A new two-dimensional allotrope of carbon, called penta-graphene, was predicted as an indirect bandgap semiconductor in 2015 [11]. Then, by using density functional theory (DFT) the energy structure and electronic properties of the two-dimensional penta-graphene nanostructure have been studied to understand the feasibility of synthesizing this nanostructure and its application in Nano-electronics. In this section, based on the principles of basic calculations related to the density functional theory, in line with new research, the optical properties of the optimized structure of the penta-graphene unit cell under buckling due to vertical compressive strain in the Z direction are investigated [12]. The final results show that according to the considered conditions, the desired nanostructure can be effective in the design of multi-purpose electro-optical devices.

The two-dimensional crystal nanostructure of penta-graphene shown in Fig. 1, has space group no. 113 and P-421m with a tetragonal lattice consisting of two types of carbon-carbon bond lengths (C1 & C2), where the atomic layer C2 with sp3 hybridization is located between two layers of C1 with sp2 hybridization. According to Fig. 1, the constants of the penta-graphene network have been optimized using the thermodynamic mode of Birch-Murganan equations [24]:

$$E\left(V\right)={E}_{0}+\frac{9{B}_{0}{V}_{0}}{16}\left\{{\left[{\left(\frac{{V}_{0}}{V}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1\right]}^{3}{B}_{0}^{{\prime }}\right\}+\frac{9{B}_{0}{V}_{0}}{16}\left\{{\left[{\left(\frac{{V}_{0}}{V}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1\right]}^{2}\left[6-4{\left(\frac{{V}_{0}}{V}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]\right\}$$

(1)

In Eq. (1), the optimum volume energy is gotten with the lattice parameters (a, b) where\({B}_{0}^{{\prime }}\)concerns the bulk modulus under pressure, *B**0* is the modulus in zero pressure, and *V**0* is the primary volume. Figure 1 (a), shows the top and side views of the Penta-graphene 2D crystal supercell, and (b) indicates the volume vs energy curve for the penta-graphene. The curve's minimum point in Fig. 1(b) shows the optimized volume in the least energy of the unit cell. By the obtained equilibrium volume, the optimized lattice parameter is proposed as a = b = 3.649A°, and the thickness between each upper and lower layer of the C1 atom is called the buckling (δ), which is equal to 0.589A°. Consequently, the total thickness of the 2D Penta-graphene nanosheet is 1.178A°.

Table 1, demonstrates the calculated optimized structural data of Penta-graphene such as lattice parameter, buckling constant, Bond length, and optimize energy by using the PBE-GGA approximation method [12].

Table 1

optimized structural data of Penta-graphene by PBE-GGA method.

Method | Optimized lattice (A0) | Buckling (A0) | Bond length (A0) | Eopt (Ry) |

PBE-GGA | **a = b = 3.649** | **δ = 0.589** | **C1-C2 = 1.543** **C1-C1 = 1.342** | **-456.735** |

Results of the investigation of the structural properties of this 2D carbon allotrope indicate that optimized penta-graphene nanostructure is known as an indirect band gap semiconductor, it has structural data in accordance with previous studies 27, 28. Also according to Fig. 2, the band gap and total energy are calculated under buckling reduction up to 12%. The results show that when the buckling constant is reduced, the total energy and consequently, stability of the system slightly decreases according to the blue curve in Fig. 2. The band gap shifts under the effect of buckling variation shown by the violet curve in Fig. 2, which illustrates by imposing vertical stress, the band gap of penta-graphene is reduced.

## 2.1 Penta-graphene optical properties

In this part, some significant optical properties such as joint density of states, real and imaginary parts of the complex dielectric function, absorption, reflectivity spectrum, and energy loss, of the penta-graphene, in optimal state and buckling variation condition up to 12 percent, are examined in the 0–30 eV energy range. Taking into account the fact that in order to make variations in the buckling, it is necessary to apply stress in the vertical direction on the nanosheet, the optical properties in the Z polarization direction are investigated and plotted.

