Insights into first-principles characterization of the monoclinic VO2(B) polymorph via DFT + U calculation: electronic, magnetic and optical properties

We have studied the structural, electronic, magnetic, and optical properties of the VO2(B) polymorph using first-principles calculations based on density functional theory (DFT). This polymorph was found to display four optimized structures namely VO2(B)PP, VO2(B)LP, VO2(B)PPD, and VO2(B)LPD using the generalized gradient approximation (GGA) PBE exchange-correlation functional by including/excluding van der Waals interaction. Our derivation provides a theoretical justification for adding an on-site Coulomb U value in the conventional DFT calculations to allow a direct comparison of the two methods. We predicted a zero bandgap of the VO2(B) structure based on GGA/PBE. However, by GGA/PBE + U, we found accurate bandgap values of 0.76, 0.66, and 0.70 eV for VO2(B)PP, VO2(B)LP, and VO2(B)PPD, respectively. The results obtained from DFT + U were accompanied by a structural transition from the metallic to semiconductor property. Here, we verified the non-magnetic characteristic of the monoclinic VO2(B) phase with some available experimental and theoretical data. However, the debate on the magnetic property of this polymorph remains unresolved. Imaginary and real parts of the dielectric function, as computed with the GGA/PBE functional and the GGA/PBE + U functional, were also reported. The first absorption peaks of all considered geometries in the imaginary part of the dielectric constants indicated that the VO2(B) structure could perfectly absorb infrared light. The computed static dielectric constants with positive values, as derived from the optical properties, confirmed the conductivity of this material. Among the four proposed geometries of VO2(B) in this study, the outcomes obtained by VO2(B)PPD reveal good results owing to the excellent consistency of its bandgap, magnetic and optical properties with other experimental and theoretical observations. The theoretical framework in our study will provide useful insight for future practical applications of the VO2(B) polymorph in electronics and optoelectronics.


Introduction
Transition metal oxides [1][2][3] have been widely studied during the last decades because of their unique physical properties that are exploitable in the areas of dielectrics, 4,5 thermoelectricity, 6-8 catalysis, [9][10][11] microelectronics, 12,13 and thin-lm transistors. [14][15][16][17][18] Vanadium oxides are of particular interest due to these materials provide outstanding advantages in many optoelectronic devices, such as smart windows, [19][20][21] sensors, [22][23][24] and resistive memories. [25][26][27][28][29] They have received considerable attention since they are studied as a metal-insulator transition (MIT) material. [30][31][32][33][34] The MIT can be induced by increasing the temperature/ pressure, which causes changes to the structural, electronic, electrical, and optical properties of the materials. [35][36][37][38][39][40][41][42] According to experimental and theoretical studies, different structures of vanadium oxides have been found at high and lowtemperature phases. So far, VO 2 , 43 54 and 55) are the most interesting compounds with well known structural properties for the MIT. Vanadium dioxides (VO 2 ) are very well known materials with several polymorphs, including tetragonal (R), 56 monoclinic (M), 57 triclinic (T), 58 tetragonal (A), 59 monoclinic (B), 60 paramontroseite 61 and the new body centered-cubic (bcc) structure. 62 At a high temperature, a metallic phase (VO 2 (R)) with a rutile structure can be achieved, while cooling to below 340 K, the R phase changes into an insulating monoclinic structure M phase. [63][64][65] Because the phase transition from the rutile VO 2 (R) and the monoclinic VO 2 (M) is associated with a huge change in resistivity, it has attracted considerable attention for electronic and optical applications. 43,[66][67][68][69] VO 2 (B) has been explored as a promising cathode material in Li ion batteries, mainly because of its prominent advantages of high discharge capacity of 323 mA h g À1 and low cost. [70][71][72][73] Moreover, the VO 2 (M) and VO 2 (R) phases can be prepared by the irreversible transformation of VO 2 (B) as a precursor. 