2.1 Theoretical Models

The hexagonal system is the crystal structure of wurtzite ZnO, with the space group of P63mc. In the structure, each zinc ion is surrounded by four oxygen ions and also the three parameters of the unit cell are as follows: $a=b=3.249\mathrm{A}^{\mathrm{o}}$, $c=5.205\mathrm{A}^{\mathrm{o}}$ ^[6]. To make the experimental results consistent, the supercells $Zn_{1-x}V_{x}O$, $3\times 2\times 2$, $2\times 2\times 2$, and $2\times 2\times 1$ have been established corresponding to $4.1\% $, $6.2\% $, and $12.5\% $ doping, where V atoms can replace Zn atoms. Fig. 2 shows the proposed supercells.

Fig. 2. Designed Models for (a) $\mathrm{ZnO}$; (b)$~ \mathrm{Zn}_{0.9583}\mathrm{V}_{0.04166667}\mathrm{O}$; (c) $\mathrm{Zn}_{0.9375}\mathrm{V}_{0.0625}\mathrm{O}$; (d) $\mathrm{Zn}_{0.875}\mathrm{V}_{0.125}\mathrm{O}$.

2.2 Calculation Method

Taking the DFT into consideration and aiming at the plane-wave-pseudo potential (PWP) approach, the first-principles calculations have been carried out using the CASTEP code. The computations were based on DFT, as applied in the Materials Studio package's CASTEP software ^[23]. The fast calculation is the major feature of the applied approach because of its relatively high efficiency in which, in advances, approximating the shape of the orbitals is absent. The assumption is that the core of the ions is both inner electrons and nuclei; on the other hand, the valence electrons will interact and thus assure fast convergence of ion-electron potential. For electron-ion interaction investigation, the ultrasoft pseudopotential (USP) has been applied in addition to generalized gradient approximation (GGA) and the electron exchange interaction proposed by Perdew-Burke-Ernzerhof (GGA-PBE) ^[23,24]. The PWP method relies strongly on an assumption of governing almost all physical and chemical characteristics of solids and molecules by valence electrons. This attempt rationalizes the real atoms as pseudo atoms; wherein just the valence electrons are involved in the self-consistent calculation (i.e., neglecting the impact of core electrons in the calculation). Therefore, the cost of computational calculation reduces considerably ^[25]. The exchange and correlation energy possess impact; herein, it has been dealt with within the GGA ^[26]. The certain valence electronic configuration used in the current calculation is $\mathrm{Zn}\colon 3d^{10}4s^{2}$, $\mathrm{V}\colon 3d^{3}4s^{2}$, and $\mathrm{O}\colon 2s^{2}2p^{4},$for the materials in a reciprocal space form.

After the self-consistent process, the structure's total energy converges, the force exerted on the atoms is smaller than 0.03 eV/nm, and the stress deviation is less than 0.05 GPa. Just to take into consideration, the K points of the Brillouin zone are $5\times 5\times 4$ ($1\times 1\times 1$ unit cells); $3\times 2\times 3$, ($2\times 2\times 1$, $2\times 2\times 2$, and $3\times 2\times 2$, supercells). The first optimization was the crystal structure, followed by the electronic structure and total and partial densities of state. Afterward, the optical characteristics have been specified using the outcomes obtained from the structural parameters. The accurate description of the electronic structure of oxides doped with transition metal V and localized oxygen components is quantitatively performed from the GGA+U ^[27,28]. However, apparent limitations of DFT in GGA make the bandgap to be lower than the experimental value (3.37~eV) ^[1]. Furthermore, the energy calculation for the first Brillouin zone has been carried out by implementing the special k-point sampling techniques of the Monkhorst-Pack scheme ^[29]. The total energy of different k-point grids has been computed for all compounds to reach a satisfying mesh to ensure the accuracy of the whole results. The electronic structure was modified by GGA+U for all systems in this study, the energy of the 2p level of O and the 3d level of Zn take the values Ud, Zn =10 eV, Us, Zn = 0 eV, Up, Zn = 0, and Up, O = 7 eV ^[2], respectively.

