Theoretical investigations of optoelectronic properties, photocatalytic performance as a water splitting photocatalyst and band gap engineering with transition metals (TM = Fe and Co) of K₃VO₄, Na₃VO₄ and Zn₃V₂O₈: a first-principles study

Abstract

First-principles density functional investigations of the structural, electronic, optical and thermodynamic properties of K₃VO₄, Na₃VO₄ and Zn₃V₂O₈ were performed using generalized gradient approximation (GGA) via ultrasoft pseudopotential and density functional theory (DFT). Their electronic structure was analyzed with a focus on the nature of electronic states near band edges. The electronic band structure revealed that between 6% Fe and 6% Co, 6% Co significantly tuned the band gap with the emergence of new states at the gamma point. Notable variations were highlighted in the electronic properties of Na₃V_(1−x)Fe_xO₄, Na₃V_(1−x)Co_xO₄, K₃V_(1−x)Fe_xO₄, K₃V_(1−x)Co_xO₄, Zn_3(1−x)V_2(1−x)Co_xO₈ and Zn_3(1−x)V_2(1−x)Fe_xO₈ (where x = 0.06) due to the different natures of the unoccupied 3d states of Fe and Co. Density of states analysis as well as α (spin up) and β (spin down) magnetic moments showed that cobalt can reduce the band gap by positioning the valence band higher than O 2p orbitals and the conduction band lower than V 3d orbitals. Mulliken charge distribution revealed the presence of the 6s² lone pair on Zn, greater population and short bond length in V–O bonds. Hence, the hardness and covalent character develops owing to the V–O bond. Elastic properties, including bulk modulus, shear modulus, Pugh ratio and Poisson ratio, were computed and showed Zn₃V₂O₈ to be mechanically more stable than Na₃VO₄ and K₃VO₄. Optimal values of optical properties, such as absorption, reflectivity, dielectric function, refractive index and loss functions, demonstrated Zn₃V₂O₈ as an efficient photocatalytic compound. The optimum trend within finite temperature ranges utilizing quasi-harmonic technique is illustrated by calculating thermodynamic parameters. Theoretical investigations presented here will open up a new line of exploration of the photocatalytic characteristics of orthovanadates.

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1. Introduction

Oxide-based semiconductors as water splitting photocatalysts have gained much attention in an effort to help with environmental preservation and the growing energy issue facing our society.¹ Owing to the world's extensive reliance on less efficient non-renewable fossil fuels (e.g., coal, oil, and ethanol), a substantial amount of carbon dioxide is produced, which causes the greenhouse effect. One of the most promising alternatives to fossil fuels is the conversion of water into elemental components using incident photons. Because of its exceptionally clean combustion and green impact on the environment, photocatalytic materials furnish a direct pathway for converting solar radiation into hydrogen fuel.² A TiO₂ photocatalyst was firstly utilized for the efficient conversion of solar energy into chemical energy and emphasized the development of a photocatalytic system having the potential to split water into H₂ and O₂.^3,4 Unfortunately, its practical applications are limited by its fairly wide band gap as UV photons are the only primary source for its photocatalytic activity.⁵ However, only 4% of solar energy is accessible. It is possible to develop photocatalytic materials that rely on less intense but more prevalent visible light that can harness a significant portion of the solar spectrum.⁶ Such a compelling rationale for obtaining a stable, efficient, and cost-effective photocatalyst remains a goal to pursue for application in the field of photochemical cells (PECs).

The major role of a photocatalyst is to absorb incident light and create charge carriers, i.e., electrons and holes. With the absorption of light having wavelengths equal to the band gap of the semiconductor, electrons are excited from the valence band (VB) to the conduction band (CB), creating holes in the VB.⁷ These charge carriers can then interact with pollutants, undergo oxidation–reduction processes, and result in products being adsorbed on the semiconductor's surface. The hole in the valence band and electrons in the conduction band can recombine at the same time. Effective light absorption, high redox potential, and limiting the photoinduced charge recombination are helpful methods of chemical conversion that improve the photocatalytic process efficiency.^8,9