74,75 Advanced theoretical and experimental techniques have been implemented to study the VO 2 (B) polymorph. While an experimental bandgap of 0.6-0.7 eV (ref. [76][77][78] was found for VO 2 near the semiconductor-metal transition, an X-ray photoelectron spectroscopy study revealed a metallic observation of this material at room temperature. 64 In agreement with the experimental results, the rst-principles calculations conrmed both metallic and insulating features of VO 2 (B). 64,74,79,80 In the study conducted by Lee et al., 80 XAS, optical spectroscopy, and DFT calculations were applied to assess the bandgap in the VO 2 (B), VO 2 (M), and VO 2 (A) structures. This study revealed that by comparing the electronic structures of the A, B, and M phases, conventional DFT calculations estimated signatures of a metallic behavior for the A, B, and M phases. Meanwhile, hybrid functional calculations indicated bandgaps of 0.6 eV and 0.5 eV for the M and A phases, respectively, and a very narrow bandgap of 25 meV for the B phase. Zhang et al. 74 investigated the phase transition process from VO 2 (B) to VO 2 (A) based on Xray absorption spectroscopy (XAS) analysis and DFT calculations. They reported on the metastability of VO 2 (B) in comparison with the VO 2 (A) and VO 2 (R) phases. The calculation of the formation energy in a different phase of VO 2 showed that the VO 2 (B) structure has less geometrical stability with À6.66 eV formation energy compared to À6.93 and À7.18 eV for VO 2 (R) and VO 2 (M), respectively. This study proposed that the different electronic structure completely depended on the different stabilities of the VO 2 phases. In a recent study carried out by Popuri et al., 81 the electron transport properties of the VO 2 (B) structure at low (25-200 K), intermediate (200-320 K), and high temperatures (320-350 K) were investigated using spark plasma sintering. They found different electronic and magnetic properties at different thermal phases. At the low and intermediate temperature phases, nonmagnetic ordering was associated with the insulating characteristic of the structure. At a high temperature, metallic behavior and antiferromagnetic property were observed. In another study, Lourembam et al. 82 used terahertz time-domain spectroscopy (THz-TDS) to investigate the temperature-dependent complex optical conductivity of the VO 2 (B) structure. They observed that VO 2 (B) transformed from an insulating system to a conducting system at 240 K. Furthermore, there was a broad intermediate state with the transition onset being much closer to room temperature, allowing this polymorph to be more suitable for optoelectronic devices near room temperature. In an extensive experimental and theoretical research recently conducted by Wan et al., 83 they indicated that pure VO 2 (B) has weak absorption in infrared light, with excellent agreement between theory and experiment.
So far, some experimental results have been determined for the geometrical data and MIT for this compound. [82][83][84][85][86][87][88] The electronic properties have been investigated with some experimental and theoretical methodologies; however, the data are still limited and variable. 64,74,79,83,89 Furthermore, magnetic and optical features are potentially important properties that have not yet been systematically studied for this polymorph with rst-principles calculations, and no detailed values have been reported yet. Since a systematic investigation of the efficacy of advanced theoretical methods for computing the chemical and physical properties of VO 2 (B) is missing, our study seeks to ll this gap in the literature. Our present work is focused on the complete theoretical description of the electronic, magnetic, and optical properties of the VO 2 (B) polymorph using GGA/PBE and GA/PBE + U functionals. Recent studies have shown how the combined use of these methods makes it possible to calculate different material properties. 90 The main objectives of this work are as follows: (i) investigation of the structural parameters of VO 2 (B) by employing different theoretical approaches. The geometry optimization of this polymorph will be assessed by different methodologies in order to nd the most accurate results of the material characterization in agreement with the experimental outcomes. (ii) Calculation of the electronic band structure and magnetic properties, (iii) and analysis of the optical properties of the proposed different geometries of VO 2 (B).