3.1 Lattice Parameters Study

A common theoretical tool used to determine crystal structure is the geometry optimization procedure. Full geometrical optimizations on the atomic positions or cell parameters are carried out as presented in Table 1. In the theoretical calculations, only a single crystal structural data of the compound of interest have been taken into account. For example, the pure ZnO, ($a=b=3.2492\mathrm{A}^{\circ}$ and $c=5.2054\mathrm{A}^{\circ}$, $c/a=1.602$) found these lattice parameters are quite in agreement with the following experimental values; ($a=b=3.25\mathrm{A}^{\circ}$, $c=5.207\mathrm{A}^{\circ}$, $c/a=1.602$) with the variance of less than $2\% $ ^[20]. Despite this, the predicted lattice parameters and unit cell volumes were somewhat out compared with the experimental findings published in the literature (within 2 percent). On the other hand, satisfactory agreement with the existing theoretical [17, 27, 30, 31] findings has been attained. It's worth noting that the GGA approximation has a built-in tendency to overestimate lattice parameters. As long as the computations are as accurate as they need to be, these types of mistakes are acceptable. Another interesting feature of the data is the tiny amount of variance between the theoretical and experimental values. First-principles computations are accurate as a consequence. The volume and lattice parameters of the ZnO system in the a and c axes increase as the quantity of V$^{5+}$ ions in doping is increased. The doped system's lattice properties and volume are somewhat higher than those of the parent ZnO system. This is caused by the indigence to calculate the electronic structure properties in which the lattice parameters are impacted.

In calculating the BG of most systems, the limitations involved in the GGA approximation, make the BG calculations to be as not reached the level of reflecting the actual results. Another cause of the present results is the repulsive interactions of the extra electrons of the V$^{5+}$ ion that exists effectively. As a result of both factors' impact, the volume of the doped sample rises as the amount of V-doping increases. And also in Table 1. Different doping elements present various impacts on the changing rate of the lattice parameters a and c. and the ratio between them floats up and down.

3.2 Band Structure and Density of State Studies

The work conducted in this research specified the band structure along a high symmetry direction of the Brillouin zone (G-A-H-K-G-M-L-H) for pure ZnO and V-doped ZnO. When determining a material's characteristics, it is crucial to accurately determine its density of states (DOS) and its bandgap structure. In addition, the likelihood of electrons occupying certain energy ranges may be calculated from the shape of the electronic band (that is, energy bands). As a result, the bandgap refers to the unusable energy ranges. The electronic BG may be measured in insulators and semiconductors by comparing the valence band maximum (VBM) with the conduction band minimum (CBM). The electronic band structure computations will be useful in determining the probable transition of electrons from VBM to CBM. Intuitively, the valence band lies below the Fermi state $\left(E_{F}\right)$ at 0 eV, and the CB locates above the BG energy.

From Fig. 3, a direct G-to-G transition BG energy of 3.331 eV for super-cells including, $\mathrm{Zn}_{0.9583}V_{0.04166667}O$ and $\mathrm{Zn}_{0.9375}V_{0.0625}O$ and the band structure calculated along the symmetry direction of the Brillion zone (G-A-H-K-G-M-L-H) are recorded. Located at the G point, the valence and conduction bands have a straight $G-to-G$ transition. This suggests the existence of p-type semiconductors of $\mathrm{Zn}_{0.9583}V_{0.04166667}O$ and $\mathrm{Zn}_{0.9375}V_{0.0625}O$. It is principally accepted that the direct nature of the BG contains a transition of an electron from the valence band to the conduction band without phonons assistance which is mostly free of losing incident energy during transitions. The bottom 10 bands (occurring around -9 eV) are related to $\mathrm{Zn}\colon 3d$ levels. The next 6 bands from -5 eV to 0 eV correspond to $\mathrm{O}\colon 2p$ bonding states ^[33,34]. Based on the DFT approach obtained in the current work it is possible to design doped ZnO with direct BG. The BG energy was reduced from 3.331 eV for neat ZnO to 2.055 eV for ZnO dopped with 12.5 percent of V atom. Ultimately, the direction of the Brillouin zone was (G-A-H-K-G-M-L-H), and the direct $G-to-G$ transition of the supercell $\mathrm{Zn}_{0.875}V_{0.125}O,$ evidence formation of p-type semiconductors.

