Stability, phonon calculations, electronic structure, and optical properties of a VO2(M) nanostructure: A comprehensive density functional theory study

This work aimed to precisely evaluate the physical properties of vanadium dioxide (M), particularly the optical characteristics. We employed different exchange-correlation functionals to determine the phase stability, band gap properties, and optical characteristics of an experimentally recognized monoclinic VO2(M) polymorph. The calculations not only correctly interpreted the VO2(M) origin but also predicted other optical properties including the extinction coefficient ( k ) and refractive index ( n ), which have not been reported in experimental measurements. Phonon dispersion calculations confirmed the presence of negative frequencies for acoustic modes in the phononic curves. When the HSE functional correctly reproduced the experimental band gap, here for the first time, our calculations based on PBE and PBEsol yielded non-zero electronic bandgaps of 0.23 and 0.15 eV for bulk VO2(M). Our predictions showed that semi-local functionals can adequately predict the semiconductor properties of VO2(M) and performed better than all previously reported theoretical works on nulled band gaps. In addition to the better prediction of the peak position in the absorption spectra with HSE hybrid functional, this method also reasonably well described the static dielectric constant of 7.54, showing an excellent match to the experimental values. In general, the results of this study reveal that hybrid functionals yield superior outcomes compared to semi-local functionals for optical properties of a VO2(M) polymorph. Our results suggest that the PBEsol + HSE approach allows the efficient characterization of smart materials for electronic and optoelectronic applications.


Introduction
Strongly correlated vanadium dioxide (VO 2 ) materials exhibit a diversity of polymorphs, leading to outstanding electric, electronic, magnetic, and optical properties (Ruzmetov et al., 2008;Popuri et al., 2015;Huang et al., 2016;Hattori et al., 2020). This type of material has attracted interest due to its reversible metal-insulator transition (MIT) at a phase transition temperature of~340 K. Through this phase transition, the VO 2 structure can transit between OPEN ACCESS EDITED BY Simone Taioli, European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECTp), Italy an insulating (low temperature) state by a monoclinic M-phase with alternative V−V distances and a metallic (high temperature) state with a tetragonal rutile phase (R) with uniform V−V distances (Goodenough, 1971;Bai et al., 2015;Zylbersztejn and Mott, 1975). The MIT near room temperature has attracted interest related to controversy regarding the transitional mechanism and the potential technological applications such as electronic and photonic operations (He et al., 2020;Pendse et al., 2020), sensors (Huang et al., 2021;Xu et al., 2021) smart windows (Kang et al., 2011;Gao et al., 2012), and Mott field effect transistors (FETs) (Sasaki et al., 2015).
The monoclinic M phase (P2 1 /c (D 5 2h )) is one of the most debated smart materials. Recent studies suggest that this phase should be considered a Peierls-Mott insulator (Babulanam et al., 1987) since both electron-electron correlations and V-V ions dimerization contributed to the opening of an insulating gap. The literature has reported semiconducting experimental bandgaps of 0.6-0.7 eV (Verleur et al., 1968;Tanaka, 2003;Haverkort et al., 2005;Pandurang et al., 2017) and optical absorption in the infrared-visible region (Eaton et al., 2018) for this polymorph.
Traditional density functional theory (DFT) calculations have predicted different outcomes of the physical features of VO 2 (M) using different codes. For instance, whereas several exchangecorrelation (XC) functionals have wrongly proposed a ferromagnetic property for this phase, these functionals also failed to reproduce the semiconducting bandgap of 0.6-0.7 eV or the optical absorption features of monoclinic VO 2 (M) in the infrared region (Grau-Crespo et al., 2012;Yuan et al., 2012;Ganesh et al., 2020). In particular, the LDA, GGA, and sometimes the meta-GGA functionals using PBE or PBEsol XC functionals have incorrectly described the non-metallic behavior of VO 2 (M) by predicting a conduction band below the Fermi level (Grau-Crespo et al., 2012;Yuan et al., 2012;Ganesh et al., 2020;Kylänpää et al., 2017). The failure to open the bandgap of this structure by GGA and LDA does not mean that it is a strongly correlated system; rather, this occurs because conventional theoretical approaches cannot correctly predict the experimentally observed properties of this system. The hybrid Heyd-Scuseria-Ernzerhof (HSE) and PBE0 functionals used in other studies (Grau-Crespo et al., 2012;Tran and Blaha., 2017;Xu et al., 2017) also indicated a strong discrepancy with the experimental characteristics of this polymorph. However, several studies have aimed to overcome these issues. For instance, Sakuma et al. (2008) predicted the electronic property of the monoclinic VO 2 (M) structure based on G 0 W 0 calculations. They reported that this approach did not display the experimental satellite features of conduction and valence bands above and below the Fermi level; however, in their computational study, Gatti et al., 2015 reported a better approach of self-consistent GW (COHSEX calculations), which successfully opened the band gap of 0.78 eV, quite close to the experimental value (0.6-0.7 eV) in the electronic properties of the monoclinic phase of VO 2 . Another approach is the dynamical mean field theory (DMFT), which provides a view of the role of electron correlation in the physical properties of the VO 2 (M) phase. Brito et al. (2016) suggested a distinct mechanism for the gap opening in the monoclinic phase utilizing different computational approaches. However, the results have led to some controversy regarding the origin of the VO 2 (M) polymorph.
To address these controversies, we modeled a VO 2 (M) polymorph to theoretically investigate the geometry, energy stability, and electronic and optical properties by first-principle calculations. We used the PBEsol XC functional to elucidate whether it can appropriately describe the correct geometry of the M phase. We also computed phonon softening using different LDA and GGA methodologies to provide information about the dynamical stability of this polymorph. We also combined the Hubbard U value in part of our calculations to explore the electronic properties and demonstrate whether the inclusion of the repulsion parameter has advantages over semi-local functionals. Similar calculations were performed with GGA/PBE to characterize the properties of the VO 2 (M) for comparison with those resulting from the PBEsol and PBEsol + U approximations. Additional HSE and PBE0 hybrid functionals were performed to estimate the reliability of the PBE, PBEsol, and PBEsol + U bandgaps and the optical properties of the dielectric function of monoclinic VO 2 (M). Finally, we compared the predicted results to those feasible to experimentally measure to demonstrate the theoretical accuracy of the semi-local GGA (PBE and PBEsol), PBEsol + U, hybrid HSE, and PBE0 functionals.