Computational methods
We carried out the atomistic calculations using the Quantum ESPRESSO (QE) 91 and QuantumATK (QATK) 92 packages. The DFT approach was implemented in the Kohn-Sham (KS) formulation 93,94 within the framework of the linear combination of atomic orbitals (LCAO) and plane-wave (PW) basis set approaches, combined with the pseudopotential (PPs) method. PseudoDojo 95 and Projector Augmented Wave (PAW) PPs 96 were used for the LCAO and PW calculations, respectively, with the aim to describe the interaction between ion cores and valence electrons. DFT-LCAO and DFT-PW calculations were performed within the GGA framework adopting the PBE exchangecorrelation (XC) functional. 97 Valence orbitals were expanded in a PW basis set with a kinetic energy cut-off of 70 Ry. Brillouinzone (BZ) 98 integrations were limited to the gamma point mesh, and a smearing parameter of 0.0001 Ry was considered for the electron population function. 99 The van der Waals corrections were included by the Grimme's DFT-D3 method, 100 and the structure was relaxed with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. 101 We implemented both PBE and PBE + U 102,103 in post-processing calculations in order to make an exhaustive comparison between the different geometries. Since previous studies have shown that the outcomes substantially depended on the magnitude of U, we tested different Hubbard U values for the vanadium d orbital (U d ) and oxygen p orbital (U p ). In this study, we have set U d ¼ 5.20 eV and U p ¼ 0.95 eV, wherein they have more similarity to the results reported by Huffman et al. 104 (U d ¼ 5.00 eV and U p ¼ 0.00 eV). However, our chosen Hubbard values show more agreement between the theoretical and experimental bandgaps, as we will discuss later.
To calculate the electronic band structure and Projected DOS (PDOS), since the KS equation is a nonlinear differential formula, we rst converged the charge density with the self-consistent eld (SCF) calculations to compute the DOS on a uniform k-mesh (we used the 6 Â 6 Â 6 k-points). Then, we ran the non-self-consistent (NSCF) calculation with a twice high kmesh with respect to the SCF calculations (12 Â 12 Â 12 Monkhorst-Pack mesh) in order to construct the Hamiltonian for the charge density by using the tetrahedron method. PDOS was also considered to account for the magnetic ordering of the compound. Optical calculations were analyzed based on the random phase approximation (RPA). 105,106 The optical properties of the VO 2 (B) structure in this study are discussed by the two components of the dielectric function (3(u) ¼ 3 1 (u) + i3 2 (u)) related to different polarizations in the electric eld. The imaginary part of the dielectric coefficient can be obtained from the direct interband transitions through Fermi's golden rule as: 107-109 where VB, CB, u, U, W k , and r ij denote the valence band, conduction band, photon frequency, volume of the unit cell, weight of the k-point, and elements of the dipole transition matrix, respectively. Moreover, the real part of the dielectric constant can be obtained using the Kramers-Kronig relation in eqn (2): where P is the principal value. The real part of the dielectric constants determines the polarization of a material subjected to an external electric eld (in this case, the light beam). In addition, the imaginary part shows the amount of light absorption. 110 The electron energy loss spectrum, L(u), can also be described using the dielectric constants by: 111,112 LðuÞ ¼ 3 2 ðuÞ 3 2 2 ðuÞ À 3 1 2 ðuÞ The energy loss function determines the loss of energy while traversing through the material. Molecular graphics were generated using the XCRYSDEN graphical package. 113

Results and discussion
Analysis of the polymeric structure VO 2 (B) with the space group C2/m was simulated in periodic boundary conditions (PBC) along with three Cartesian coordinates. As shown in Fig. 1, the base-centered monoclinic unit cell has dimensions of 11.85 Â 3.74 Â 6.49Å 3 . This included 12 atoms in the primitive unit cell (4 vanadium and 8 oxygen) and 24 atoms in the conventional unit cell (8 vanadium and 16 oxygen atoms) (see Fig. 1(b)). This unit cell dimension is in good agreement with the previous experimental patterns 64,81,85 and theoretical studies. 74,79,83 As shown in Fig. 1(a) and (b), the VO 2 (B) structure can be considered as two identical atom layers including 3D frameworks of VO 6 octahedra. These octahedra packings of VO 6 are only linked by oxygen atoms in the corners.
