**3. Results and discussion**

#### **3.1 Structural properties of graphene supercells**

A graphene unit cell of two carbons was constructed with a 1.42 Å C-C bond lengths, 120 bond angles, and the lattice parameters a = b = 2.46 Å and c = 6.8 Å (see **Figure 1a**). The unit cell was then extended to construct n n supercells (where n is an integer number). A total of seven graphene supercells were constructed. A 4x4 supercell is shown in **Figure 1b**. A space group of P6/mmm was used for all supercells. The constructed supercells are listed in **Table 1**.

#### **3.2 Electronic properties of graphene supercells**

**Table 1** shows all the possible combinations of *n* x *n* graphene supercells constructed in this work along with the number of carbons of that supercell. We use C# to denote the number of carbons of an *n n* supercell, where # is an integer number representing the number of carbons. The calculated band structures are shown in **Figures 2** and **3**. The Fermi level was set to zero and is indicated by the red dashed lines.

The band structures of the supercells studied show Fermi levels at the Dirac points, showing a bandgap of zero, which agrees with other literature [6–12]. It is observed

**Figure 1.** *A four-atom unit cell of graphene.*


#### **Table 1.**

*Different graphene supercells.*

**Figure 2.** *Calculated band structures of (a) C2, (b) C8, (c) C18 and (d) C32.*

that the bandgap energy of graphene is not affected when the size of the supercells is changed.

To investigate the nature of the states that comprise the conduction and valence band edges, we calculated the contributions of all atomic orbitals in the total density of states (TDOS) and the unique atomic shells in the partial density of states (PDOS) band edges. **Figures 4** and **5** show the density of states (DOSs), which describes the number of states per energy interval. The DOS agrees with the calculated band structure. At the Fermi level, the DOSs are very low, which are consistent with the calculated band structure.

*Structural, Electronic, and Optical Properties of Mono- and Co-Doped Graphene with Ti… DOI: http://dx.doi.org/10.5772/intechopen.106143*

**Figure 3.** *Calculated band structures of (a) C50, (b) C72, and (c) C98.*

**Figure 4.** *TDOS and PDOS for the (a) C2, (b) C8, (c) C18, and (d) C32 graphene supercells.*

**Figure 5.** *TDOS and PDOS for the (a) C50, (b) C72, and (c) C98 graphene supercells.*

It was found that the electron distribution in graphene is due to the contribution of the s and p atomic shells, which are responsible for the energy transfer in graphene. The results show that the s and p states are dominant in both the conduction and valence bands. However, at the Fermi level or near the Fermi level, only the p state is dominant.

#### **3.3 Optical properties of graphene supercells**

To investigate the optical response of graphene, we calculated its absorption, dielectric function, and refractive index. **Figure 6** illustrates the optical absorption calculations for different supercells. It can be observed that C8, C32, and C50 supercells exhibit strong absorption in the ultraviolet-visible range that extends into the infrared. The C2 and C18 supercells absorb more light in the UV area and dissipate in the visible region around the wavelength of 600 nm. C98 enhances the absorption activity to 700 nm, while C72 enhances it to 900 nm.

Dielectric materials tend to become polarized when exposed to an external electric field. The term "dielectric function" refers to the property of a substance that determines its polarization. The dielectric function is defined as follows:

$$
\varepsilon = \varepsilon\_1(a) + \varepsilon\_2(a) \tag{1}
$$

where *ε*1ð Þ *ω* and *ε*2ð Þ *ω* are the real and imaginary parts of the dielectric function. The real part of the dielectric function is connected to the material's polarization, whereas the imaginary part is related to the electronic absorption. **Figure 7** shows the calculated dielectric function of the seven graphene supercells up to a photon energy of 10 eV. In the limit of zero photon energy, the findings showed dielectric constant *ε*<sup>0</sup> values of 4.97, 6.82, 4.23, 11.92, 7.44, 6.07, and 6. 03 for the supercells C2, C8, C18, C32, C50, C72, and C98, respectively. The dielectric constant is proportional to the

*Structural, Electronic, and Optical Properties of Mono- and Co-Doped Graphene with Ti… DOI: http://dx.doi.org/10.5772/intechopen.106143*

**Figure 6.** *Calculated absorption properties of the graphene supercells.*

**Figure 7.** *Calculated dielectric function properties of the graphene supercells.*

electric displacement, which is proportional to the polarization of the material. The imaginary part of the dielectric function shows that the low-frequency peaks are located at 1.93 to 3.62 eV.

The real part *n*(ω) and imaginary part *k*(ω) (extinction coefficient) of the refractive index are determined by the dielectric function using the Kramers-Kronig transformation and are defined as:

$$m(\alpha) = \left(\frac{|\varepsilon(\alpha)| + \varepsilon\_1(\alpha)}{2}\right)^{\frac{1}{2}}\tag{2}$$

$$k(o) = \left(\frac{|\varepsilon(o)| - \varepsilon\_1(o)}{2}\right)^{\frac{1}{2}}\tag{3}$$

The extinction coefficient is relative to the amount of light absorbed. **Figure 8** shows the calculated refractive index of graphene supercells. The refractive index *n*<sup>0</sup> is equal to the ffiffiffiffiffi *ε*0 p in the limit of zero photon energy. These findings reveal refractive index *n*<sup>0</sup> values of 2.23, 2.61, 2.06, 3.45, 2.73, 2.46, and 2.46 for supercells C2, C8, C18, C32, C50, C72, and C98, respectively. The extinction coefficient has low-frequency peaks located from 2.28 to 3.97 eV.