Computational methods
All calculations were carried out using Quantum ESPRESSO (QE) (Giannozzi et al., 2009) and QuantumATK (QATK) (Smidstrup et al., 2019) ab initio simulation packages. The polymorph was fully relaxed using the generalized gradient approximations (GGA) and Perdew-Burke-Ernzerhof for solids (PBEsol) XC functional. Compared to the PBE XC functional, the PBEsol XC functional could be important since it handles V-V medium-ranged bonding in VO 2 (M) associated with the smallest mean error relative to the experimental values. Numerical studies have suggested that PBEsol can accurately predict the bulk exchange energies of materials within the pseudopotential approximation. Projector augmented wave (PAW) (Blöchl et al., 1994;Kresse and Hafner, 1994) pseudopotential (PP) types were chosen to treat the V-3d and O-2p as valence orbitals. The valence orbitals were expanded in a planewave (PW) basis set with a kinetic energy cut-off of 70 Ry. The Brillouin-zone (BZ) (Brillouin, 1930) integrations were limited to the gamma point mesh, and the lattice constant and internal coordinates were fully optimized until Hellmann-Feynman forces become <0.01 Ry (Vanderbilt, 1990). This set of parameters assured that the total energies converged to 1 × 10 −5 eV/Å. The phonon frequencies were calculated using the local density approximation/ Perdew-Wang (LDA/PW), GGA/PBE, and GGA/PBEsol functionals according to the density functional perturbation theory (DFPT) (Baroni et al., 2001) implemented in the QE package.
The electronic properties of VO 2 (M) were computed using the semi-local PBE, PBEsol, PBEsol + U, and hybrid HSE and PBE0 functionals with post-processing calculations to allow exhaustive comparisons between functionals. As the strong onsite Coulomb repulsion for V-3d electrons is often adjusted to match existing experimental quantities, the Hubbard parameter Frontiers in Materials frontiersin.org 02 U was added to the PBEsol functional. The DFT + U method has been shown to accurately describe the electronic structure and strong correlation of VO 2 (Zayed et al., 2020). As the results of previous studies have underscored that the outcomes are substantially dependent on the magnitude of U, we tested different Hubbard U values for V-3d and O-2p. We set V-3d = 5.20 eV and O-2p = 0.95 eV since these parameters have shown very good agreement with the experimental results, as reported by Mohebbi et al. (2022).
The hybrid XC functionals are constructed by a fraction of nonlocal exact exchange instead of a fraction of local/semi-local exchange as Eq. 1: ( 1 ) The non-local exact exchange is often computed according to the Hartree-Fock (HF) method. The type of DFT local/semi-local functional and the fraction of non-local exact exchange substituted in (α) vary depending on the hybrid functional type.
The hybrid functional HSE (Heyd and Scuseria, 2004;Heyd et al., 2005), using an error-function-screened Coulomb potential, has the following form: where α is a mixing parameter and ω is an adjustable parameter controlling the short range of the interaction.  (Krukau et al., 2006).
The hybrid functional PBE0 has the following formula: The PBE0 hybrid XC functional degenerates ω to zero (Perdew et al., 1996;Yang et al., 2018).
The optical properties of the VO 2 (M) nanostructure are discussed based on random-phase approximation (RPA) independent-particle (IP) to the dielectric constant using the KS states as a useful tool to calculate the screening and optical response of materials (Das, 1975;Wolterink et al., 2002;De Conti et al., 2001;Paier et al., 2008).
The imaginary part (ε 2 (ω)) of the dielectric coefficient can be determined from Eq. 4 (Dinh et al., 1996;Mortazavi et al., 2017;Mortazavi et al., 2019): where Ω and q denote the volume of the unit cell and Bloch vector of the incident wave, respectively, w k is the weight of the k-point, and c and v correspond to the conduction and the valence bands, respectively. The vectors e α refer to the unit vectors of the three Cartesian directions, and u ck is related to the cell periodic part of the orbitals at the k-point mesh. Moreover, the real part of the dielectric constant can be obtained using Eq. 5: where P is the principal value and η denotes the complex shift. The absorption coefficient (α(ω)) of this structure is determined according to Eq. 6 (Kolwas and Derkachova, 2020): where c is the speed of light and n (ω) refers to the refraction index. The real part of the optical conductivity Re[σ(ω)] of the VO 2 (M) structure can be determined as follows: ].
( 7 ) We next assessed the reflectivity of the VO 2 (M) structure using Eq. 8: Finally, we evaluated the refractive index (n) and extinction coefficient (k) of the VO 2 (M) based on Eqs 9, 10: Material visualization was performed using XCrySDen graphical software (Kokalj, 1999).