The second layer is shied with respect to the rst one by 1/2, 1/ 2, 0. This polymorph is distorted because of the out-of-center vanadium atoms, resulting in the presence of short/long V-V and two different types of octahedra.
The lattice energy minimized for VO 2 (B) was obtained by employing geometry optimizations of the atomic positions and altering the size and angle of the unit cell, systematically. Aer the optimization of the lattices, we evaluated the structural parameters and bond lengths on different types of vanadium atoms in the 3D structure. As shown in Fig. 1(b), there are 4 types of V-V bonds in the VO 2 (B) structure. From the experimental bond lengths (see Table 1), they are as follows: (i) the shortest V 1 -V 2 bond in the center of the xy plane with a bond length of 2.89Å, (ii) V 2 -V 3 vanadium atoms in the xy plane with a length of 3.24Å, (iii) long V 3 -V 4 vanadium bonds with an average distance of 3.33Å, and (iv) medium bond length of V 1 -V 4 characterized by 3.06Å. Table 1 indicates that the VO 2 (B) compound exhibits different ranges of the V-V bond distance with the four optimized geometries obtained from the PW and LCAO approaches by including/excluding the dispersion corrections (DFT-D3) in the GGA/PBE calculations. For the convenience of discussion, the experimental and theoretical V-V bond lengths and their differences are listed in Table 1. In comparison to the experimental results in ref. 81, the VO 2 (B) optimized structure obtained from the PW approach and PBE XC functional without including the dispersion corrections (PW(PBE)) (the material named VO 2 (B) PP ) exhibited V 1 and V 3 atoms that were greatly displaced away from the central vanadium atom V 2 with V 1 -V 2 ¼ 3.33Å and V 2 -V 3 ¼ 3.79Å. Moreover, V 4 became closer to V 3 and V 1 with distances of 3.18Å and 2.94Å, respectively. In the case of PW(PBE-D3) (VO 2 (B) PPD ) including the van der Waals interactions, the V 1 -V 2 and V 2 -V 3 bond distances were shortened to 2.98Å and 3.34Å in more agreement with the experimental values of 2.89Å and 3.24Å, respectively. However, the V 3 -V 4 bond distance reached the length of 3.45Å, which is far from the experimental distance of 3.33Å. In the VO 2 (B) LP structure obtained from the LCAO(PBE) method, the V 3 -V 4 and V 1 -V 4 bond distances were elongated from 3.18Å and 2.94Å in VO 2 (B) PP to 3.45Å and 3.05Å in VO 2 (B) LP , respectively, while the V 2 -V 3 bond distance decreased from 3.79Å in VO 2 (B) PP to 3.27Å in this geometry. In this case, V 1 and V 2 signicantly became closer together by 2.81Å, which is less than the experimental value of 2.89Å. Finally, the optimized structure of VO 2 (B) LPD obtained from LCAO(PBE-D3) including the dispersion corrections exhibited the greatest similarity to the results obtained from VO 2 From the structural parameters computed by the two different PW and LCAO approaches presented in Table 1, we can understand the trend with regards to the treatment of the chosen basis set and XC functional. By using the PW basis set and the PAW XC functional, the approximation tends to overestimate V 1 -V 2 and underestimate V 3 -V 4 . Conversely, the LCAO basis set associated with the PseudoDojo XC functional leads to an underestimate of V 1 -V 2 and overestimate of V 3 -V 4 . However, including the van der Waals interactions moderated the system in both approximations. From these results, it is important to note that the basis set approximation can have a signicant effect on the computed structures. Calculations on the bulk of similar materials have demonstrated that the LCAO approximation tends to give results that agree less with experimental results compared to the PW. In addition, it is worth noting that van der Waals interactions have a remarkable role in reducing the V 1 -V 2 and V 2 -V 3 interlayer distances when we consider our system by the PW basis set (Table 1). Meanwhile, it has the opposite effect when the LCAO basis set is used by enhancing the corresponding bond lengths of V 1 -V 2 and V 2 -V 3 . As we shall discuss later, since the shortest V 1 -V 2 bond distance plays a critical role in the electronic structure of the VO 2 (B) polymorphs, the PW(DFT-D3) method gives us more accurate results related to the different physical properties of the VO 2 (B) nanostructure.