#### **3.4 Electronic properties of doped graphene**

According to the findings of research conducted on the optical properties of various graphene supercells, the 4x4 graphene supercell demonstrates a superior optical response compared to other graphene supercells. These results have contributed to the author's decision to use the 4x4 graphene supercell and dope it with titanium and ruthenium atoms to study its electronic and optical properties. In the case of monodoped graphene, the doping was accomplished by exchanging one of the graphene's carbon atoms for either titanium or ruthenium as shown in **Figure 9(a)**. On the other hand, in the case of co-doped graphene, two carbon atoms were exchanged for titanium and ruthenium, as illustrated in **Figure 9(b)**.

**Figure 10** presents an illustration of the band structure that was computed for the doped graphene 4x4 super cell. The results show that doping graphene causes an increase in the bandgap, which can be seen near the Fermi level in **Figure 10**. While the energy of the bandgap of Ti-doped graphene is 0.555 eV, the energy of the bandgap of Ru-doped graphene is 0.786 eV, and the energy of the bandgap of codoped graphene is 0.272 eV. As a result of these properties, graphene can now be categorized as a material that falls into the category of semiconductor. The band

**Figure 8.** *Calculated refractive index properties of the graphene supercells.*

*Structural, Electronic, and Optical Properties of Mono- and Co-Doped Graphene with Ti… DOI: http://dx.doi.org/10.5772/intechopen.106143*

**Figure 9.** *Structures of doped graphene: (a) mono-doped and (b) co-doped.*

**Figure 11.** *TDOS and PDOS for the (a) Ti*-*doped, (b) Ru-doped and co-doped graphene.*

structures shown in **Figure 10a** and **b** both exhibit an indirect bandgap, whereas the bandgap shown in **Figure 10c** exhibits a direct bandgap.

**Figure 11** shows the contribution made by doping graphene to the atomic shells. Doping graphene with the selected elements results in the introduction of numerous minor state density peaks, which can be clearly seen in the Fermi level region, as shown in **Figure 11**. This can be clearly observed by comparing **Figure 11** with the density of states of the 4x4 graphene supercell as shown in **Figure 5(d)**. Ru is a transition metal having a 4d electron configuration, whereas Ti has a 3d electron configuration. The 3d contribution from Ti is highlighted by the cyan color in **Figure 11a**, and the 4d contribution from Ru is highlighted by the cyan color in **Figure 11b**. Both of these contributions are in the energy range of 6 eV to 6 eV. In the vicinity of the Fermi level, the 3d state density has a modest peak, whereas the 4d state density has obvious peaks.

#### **3.5 Optical absorption of mono and co-doped graphene**

The calculated results of the optical absorbance of the doped graphene are illustrated in **Figure 12**. The results demonstrate that, when graphene is doped with titanium and ruthenium, there is no change in the activity of the absorption in the ultraviolet area. When Ti and Ru are added to graphene, a blue shift occurs in the

*Structural, Electronic, and Optical Properties of Mono- and Co-Doped Graphene with Ti… DOI: http://dx.doi.org/10.5772/intechopen.106143*

**Figure 12.** *Calculated absorption properties for pristine and doped graphene.*

visible area. This blue shift is found for both mono-doped and co-doped graphene. Graphene with one dopant also exhibits this blue shift. Because of the dopants that were introduced into graphene, there is a red shift that occurs in the infrared region, which is located above 800 nm.

## **4. Conclusion**

Using density functional theory as it is embodied in Material Studio, a number of electrical and optical properties were successfully explored. The results of our research indicate that the electrical structure of different graphene supercells is the same. The band structure computations showed that the Fermi level in all graphene supercells is located at the Dirac point, which indicates that there is no bandgap energy. According to the results of the density of state calculations, the s and p orbitals predominated in the valence band as well as the conduction band. According to the findings of the absorption, certain graphene supercells have minimal activity in the visible region, but other graphene structures have high absorbance in the visible area reaching all the way to the infrared sector of the spectrum. The findings suggest that graphene has a high dielectric constant, which qualifies it as an excellent candidate for application in various electrical devices. When graphene is mono-doped or co-doped with either titanium or ruthenium, a bandgap opening occurs. These results demonstrate a change from semi-metallic to semi-conducting behavior. This paves the way for the exploration of novel applications for graphene as a semiconductor.

## **Acknowledgements**

The authors acknowledge the Centre for High Performance Computing (CHPC) for computing.

Resources. LP thanks the National Institute for Theoretical and Computational Sciences (NITheCS) and Armaments Corporation of South Africa SOC Ltd. (ARMSCOR) for financial support.