Analysis of geometry and stability
The unit cell of the VO 2 (M) polymorph consists of a unit cell of 24 atoms, including eight vanadium atoms. The lattice parameters were calculated as follows: a = 5.1836 Å, b = 5.0504 Å, and c = 9.0187 Å. Figure 1 shows the modeled possible planes along the xy, xz, and yz directions. The VO 2 (M) nanostructure comprised two layers including the 3D frameworks of VO 6 octahedra ( Figure 2). These octahedra packings of VO 6 are only linked by oxygen atoms (plane xy). The octahedra in the VO 2 (M) phase are orthorhombically distorted, making it possible to differentiate the apical and equatorial V-O bonds resulting in alternate long and short V-V bonds (plane yc). Plane xc indicates how V atoms in the octahedra frameworks share their edges with oxygen atoms along the zigzag direction. We investigated the atomic positions, cell parameters, and V-V and V-O bonds in this bulk structure. In our DFT calculations, the VO 2 (M) nanostructure showed the longest/shortest V-V bond distances of 3.04 and 2.76 Å, respectively. Moreover, the V-O bond lengths in the octahedra Frontiers in Materials frontiersin.org 03 VO 6 were 2.10, 1.85, 1.83, 1.81, 1.97, and 2.18 Å. These parameters were in excellent agreement with available experimental observations (Goodenough, 1971;Zylbersztejn and Mott, 1975;Zhang, 2016) and theoretical studies Chen et al., 2021). The calculated formation energy of the VO 2 (M) structure in this study by PBEsol (8.00 eV) was in fair agreement with other theoretical values using PBE + U approximation (7.14 eV), HSE functional (8.17 eV (Abdellaoui et al., 2016) and 7.18 eV ), and the experimental value of 7.38 eV (Melnik et al., 2012;Lee et al., 2016), indicating that our method for semi-local XC functional calculation is reasonable.
The phonon dispersion of a VO 2 (M) polymorph was investigated using the different methodologies of LDA and GGA and the PW, PBE, and PBEsol XC functionals to provide useful information about the dynamic stability of this polymorph. All phonon calculations were performed after full relaxation of the atomic positions and the lattice volume. The phonon bands were computed for the VO 2 (M) nanostructure along with the highsymmetry points of G-M-K-G in the first BZ (see Figure 3). Inspection of the dispersion plots in Figure 3 shows that the dynamical matrix has negative frequencies for the three approximations, which indicates the dynamic instability of this  polymorph. The most noteworthy was the difference in phonon dispersion curves between different XC functionals. Figure 2 shows that in the graph obtained by LDA/PW functional phonon, softening of the instabilities occurs around −177 and −52 cm -1 at the G and M points in reciprocal space, while in GGA/PBE, the negative frequencies were suggested to occur around −62 and −86 cm -1 at the G and K points, respectively. More interestingly, in the case of GGA/PBEsol, the imaginary frequencies at the G point were limited to one acoustic mode, with no additional negative frequencies like those in the LDA/ PW (−177 cm -1 ) and GGA/PBE (−62 cm -1 ) functionals. Since there are no first-principle DFT calculations of phonon dispersions of VO 2 (M) for comparison, the results of previous experiments (Li et al., 2017;Wu et al., 2020) confirm the phase instability of this polymorph. According to these experimental observations, among different polymorphs of the VO 2 family, the VO 2 (M) and VO 2 (B) phases are metastable in their bulk structures; hence, the preparation of phase-pure and high-quality materials has been a major challenge in VO 2 research. These two monoclinic polymorphs can undergo an irreversible phase change into VO 2 (R) (rutile phase of VO 2 ) upon heating, indicating the thermodynamic instability of these non-equilibrium materials (Chernova et al., 2009;Sims et al., 2017). One solution to improve the stability of these materials could be the deposition of VO 2 (M) on perovskite substrates like SiTiO 3 (Sims et al., 2017) or sapphire (Jian et al., 2016;Lee et al., 2016).