In the next step, we calculated the post-processing computations of the electronic, magnetic, and optical properties of four different geometries of the VO 2 (B) polymorph in order to discover which geometry indicates better accordance to experimental and theoretical studies for this material.

Electronic and magnetic properties
We studied the electronic properties of the VO 2 (B) polymorph by computing the electronic band structure and the corresponding PDOS curves for total, V-3d, and O-2p for VO 2 (B) PP , VO 2 (B) LP, VO 2 (B) PPD , and VO 2 (B) LPD geometries based on the  PBE and PBE + U approximations. The band structure was analyzed along the high symmetry G-M-G-X-Y-I-L-G directions in the rst BZ. The results calculated by GGA/PBE revealed the zero bandgap for all four geometries of the VO 2 (B) polymorph. Since conventional XC functionals such as PBE oen underestimate the bandgap in semiconductors, 114 we also employed the DFT + U method to provide a more accurate prediction of the V d-d orbital correlations and bandgap. So far, the DFT + U method was successful in the prediction of the bandgap for different polymorphs of vanadium oxides. Furthermore, the outcomes agreed relatively well with experimental results. 83,[115][116][117][118] Fig. 2 and 3 describe the band structure and DOS predicted by the PBE + U functional, respectively. From these results, we found that the Hubbard method The magnetism has been reported as two values of the total magnetization and absolute magnetization. While the total magnetization indicates the same value of 4.00 m B for all four geometries of the VO 2 (B) polymorph, the absolute magnetization (Table 2) shows different values of magnetism. As expected, Table 2 reveals that with the Hubbard approximation, the absolute magnetization resulted in higher values than that observed using the conventional DFT, while VO 2 (B) PP and VO 2 (B) LPD showed higher and lower magnetization, respectively.
To elucidate the amount of magnetism contribution of vanadium and oxygen orbitals in the unit cell, we collected the Lowdin charges and the magnetic moment (MM) of the V-3d and O-2p orbitals (s and p orbitals for vanadium atoms and s orbitals for oxygen atoms can be neglected because these orbitals have negligible MM contributions). Table 3 indicates the total MM per unit cell (MM/cell) for different geometries of the VO 2 (B) polymorph predicted by GGA/PBE and GGA/PBE + U approximations. With both approaches, the main contribution of the total MM is related to the vanadium 3d orbitals. Inspecting Table 3    According to the data in Tables 2 and 3 Therefore, it can be concluded that the metallic state of this geometry is composed of dispersive bands of Vanadium 3d electrons.
Since the vanadium 3d orbitals contribute the most to the magnetism of the VO 2 (B) polymorph, we considered the detail of the total MM/cell of the V-3d orbitals. In the transition metal oxides, the d level is vefold degenerate. The degeneracy of the d level is split into the lower energy t 2g level and higher energy e g level by the crystal eld splitting in an octahedral eld. In this system, the vanadium atom is octahedrally coordinated by oxygen. In the earlier study by Zhang et al., 74 the semiconducting band structure diagram of VO 2 (B) was precisely explained. It is worth noting that V-V localized pairing interactions inuenced the p band and consequently the 3d xz , 3d yz and 3d xy orbitals in t 2g level. Meanwhile, the 3d z 2 and 3d x 2 Ày 2 orbitals (both in the e g level) that are involved in the s band, are mainly affected by the indirect V-O-V metal-ligand interactions. Table 4 reveals that the electrons predominantly occupy the p band. In contrast, very few electrons occupy the s band. Based on the outcomes collected in Table 4, it can be observed that the 3d xz and 3d yz orbitals have the prevailing contribution in the MM/cell, 3d xy has some contribution to a lesser extent, while the 3d z 2 and 3d x 2 Ày 2 orbitals have negligible contributions. Among the four geometries, VO 2 (B) LP produces quite a different effect. The charge accumulation in the 3d xy orbital is more than that for the 3d xz and 3d yz orbitals (MM ¼ 1.8746 m B for 3d xy in comparison to MM ¼ 1.2118 m B and MM ¼ 1.2444 m B for the 3d xz and 3d yz orbitals, respectively, with PBE + U approximation). In this case, the accumulation of the charge in the 3d xy orbitals are greater than those of other geometries. This can be interpreted from the existence of the very short distance V 1 -V 2 ¼ 1.81Å (Table 1) in this structure. By comparison, in the other three considered geometries, this bond distance is about 3Å. Taking into account that the GGA + U method adds a Hubbard-type term to the density functional that increases the electron localization in the correlated orbitals, it is generally believed to provide better results.