This model tells that the BG can be reduced to the energy of $2.055\mathrm{eV}$. Fig. 3(a) demonstrates that the value of the direct band in parent ZnO is lower than that of the experimental value (3.37 eV).In other words, the theoretical values of the BG energy are lower than the experimental values in all cases. This might be due to the lack or poor description of strong Coulomb correlation and exchange interaction between electrons. Notably, the prominent notice is that the calculated BG of the present and previous studies are in good agreement, as presented in Table 2.

The DOS and partial density of states (PDOS) of both parent $ZnO$, $\mathrm{Zn}_{0.9583}V_{0.04166667}O$, $\mathrm{Zn}_{0.9375}V_{0.0625}O$, and $\mathrm{Zn}_{0.875}V_{0.125}O$ supercells are plotted and exhibited in Fig. 4. Overall, introducing the V atoms into the lattice results in existing of primarily $\mathrm{O}\colon 2s^{2}2p^{4}$ state, $\mathrm{V}\colon 3\mathrm{d}^{2}$ in the top of valence, forming $V-O$ bonds besides $Zn-O$ bonds. In Fig. 4, PDOS was applied to analyze the electronic structure deeply providing a contribution of the energy bands in a specific atomic orbital.

In parent ZnO, both $\mathrm{Zn}\colon 3d^{10}4s^{2}$, and $\mathrm{O}\colon 2s^{2}2p^{4}$ states contribute mainly to the valance bands between ${-}$5.5 and 0 eV, and $\mathrm{Zn}\colon 3d^{10}4s^{2}$, $\mathrm{V}\colon 3d^{3}4s^{2}$, and $\mathrm{O}\colon 2s^{2}2p^{4}$ states are involved in the conduction bands ranging between 3.318 and 12.4 eV. For super-cells, there are additional contributions of $\mathrm{V}\colon 3d^{3}4s^{2}$ states. The localized $\mathrm{V}\colon 3d^{2}$ energy energy state substitutes dopant V ions for Zn, localizing within the Fermi level between -0.7 and 0.5 eV. The presence of $\mathrm{V}\colon 3d^{3}4s^{2}$ also also within, the energy gap from 1.6 to 3 eV reduces the energy gap. The existence of these states between 4 to 8~eV is evidenced. Consequently, the PDOS of all these states increases with increasing the quantity of V in the doping process.

Fig. 3. The E-K (Band Structure) diagram for (a) ZnO; (b) $\mathrm{Zn}_{0.9583}\mathrm{V}_{0.04166667}\mathrm{O}$; (c) $\mathrm{Zn}_{0.9375}\mathrm{V}_{0.0625}\mathrm{O}$; (d) $\mathrm{Zn}_{0.875}\mathrm{V}_{0.125}\mathrm{O}$.

Fig. 4. The PDOS and total DOS for (a) ZnO; (b) $\mathrm{Zn}_{0.9583}\mathrm{V}_{0.04166667}\mathrm{O}$; (c) $\mathrm{Zn}_{0.9375}\mathrm{V}_{0.0625}\mathrm{O}$; (d) $\mathrm{Zn}_{0.875}\mathrm{V}_{0.125}\mathrm{O}$.