Band gap properties and state densities
Next, we examined the electronic properties of the VO 2 (M) structure. Table 1 summarizes the band gaps for the VO 2 (M) polymorph predicted by the PBE, PBEsol, PBEsol + U, HSE, and PBE0 functionals. Our calculation with PBE showed a bandgap of 0.23 eV for bulk VO 2 (M), close to the value predicted using PBEsol (0.15 eV) by upshifting of the Fermi level to the higher energies. Thus, the inclusion of the Hubbard U value produced a larger band gap (0.96 eV), while the position of the Fermi level did not change significantly. The electronic structures were also calculated using the QE code based on the hybrid HSE and PBE0 functionals. When the band gap of the semiconducting VO 2 (M) computed using HSE showed excellent agreement with photoemission experiments (E g~0 .6-0.7 eV) (Verleur et al., 1968;Tanaka, 2003;Haverkort et al., 2005;Pandurang et al., 2017) by 0.60 eV, the hybrid  PBE0 approximation yielded an unexpectedly high band gap of 1.39 eV. The HSE calculation produced an accurate band gap, in contrast with the other DFT approximations.
The electronic band diagram of monoclinic VO 2 (M) showed that each V-3d is surrounded by oxygen's octahedron, resulting in the splitting of the d level to triply degenerate the t 2 g and doubly degenerate the e σ g states. The former further splits into e π g (π; π*) and a 1g (d ⫽ ) orbitals. In this polymorph, the dimerization of V atoms causes splitting of the a 1g (d ⫽ ) bands into bonding/antibonding states, energetical upshift of π* bands, and a charge transfer from the e π g orbitals, leading to the separation of the bonding d ⫽ and e π g subbands at the Fermi level with a band gap of approximately 0.7 eV. Figure 4 shows the calculation of the predicted band structure and partial density of states (PDOS) of VO 2 (M) using the PBEsol + U functional. The band structure was considered along with the directions of the high-symmetry points of G-M-K-G in the first BZ ( Figure 4A). According to the PDOS calculation of this polymorph, the main contribution of total DOS belonged to the V-3d orbitals associated with the lower contribution of the O-2p states ( Figure 4B). The DFT + U system shows a semiconducting characteristic due to the a 1g (d ⫽ ) bands splitting into bonding/ antibonding states and the π* states experiencing energetical upshift, leading to strong hybridization with the O-2p states.
While all previous theoretical studies have reported a metallic property for VO 2 (M), with a band gap of 0.0 eV (Tran and Blaha, 2017) using HLE16 (a local Kohn-Sham gradient approximation), 0.07 eV (Zheng and Wagner, 2015) using quantum Monte Carlo (QMC), and 0.06 eV (Varadwaj and Miyake, 2022) using first-principle PBE calculations, our PBE and PBEsol calculations using the QE code revealed the semiconductor properties of this polymorph. Similarly, a DFT study of the geometrical and bandgap properties of VO 2 (M) using another semi-local functional in the WIEN2k (Grau-Crespo et al., 2012) package showed the zero band gap of this structure, demonstrating that the Hubbard U value is essentially required to reproduce the experimental band gap. In contrast, in a theoretical study, Tran and Blaha (2017), reported that all semi-local functionals implemented in various first-principle simulation codes all failed to provide a bandgap for monoclinic VO 2 , with high band gap values of 1.03 and 1.35 eV obtained by HSE and B3PW91 approximations, respectively. In their recent theoretical study, Varadwaj and Miyake (2022) implemented SCAN, SCAN-rVV10, and G 0 W 0 functionals in the VASP package to study the bandgap property of the monoclinic M phase of VO 2 . Their results showed that VO 2 (M) had semiconductor properties, with band gaps of 0.58, 0.57, and 0.68 eV, respectively. Meanwhile, Grau-Crespo et al. (2012) reported an overestimated band gap value of 0.98 eV from HSE hybrid functional using the same code as that used by Varadwaj and Miyake (2022). The comparison of our results to those of other methodologies implemented in different quantum mechanics packages suggests that the PBE and PBEsol functionals employing PAW PPs in QE code can polarize the Fermi level and open the band gap, resulting in a non-zero band gap and the semiconductor property of this polymorph. Furthermore, the HSE code in QE showed an excellent ability to reproduce the experimental band gap of this polymorph, while PBE0 showed some unacceptable results. Overall, our results reveal that the opening of the band gap in the band structure of VO 2 (M) may first have a strong dependency on the optimized geometry (in our study, we used PBEsol XC functional) and then is more correlated with the functional used to investigate the postprocessing electronic calculations.