According to the experiments carried out by Popuri et al., 81 macroscopic magnetic measurement results showed that the interactions for the vanadium ions were antiferromagnetic during the high temperature phase. A very weak ferromagnetic property of the VO 2 (B) polymorph can be observed at low temperature. As proposed in this study, the Curie constant (the contribution percentage of the half-spin (S 1/2 )) in the vanadium cation is varied in different phases. The obtained curie constant at the low-temperature phase of the VO 2 (B) structure was 12% for S 1/2 in the V-3d cation (spin singlets). This contribution increased to 50% and 100% at the intermediate temperature and high temperature phases (free spins), respectively. Furthermore, experimental X-band EPR spectra in this work revealed a broad resonance line related to the weak interaction of the V-V pairs in the low temperature phase. In contrast, this line became signicantly narrower in the intermediate temperature and high temperature phases because of the unlocalized interactions. Similar observations were made by Oka et al., 85 with the paramagnetic vanadium ions in the high temperature phase and the formation of nonmagnetic V-V pairs in the low temperature phase. In agreement with the outcomes obtained for these studies, our calculations based on GGA/PBE and GGA/PBE + U conrmed the total contribution of 12.5-15.5% for V-3d (as see in Table 3, the MM/cell for V-3d altering between $3.95-4.95 m B ), instead of 32 m B for eight vanadium atoms in the unit cell. These outcomes suggested the presence of less free spins in the VO 2 (B) polymorph, resulting in weak interactions of the vanadium atoms and very poor magnetic (not-magnetic) property of this material. However, the magnetic description of the VO 2 (B) structure has been controversial. Conicting experimental reports of ferromagnetism, 121,122 nonmagnetic/antiferromagnetic, 85 paramagnetic/antiferromagnetic, 81 and paramagnetic 123 properties suggest that this material probably has a negligible magnetic susceptibility. We therefore designate it as non-magnetic, as previously reported. 79,81,85,123 Table 3 Total MM/cell (in m B ) for the V-3d and O-2p orbitals of VO 2 (B) PP , VO 2 (B) LP , VO 2 (B) PPD , and VO 2 (B) LPD predicted by GGA/PBE and GGA/PBE + U

Optical properties
Once the electronic structure calculations conrmed the semiconducting character of the VO 2 (B) polymorph, we probed their optical properties for possible optoelectronics applications. The imaginary (3 2 (u)) and real parts (3 1 (u)) of the dielectric function, as well as the energy loss function for the VO 2 (B) PP , VO 2 (B) LP , VO 2 (B) PPD , and VO 2 (B) LPD structures are presented in Fig. 4-6 as functions of photon energy. We considered the parallel (inplane) and perpendicular (out-of-plane) polarization directions within RPA + PBE and RPA + PBE + U. According to Fig. 4 and 5, the rst main peak of 3 2 (u) shows a weak absorption in the infrared range (1.24 meV to 1.7 eV) for the VO 2 (B) PP structure along the in-plane/out-of-plane polarizations. However, the situation changes remarkably for the VO 2 (B) LP , VO 2 (B) PPD , and VO 2 (B) LPD geometries, in which they indicate that the adsorption peaks in the infrared light are only along the out-of-plane polarizations. Based on the GGA/PBE calculations (Fig. 4) For the real part of the dielectric function related to the static dielectric function, it was found that the 3 1 (u) part for VO 2 (B) PP , VO 2 (B) LP , VO 2 (B) PPD , and VO 2 (B) LPD geometries shows the positive values of 38.53, 4.60, 5.08 and 5.06 along the in-plane polarization, and 65.99, 19.50, 20.46 and 18.43 for the out-ofplane polarization directions, respectively. From the predicted data based on the DFT + U calculations, as shown in Fig. 5, the adsorption