3.3 Optical Properties

Crystals' band structure and optical qualities are reflected in the dielectric function, which may be considered the point at which electronic transition and electronic structure meet. The solid-state technique states that the complex dielectric function $\varepsilon \left(\omega \right)=\varepsilon _{1}\left(\omega \right)+i\varepsilon _{2}\left(\omega \right)$ can express the optical properties of semiconductor materials ^[36]. On the one hand, the imaginary part results from transitions between the CB and the VB. Calculating momentum matrix components for filled and non-filled levels and Kramers-Kronig dispersion relations theoretically yields both the real $\varepsilon _{1}\left(\omega \right)$ and imaginary $i\varepsilon _{2}\left(\omega \right)$ portions of dielectric functions, respectively ^[36,37]. According to quantum mechanics, the system's photon-electrons interact through time-dependent disturbances in the system's ground electronic state. The absorption or emission of photons triggers transitions between occupied and unoccupied states in a quantum system. In terms of DOS, excitation spectra may be thought of as a conduction-valence band DOS ^[38]. Furthermore, from $\varepsilon _{1}\left(\omega \right)$ and $i\varepsilon _{2}\left(\omega \right)$, extraction of other optical properties is obtainable. The optical properties from the density functional theory are achievable, for instance, the dielectric function, absorption coefficient, and refractive index ^[12] equation (1,2,3,4,5) ^{[12, 31, 38-41]}.

where $~ \mathrm{\hslash }\omega $ is a photon energy, $k$ is the inverted lattice, respectively. $\omega ~ $ and $\rho _{0}$ are frequency and the density of the medium, respectively. The$\delta \left(E_{C}^{k}-E_{V}^{k}-\mathrm{\hslash }\omega \right)$ represents the delta function imposes energy conservation in the third step corresponding to the electron's escape through the crystalline surface. $\left| M_{CV}\left(k\right)\right| ^{2}$ refers to the momentum transition matrix element between initial and final states. $E_{V}^{k,}$ and $E_{C}^{k}$ are the VBM and CBM energy, respectively.

Fig. 5(a) Show that pure ZnO and V-doped systems ($\mathrm{Zn}_{0.9583}V_{0.04166667}O$; $\mathrm{Zn}_{0.9375}V_{0.0625}O$; $\mathrm{Zn}_{0.875}V_{0.125}O$.) have static dielectric constants of around 2.302, 2.553, 2.572, and 2.657, respectively. The band gap is inversely proportional to the $\varepsilon _{1}\left(0\right)$. As a result, the decrease in the band gap can be attributed to the rise in the static dielectric constant, which is consistent with the results of band structure calculations. The band illustrates that V-doped ZnO has a metallic characteristic, yet the $\varepsilon _{1}\left(\omega \right)$ shows that it is dielectric. $\varepsilon _{1}\left(\omega \right)$ is fluctuated between 0 to 12 eV after that it decreases rapidly to -0.3 at 17.4~eV. Evidence of poor light transmission and energy losses within the media with significant reflection can only be obtained using the negative component of the dielectric constant $\varepsilon _{1}\left(\omega \right)$.

The imaginary part of electronic transitions between occupied and unoccupied states accounts for the peaks of $\varepsilon _{2}\left(\omega \right)$. In Fig. 5(b), we can see that there are three distinct peaks of the $\varepsilon _{2}\left(\omega \right)$ for pure and V-doped ZnO. ZnO's DOS and energy band structure interpretation suggests that the peak at 9.5 eV is mostly the optical response to the transition between the $\mathrm{O}\colon 2p$ (higher valence band) and $\mathrm{Zn}\colon 3d$ (lower conduction band) states. The transitions in the valence band, which are between $\mathrm{Zn}\colon 3d$ and $\mathrm{O}\colon 2p$ states, give rise to the peak at 13.17 eV. Transitions between$~ \mathrm{~ Zn}\colon 4d$ (lower valence band) and $\mathrm{O}\colon 2s$ (lower conduction band) states produce a peak of about 16 eV. The electronic transition between the $V\colon 3d$ states in the higher valence band and the $Zn\colon 3d$ levels in the lowest conduction band produces the first peak of about 5 eV near the energy gap. At the high energy range, the remaining three primary peaks stay in their original locations, indicating that there has been no transition.