Optical features
In this section, the optical properties of VO 2 (M) were calculated, including the imaginary and real parts of the dielectric constant, the absorption coefficient [α(ω)], the optical conductivity Re[σ(ω)], the reflectivity spectra (r), the extinction coefficient (k), and the refractive index (n) which were computed using Eqs 4-9. Figure 5 compares the real and imaginary parts of the dielectric constant of the VO 2 (M) polymorph predicted by the PBE, PBEsol, PBEsol + U, and HSE functionals. For the real part of the dielectric function, the ε 1 (ω) for the VO 2 (M) geometry showed predicted positive values of 172. 57, 77.22, 82.44, and 7.54 for the PBE, PBEsol, PBEsol + U, and HSE functionals, respectively. According to these results, when we apply the PBEsol functional including/excluding the Hubbard U value, the similarity in the ε 1 (ω) spectra between PBEsol and PBEsol + U indicates a nearly unchanged dielectric constant value. The value of the dielectric function approximated by the HSE functional (7.54) was very similar to that for the experimental spectra [ε 1 (ω) =~7.80] in the study by Verleur et al. (1968). More specifically, the ε 1 (ω) spectrum obtained in their study had a reported maximum value of 12.00 for a static dielectric constant for this material, indicating that other functionals are needed to predict the far values of the dielectric constant.
As shown in Figure 5, the first main peak of imaginary of the dielectric constant (ε 2 (ω)) showed strong absorption in the infrared range in the energy range of 0-0.4 eV for the VO 2 (M) nanostructure predicted by the PBE, PBEsol, and PBEsol + U functionals. However, the situation changed remarkably for HSE, which showed a significant low-intensity adsorption peak in the infrared-visible region with a frequency range of 0.15-1.95 eV. Figure 5 also demonstrates that the simulated absorption peaks and shape of the ε 2 (ω) spectrum calculated by HSE functional qualitatively/ quantitatively showed excellent consistency with the experimental results (Verleur et al., 1968), especially the optical band gap. Figure 6 shows the absorption spectra for VO 2 (M) calculated using the PBE, PBEsol, PBEsol + U, and HSE functionals. The maximum adsorption spectra approximated by four functionals were 6.1 × 10 5 cm -1 , 4.0 × 10 5 cm -1 , 4.8 × 10 5 cm -1 , and 3.4 × 10 5 cm -1 for PBE, PBEsol, PBEsol + U, and HSE, respectively. Based on this figure and the range of photon energy (eV), the prominent main absorption peaks for PBE were located at 0.35 and 1.10 eV, while the corresponding values predicted by PBEsol were 0.25 and 1.05 eV. The outcomes also indicated that the main peaks of absorption spectra of VO 2 (M) were situated at 0.35, 0.70, and 1.05 eV based on PBEsol + U, while HSE showed peaks around 0.85, 1.45, 1.8, and 2.2 eV, respectively. Although several peaks were related to optical adsorption data on monoclinic VO 2 (M) at photon energies <2.5 eV, if a band scheme is chosen as the basis for interpretation, it can only be concluded at present that the absorption peaks near 1.0 (0.85 eV predicted by the HSE approximation) can interpret the electronic band gap for this polymorph. Moreover, our results revealed that PBE/HSE showed intense/dropped adsorption peaks for this structure.