The joint density of states (joint DOS) for the study of the optical inter-band transitions from the occupied states to the unoccupied states is investigated and shown in Fig. 3. This figure illustrates the joint DOS spectrum of the penta-graphene in the Z direction, without stress and under vertical stress conditions with 4, 8, and 12 percent to reduce buckling constant. Figure 3 shows that the joint DOS spectrum decreases with application stress in the nanomaterial. According to Fig. 3, due to the active stress, the buckling parameter decreases in the z-direction. In such a way that the sharp peak with the energy of about 11 eV in the non-stressed state, with stress, applied up to 12 percent on the penta-graphene, decreases the slope and moves towards the energy of about 14 eV.

Complex dielectric function, which consists of two parts, real and imaginary components as an operational factor to recognize all optical aspects such as absorption and reflectivity spectrum, is given by the following equation [25]:

$${\varvec{\epsilon }}_{\varvec{c}\varvec{o}\varvec{m}\varvec{p}\varvec{l}\varvec{e}\varvec{x}}= \mathfrak{R}+\mathfrak{ }\mathbf{i} \mathfrak{I}\varvec{\epsilon }$$

2

The real component of the dielectric function obtained using the krmaers-kroning relations is as follows:

$$\mathfrak{R}{\epsilon }^{\alpha \beta }\left(\omega \right)={\delta }_{\alpha \beta }+\frac{2}{\pi }Pr.{\int }_{0}^{\infty }\frac{\mathfrak{I}\mathfrak{ }{\epsilon }^{\alpha \beta }\left(\omega {\prime }\right)}{{\omega {\prime }}^{2}-{\omega }^{2}} \omega {\prime }d\omega {\prime }$$

3

Where \(Pr.\) indicates the Cauchy principal value. Also, the effective factor in the inter-band optical transmissions between occupied (*ik*) and unoccupied electron states (*fk*) can be considered as the imaginary part of the complex dielectric function, the equation of which is:

$$\mathfrak{I}{\epsilon }^{\alpha \beta }\left(\omega \right)=\frac{4\pi {e}^{2}}{{m}^{2}{\omega }^{2}} \sum _{i,j}\int \frac{2{d}^{3}k}{{\left(2\pi \right)}^{3}}{\left|⟨ik|{P}_{\alpha }|fk⟩\right|}^{2}{f}_{i}^{k}(1-{f}_{f}^{k})\delta ({E}_{f}^{k}-{E}_{i}^{k}-\stackrel{-}{h}\omega )$$

4

The real and imaginary plots of the dielectric function related to the penta-graphene in the Z direction, without stress and under vertical stress conditions with 4, 8, and 12 percent to reduce buckling constant, are displayed in Figs. 4 and 5, respectively. As shown in Fig. 4, a sharp peak of transition occurred at about 6 eV in the non-stressed state, which decreases by increasing the stress up to 12%. Next sharp peaks are formed for modes affected by bucking for stresses of 8 and 12 percent, at higher energies in the range of 10 to 12 eV. Also According to Fig. 5, by applying stress in the imaginary part, in addition to reducing the peak of transition in the energy of about 10 eV, we see a rebound of transition peaks for stress states of 8 and 12 percent in energy of about 13 to14 eV. These indicate the matching of the optical and electronic behaviors of this semiconductor under these conditions.

Two important aspects for the exploration of optical properties are undoubtedly absorption and reflectivity. Figures 6 and 7 respectively, illustrate the 2D nanostructure of the penta-graphene absorption and reflectivity spectrums in the Z direction, which are affected by changes in buckling on this material by applying stresses with values of zero, 4, 8, and 12 percent. According to the figures, the absorption and reflectivity spectrums, from visible light energy to about 11 eV, are reduced by applying stresses up to 12%. This trend continues for a 4% stress state for higher optic energies up to 30 eV. But in energies between 13–15 eV for states of 8 and 12% stress, a rapid rise in the absorption and reflectivity amplitudes is observed by creating sharp peaks at 14 eV. These changes in optical behavior in these conditions are related to the electronic behaviors of this material, in the transformation of the indirect band into a direct band semiconductor, along with reducing the band gap.

At the end of this section, the energy loss spectrum (ELOSS) of penta-graphene has been examined under the conditions of this research. According to Fig. 8, the energy loss spectrum of this structure under compressive strain is relatively reduced by 8%. But at 12% buckling reduction, sudden and rapid changes occur in the sharp peak, which indicates the plasma frequency of the structure, which shifts from the energy of 21 to 22 electron volts to the energy of 15 electron volts.