It is notable, however, that just the first primary peak has altered its location, indicating that ZnO's low-energy area has been effectively doped. However, even if the final peak is due to a mixed transition, it is seen that the loaded sample not only moves absorption peaks to lower energies but also reduces the sharpness of their appearance. Fig. 5 indicates that the BG has narrowed dramatically, resulting in a reduced absorption edge. V-doped ZnO, on the other hand, results in a decrease in energy BG. The red arrow intersection on photon energy (Fig. 5(b)) gives the energy BG (3.32). It is close enough to that estimated (3.331) from the E-K diagram (Fig. 3(a)) for pure ZnO.

The optical absorption spectrum is the vital tool that is mainly focused on. The absorption coefficients of ZnO correspond to various quantities of V are presented in (Fig. 6). The absorption edge value is recorded as 0.68~eV, (in agreement with the paper ^[42] for pure ZnO, and there are three distinct absorption peaks at 9.89 eV, 13.5~eV, and 16.12 eV (10.07 eV, 16.87 eV, and 20.69~eV) respectively. The last two peaks are created by the electronic transition between the O-2p state and Zn-4s state. A red shift is seen in the absorption of doped ZnO with V compared to parent ZnO at the lower energy state, originating from the BG reduction. The parent ZnO is not a strong absorber in the IR region; however, it is a good absorber in the UV region. It is well-known that the energy of visible light lies between 1.62 and 3.11 eV. From (Fig. 6) we can see that V-doped ZnO has an intense absorption in the ultraviolet region. The right side of (Fig. 6) shows that the pure and V-doped ZnO has a weak absorption in the visible and infrared region. The intense absorption in the visible and near-ultraviolet region is because of the electronic interband transitions in the V-3d states. The electronic intraband transitions in V-3d and Zn-4s states in the conduction band can cause a weak absorption in the infrared region (lower photon energy) ^[36]. The system has intense absorption at (13.5~eV-16.59 eV) for pure and V-doped ZnO. The absorption peaks indicate that V-doped ZnO could be a good choice for lasers, detectors, or diodes in the infrared and vacuum ultraviolet regions.

The refractive index $n\left(\omega \right)$ is an important optical characteristic that allows material transparency to be measured. The calculated refractive index $n\left(\omega \right)$ and extinction coefficient $k\left(\omega \right)$ of undoped, V-loaded ZnO are shown in (Fig. 7). In (Fig. 7(a)) the $n\left(0\right)$ of parent ZnO is around 1.51 and becoming 1.59, 1.60, and 1.62 for ($\mathrm{~ Zn}_{0.9583}V_{0.04166667}O$; $\mathrm{Zn}_{0.9375}V_{0.0625}O$;$\mathrm{Zn}_{0.875}V_{0.125}O$), respectively after doping. The refractive index in the visible region increases with increasing V concentration. The refractive indices are smaller in the deep ultraviolet band (8-11 eV). Because electrons in the deep level orbital are sheltered by electrons in the shallow level during the transition, electrons in the deep level orbital have a harder time transitioning in the orbit, resulting in a lower refractive index. when it shifts to the higher energy level the refractive index becomes 0.65 at 18.5 eV. As demonstrated in (Fig. 5(b)) the computed imaginary part $\varepsilon _{2}\left(\omega \right)$ of the dielectric constant of undoped and V-doped ZnO are similar. The extinction coefficient $k\left(\omega \right)$ for $\mathrm{Zn}_{0.875}V_{0.125}O$ at the first increase slowly with rising the photon energy but parent ZnO, $\mathrm{Zn}_{0.9375}V_{0.0625}O$ and $\mathrm{Zn}_{0.9583}V_{0.04166667}O$ remains unchanged. We can use the $k\left(\omega \right)$ to determine the energy band gap of the systems.

Fig. 5. (a) Real part of Dielectric Function $\varepsilon _{1}\left(\omega \right)$; (b) Imaginary part of Dielectric Function.

Fig. 6. Absorption coefficient of pure and V-doped ZnO.

Fig. 7. Refractive index (a) n(${\upomega}$) and (b) extinction coefficient k(${\upomega}$), for pure and v-doped ZnO with different contents.