FIGURE 5
Real and imaginary parts of the dielectric constants of the VO 2 (M) polymorph as a function of the energy (eV), predicted by the PBE, PBEsol, PBEsol + U, and HSE functionals.
Frontiers in Materials frontiersin.org 07 Figure 6 shows that by employing the functional from PBE to HSE, the adsorption spectrum upshifted to higher energies from the far-infrared to the near-infrared ranges of light, while the corresponding adsorption coefficient decreased from 6.1 × 10 5 cm -1 to 3.4 × 10 5 cm -1 (Figure 6, adsorption spectra predicted by PBE and HSE), respectively. Overall, the comparison of the four functionals showed that the absorption coefficient was more strengthened by semi-local PBE compared to hybrid functionals, resulting in low reflectivity, as we discuss later. These results indicate that the optical absorption peaks predicted using computationally

FIGURE 6
Optical absorption spectra of the VO 2 (M) polymorph, as a function of energy (eV), predicted by the PBE, PBEsol, PBEsol + U, and HSE functionals.

FIGURE 7
Optical conductivity of the VO 2 (M) structure as a function of the energy (eV), predicted by the PBE, PBEsol, PBEsol + U, and HSE functionals.

FIGURE 8
Reflectivity spectrum of the VO 2 (M) polymorph as a function of the energy (eV), predicted by semi-local PBE, PBEsol, PBEsol + U, and hybrid HSE functionals.

FIGURE 9
Extinction coefficient of the VO 2 (M) polymorph as a function of the energy (eV), predicted by the PBE, PBEsol, PBEsol + U, and HSE functionals.

