gap is de- creased. By applying the electric field to the (10, 5) carbon nanotube in the direction of x-axis, the size of band gap is first increased and then decreased [21-22]. 4. Conclusion In the study, the zigzag (2, 2), (3, 3), (4, 4) and armchair (2, 0), (3, 0), (4, 0) Graphdiyne nanotubes were investigated. The results from the investigating the carbon-carbon bond lengths in Graphdiyne nanotubes are in agreement with those of other researchers on Graphdiyne nano-ribbons and nanotubes. All these nanotubes are semiconductors with direct band gap at gamma point of the first Brillouin zone. Because of the quan- tum confinement effect, the value of band gap of the znt and 7 Abdulkadhim et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1083 8 Table 5. The Size of Band Gap (Eg), Shift of Energy Bands and Diameter (d) of the Zigzag Graphdiyne Nanotubes with Different Diameters in the absence and presence of the External Electric Field along the X-Axis and with the Sizes of 0.1V/Å and 0.2V/Å Shift (eV) d (Å) Eg (eV) structure E=0.2 E=0.1 E=0.2 E=0.1 E=0 E=0.2 E=0.1 E=0 Structure changed 0.0 10.33 10.286 10.287 10.25[3,4] 0.42 0.63 0.64 0.65[3-5] znt2 0.04 0.0 15.6 15.542 15.541 15.56[3,4] 0.53 0.54 0.55 0.55[3,4] znt3 Structure changed 0.02 12.4*24.93 20.803 20.808 20.81[3,4] 0.34 0.50 0.51 0.50[3,4] znt4 Figure 18. Comparison of the Band Structure of znt3 Nanotubes in the absence (E=0) and presence (E=0.1 and 0.2V/Å) of the Electric Field along the X-Axis Figure 19. Comparison of the Band Structure of znt4 Nanotubes in the absence (E=0) and presence (E=0.1 and 0.2V/Å) of the Electric Field along the X-Axis ant nanotubes is obviously dependent on the nanotube diame- ter, and by increasing the nanotube diameter, it decreases. In addition, the armchair nanotubes have a smaller diameter and the larger band gap compared to the zigzag nanotubes. As the diameter of the nanotubes increases, the band gap decreases and the DOS increases around the Fermi level. This means that the electrons require less energy to transition from the valence band to the conduction band. The optical properties of all nanotubes above-mentioned were calculated and investigated using SIESTA package and based on the Kramers–Kronig relations. Dielectric function, absorption coefficient, optical conductivity, reflection, refrac- tive index and extinction coefficient of all samples were calcu- Figure 20. Geometric Structure of Zigzag Graphdiyne Nanotube (znt4) in the Presence of the Electric Field of 0.2V/Å after Relaxation lated. The results show that all these functions are inversely re- lated to the diameter of the nanotubes and are directly related to the band gap, so that the highest intensity is related to the arm- chair (2, 0) nanotubes and the lowest intensity is to the zigzag (4, 4) nanotubes. In addition, the effect of external electric field on the geo- metric structure and band structure of the ant and znt Graphdiyne nanotubes with different diameters is also studied. The results show that by applying the electric field of 0.1 V/Å along the x- axis to the armchair and zigzag Graphdiyne nanotubes, there is little change in the geometric structure and band structure. 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