peaks of 3 2 (u) along the inplane polarization show similar results to the PBE calculations. However, the peaks existing in the optical spectrum of the out-of-plane direction exhibit a blue shi in the light energy range of 1.43 eV for VO 2 (B) LP and an intense peak at 3.10 eV. By applying the U correction in the PBE calculations, the light polarization becomes more intense in VO 2 (B) PP , whereas the other three geometries exhibit the opposite behavior by decreasing the peak intensity. Moreover, our theoretical calculations indicate that the optical bandgaps of the VO 2 (B) PPD and VO 2 (B) LPD geometries slightly increase by $0.95 eV. Meanwhile, the optical bandgap of VO 2 (B) LP is situated at higher energies at 1.20 eV. A strange behavior is represented by the zero optical bandgap of VO 2 (B) PP at low photon energy. This distinct difference might occur because VO 2 (B) PP contains a longer V 1 -V 2 ¼ 3.33Å (more weakly bonded) than the three other congurations with shorter V 1 -V 2 bond distances of 2.81, 2.98 and 3.02Å for VO 2 (B) LP , VO 2 (B) PPD , and VO 2 (B) LPD (Table 1)   plane polarization direction, with a drop in comparison to the PBE functional. The static optical spectra with the positive value of both in-plane/out-of-plane dielectric constants are further proof of the VO 2 (B) conductivity. Lourembam et al. 82 and Lee et al. 80 experimentally conrmed the non-zero frequency of the real part of the optical conductivity of this polymorph.
As reported in the literature, the different experimental values of the static dielectric constant of VO 2 have been observed. Yang et al. 124 investigated the temperature dependence of the dielectric constant and carrier conduction in VO 2 thin lms. They outlined that the dielectric constant of VO 2 can be increased from $36 at room temperature to a value exceeding 6 Â 10 4 at 100 C. In another study, Hood et al. 125 measured the dielectric constant of the VO 2 structure across the phase transformation at 68 C. In this work, the real part of the dielectric constant increased from less than 1000 to higher than 90 000 by elongating the lm thickness. Furthermore, the outcomes obtained by Mansingh et al. 126 showed the approximated value of 100 for the static dielectric constant of VO 2 single crystals in the frequency range of 30 to 10 5 Hz, and in the temperature range 77 to 250 K. From the theoretical side, Wan et al. 83 used both experiment and rst-principles PBE + U calculations to investigate the optical property of the VO 2 (B) structure. They observed the weak adsorption of this polymorph in the infrared light along the in-plane/out-of-plane polarization directions. According to the data presented in the literature for the other 2D oxides, VO 2 (B) possesses an excellent dielectric constant along the in-plane and out-of-plane directions. Its dielectric constant is higher than that for Al 2 O 3 with a value of 8-10 and SiO 2 with 3.9, 127 and is comparable with that for HfO 2 with a dielectric constant of 20-25. 128 Our calculations indicate that VO 2 (B) can be a good replacement for SiO 2 with a higher dielectric constant for application in eld effect transistors (FETs) and capacitors of dynamic random-access memories. Meanwhile, the stronger infrared absorption of the VO 2 (B) polymorph is favorable for achieving the maximum sensitivity for the applications in uncooled infrared bolometer. 