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frontiersin.org 09 expensive HSE are more accurate than those of semi-local methods, particularly when the optical spectrum is compared to the experimental observations. The HSE hybrid functional slightly overestimated the lower energy transition 1.3 eV (experimental value) at 1.45 eV (curve in Figure 6), whereas this functional accurately predicted the first transition of 0.85 eV (the same value as the experimental value).
Next, we assessed the optical conductivity spectra of the VO 2 (M) polymorph based on the PBE, PBEsol PBEsol + U, and HSE functionals ( Figure 7). As expected, the optical conductivities began with a high intensity of 5710 A. V −1 . cm -1 for the PBE calculations, while the intensity of peaks dropped in PBEsol + U and PBEsol (2938 A. V −1 . cm -1 and 2620 A. V −1 . cm -1 ), respectively. The results in Figure 7 indicate that the optical conductivity of 1056 A. V −1 . cm -1 was predicted by the HSE functional. The VO 2 (M) polymorph showed strong conductivity in the infrared range in the non-hybrid calculations, while the HSE calculations indicated intense optical conductivity between visible-infrared regions. The results of our HSE calculations correctly verified the experimental optical observations reported by Cai et al., 2016. According to this study, the VO 2 (M) structure showed its first optical conductivity peak in the energy range of 0.3-1.5 eV, with no peak observed below 0.3 eV. Figure 8 shows the reflectivity spectra of the VO 2 (M) nanostructure computed using semi-local and hybrid functionals. The percentages were 73%, 63%, 64%, and 22%, respectively, for the PBE, PBEsol, PBEsol + U, and HSE functionals. Experimentally, Verleur et al. (1968) reported a reflectivity of around 10% for the monoclinic VO 2 monoclinic lattice. Although the calculated reflectivity percentages for the VO 2 (M) nanostructure in this study by semi-local functionals excluding and including the U parameter are higher than the experimental value (Verleur et al., 1968), the HSE functional provided a more accurate result compared to previous methodologies. For instance, the theoretical works of Chen et al. (2017 andChen et al. (2021) reported calculated reflectivity values of this material of 32% and 35% using the same PBE + U functional. Another study by Gatti et al. (2015) based on the Bethe-Salpeter equation (BSE) and G 0 W 0 approximations reported reflectivity values of 30% and 28%, respectively.
Finally, we calculated the other optical features of the VO 2 (M) polymorph, including the extinction coefficient (k) and refractive index (n) (Figures 9, 10), based on the semilocal and hybrid functionals. As shown in Figure 9, the maximum values of the extinction coefficients were about 5.43/3.67 for the energy values of 1.05 and 1.00 eV when the systems were predicted using the PBE and PBEsol functionals, respectively. The corresponding computed extinction coefficient value for PBEsol + U was 3.56 for 1.00 eV, while the value was 1.75 for the HSE functional in the high energy range of 1.85 eV. Figure 10 shows that the refractive index (n) of this nanostructure is dependent on the DFT functional. From Figure 10, the static refractive indices are 13.13, 8.78, and 9.08 by PBE, PBEsol, and PBEsol + U, respectively, whereas the value for the HSE calculation was 2.74, significantly lower than those for the semi-local functionals.

FIGURE 10
Refractive index of the VO 2 (M) nanostructure as a function of energy (eV), as predicted by the PBE, PBEsol, PBEsol + U, and HSE functionals.
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Conclusion
In this study, we optimized the geometry of a VO 2 (M) polymorph using the PBEsol XC functional. Our results indicated that this approximation accurately reproduced the experimental properties of the lattice constants and structural parameters. The phonon dispersion calculations using LDA and GGA methods predicted some imaginary eigenvalues and instability of this nanostructure, revealing that the lowtemperature monoclinic VO 2 (M) polymorph lacked dynamical stability, consistent with some experimental observations. We also investigated the electronic band gap calculations for VO 2 (M) by applying the PBE, PBEsol, PBEsol + U, HSE, and PBE0 functionals. For the first time, both the PBE and PBEsol functionals confirmed the semiconductor property of the VO 2 (M) polymorph, whereas the hybrid HSE functional generated the experimental bandgap of this phase (0.60 eV). The results based on PBE0 revealed the overestimated prediction of 1.39 eV for the electronic band gap of the VO 2 (M) structure. In contrast, the HSE functional accurately approximated the optical features of this polymorph. Although the magnitudes differed between the static dielectric constants obtained from the four functionals, the real part of the dielectric constant, with a calculated value of 7.54, showed very good agreement with the experimental value of 7.80. Furthermore, the absorption peaks centered at 0.85 and 1.45 indicated good consistency with the experimental observations of prominent absorption peaks centered near the photon energies of 0.85 and 1.3, respectively.
The excellent electronic and optical properties of the VO 2 (M) polymorph suggest its promising advantages and desirable prospects in energy engineering applications. Windows' materials like VO 2 polymorphs are well-known as among the most energy-efficient building components since they can reduce air-conditioning load to save energy as they can modulate solar radiation passing through the windows of buildings Kang et al., 2015). Furthermore, VO 2 (M) materials are exploitable in highperformance electronic and photonic device applications like thermoelectric (Byon et al., 2012), field-effect transistor , terahertz (Luo et al., 2016), and photoelectric devices (Lu et al., 2014).

Data availability statement
The original contributions presented in the study are included in the article Supplementary Material. Further inquiries can be directed to the corresponding author.

Author contributions
Conceptualization, EM, EL, and PS; methodology, EM and EL; validation, EM, EL, and PS; investigation, EM; data curation, EM; writing-original draft preparation, EM; writing-review and editing, EM, EL, EP, and PS; visualization, EM; supervision, PS; project administration, EM, EL, PS, LP, and DM. All authors have read and agreed to the published version of the manuscript.