129,130 The theoretical energy loss function computed by GGA/PBE and GGA/PBE + U is presented in Fig. 6(a)-(d). The energy-loss spectrum is important for describing the energy loss of electrons passing through the materials. While the spectrum calculated by GGA/PBE indicated broad peaks for the in-plane polarization in the energy range of 14-20 eV, GGA/PBE + U indicated in the high intensity peaks along the in-plane and outof-plane polarization directions. The results reveal that the maximum energy loss peak value predicted by GGA/PBE for VO 2 (B) PP , VO 2 (B) LP , VO 2 (B) PPD  According to the theoretical study by Wan et al., 83 they found an electronic bandgap of 0.60 eV for the VO 2 (B) polymorph. However, the zero optical bandgap was observed in the 3 2 (u) optical graph. This disagreement also occurred in our calculations, in which the VO 2 (B) PP structure showed an 0.76 eV electronic bandgap and zero optical bandgap. Conversely, VO 2 (-B) PPD indicated a zero bandgap in the band structure calculations and a semiconductor optical property. On the other hand, VO 2 (B) LP was not able to support the correct optical bandgap when the U value was included in the PBE calculations. In conclusion, the subtle interplay between the electronic, magnetic, and optical properties leads to the VO 2 (B) PPD conguration describing the semiconductor electronic and optical bandgap well, and shows excellent agreement between experimental and theoretical observations. Therefore, on the basis of the DFT calculations with the PW approach and PBE-D3 method, this conguration strongly suggests a VO 2 (B) polymorph.

Conclusions
We have successfully reproduced the experimental electronic, magnetic and optical properties of the VO 2 (B) polymorph via DFT calculations. In this study, we optimized the geometry of the VO 2 (B) polymorph on the basis of the PW and LCAO approaches using the GGA/PBE functional and with exclusion/inclusion of the dispersion corrections. The analysis of the structural parameters showed the existence of four different geometries of VO 2 (B), namely VO 2 (B) PP , VO 2 (B) LP , VO 2 (B) PPD , and VO 2 (B) LPD obtained from different methods. In order to check for the reliability of the computational methods, particularly for the selected energy functional (PBE + U) with the Coulomb correlation effect, we calculated the electronic and optical bandgaps and magnetic state of the VO 2 (B) congurations for comparison with experiments. The electronic band structure and DOS revealed a zero bandgap for all considered geometries by using the conventional GGA/PBE approximation. However, applying a Hubbard U value of 5.20 eV for the V-3d orbitals signicantly opened the bandgap up to 0.76, 0.66 eV and 0.70 eV for VO 2 (B) PP , VO 2 (B) LP and VO 2 (B) PPD , respectively. From these numerical calculations, we indicated that the DFT + U method can be used to change the gap size and induce a metal-semiconductor transition. PDOS solution was used in our potential energy scan and the magnetic properties were assessed. The PBE and PBE + U predicted the nonmagnetic state of the ground-state VO 2 (B) phase, which is consistent with the magnetic moment observed in experiments. Moreover, the optical properties including the imaginary and real parts of the dielectric function for the in-plane and out-of-plane polarizations for the VO 2 (B) geometries were evaluated. The rst absorption peaks revealed that all considered geometries can perfectly absorb infrared light along the out-of-plane polarization. Notably, PBE and PBE + U conrmed its VO 2 (B) semiconducting feature with the static dielectric constants having positive values. The DFTbased verication of the nonmagnetic feature as well as the electronic and optical measurements of VO 2 (B) PPD , provide the important future research lines to physical characterization of other VO 2 polymorphs. data curation, E. M.; writing-original dra preparation, E. M.; writing-review and editing, E. M., E. L., E. P. and P. L. S.; visualization, E. M.; supervision, P. L. S.; project administration, E. M., E. L., P. L. S., L. P. and D. M. All authors have read and agreed to the published version of the manuscript.

Conflicts of interest
There are no conicts to declare.