Recent studies on photosensitive materials for solar energy conversion have mainly focused on two primary goals: modification of conventional photocatalysts and development of a novel photocatalyst or photocatalytic system.¹⁰ Moreover, due to their relative stability against photocorrosion (photo-oxidative disintegration), environmental friendliness, and ease of production, metal oxide semiconductors had attracted a lot of attention.¹¹ To date, the most efficient semiconductor photocatalysts to be reported are Fe₂O₃,¹² ZrO₂,¹³ ZnO,¹⁴ SnO₂,¹⁵ TiO₂ (ref. 16) and bismuth halides.¹⁷

Vanadates have gained much attention due to their interesting physicochemical properties and exceptional performance.¹⁸ They are preferred over titanium and zinc oxide photocatalysts due to their narrower band gap,¹⁹ lower recombination rate,^20,21 and higher quantum efficiency.²² Simultaneously, the complex structural chemistry of vanadates, which readily adapt to various shapes, including tetrahedral, distorted octahedral and dodecahedral, demonstrates their unique character. Due to the structural differences, they may be applied as photocatalysts or optoelectronic materials. The V-3d orbital usually offers a lower conduction band in the band gap structure, which is why vanadates are regarded as potent visible light-driven photocatalysts.²³ Having isolated discrete units, the orthovanadates (VO₄⁻³) class is the simplest of all vanadates. Numerous visible-light activated photocatalysts, such as YVO₄,²⁴ InVO₄,²⁵ and BiVO₄,²⁶ have been reported for hydrogen production and the photodegradation of dyes.

This research work focuses on the systematic theoretical comparison of alkali-metal orthovanadate (K₃VO₄, Na₃VO₄) with transition metal orthovanadate (Zn₃V₂O₈). Having exceptional chemical strength, low band gap, economical, and an ecofriendly nature, one of the most promising photocatalysts for pollutant degradation and water splitting process is Zn₃V₂O₈. Sajid et al. reported that Zn₃(VO₄)₂ and its hydrates are effective photocatalysts and solar energy conversion materials.²⁷ Slobodin et al. proposed the color features and luminescence behavior of AVO₃ (A = K, Rb, and Cs), and reported on the synthesis of M₃V₂O₈ (M = Mg, Zn) by solid-state reaction method.²⁸

Cobalt (Co) has been reported as an efficient dopant with a suitable valence band structure to photo-oxidize water to H₂, while having a band gap covering the entire range of visible light. Examples include the Co-doped Fe₂O₃–TiO₂,²⁹ Co²⁺-doped-TiO₂,³⁰ LaCo_xFe_1−xO₃ (x = 0.01, 0.05, 0.1),³¹ and BiFe_1−xCo_xO₃ (x = 0, 0.025, 0.05, 0.075 & 0.10).³² In the same way, iron (Fe) has also been found as an effective dopant in various water splitting photocatalysts, such as Fe³⁺-doped TiO₂,³³ Fe-doped TiO₂ and SrTiO₃ nanotubes,³⁴ Fe-doped BiVO₄,³⁵ and many others. Subsequent to the reported methodical photocatalytic efficiency of Fe and Co dopants, this research has also explored the enhancement of the photocatalytic efficiency with 6% Fe and 6% Co doped on Na₃VO₄, K₃VO₄ and Zn₃V₂O₈.

A review of the literature revealed that neither theoretical nor experimental studies of the electronic, elastic, optical, and thermodynamic parameters of these compounds have been reported yet. Herein, we report the first attempt to computationally design these water spitting photocatalysts with theoretical insights into the evolution of (OH⁻˙) and O₂⁻˙ radicals. The aforementioned motivation inspired us to carry out all these calculations using the full potential linear augmented plane wave (FPLAW) approach to ascertain the previous efforts on this subject, and offer reference data to experimentalists. Therefore, our investigation of all these parameters and band structure analysis with doping effects exhibited a reasonable explanation for the photocatalytic water splitting and degradation activity, which can assist in designing potentially effective vanadium-based photocatalyst materials.

2. Computational details

First-principles calculation of orthovanadates was accomplished through Cambridge Sequential Total Energy Package (CASTEP) code³⁶ in DS-BIOVIA Material Studio software.³⁷ Geometric optimization, electronic density of states, α (spin up) and β (spin down) magnetic moments, optical, elastic, and thermodynamic parameters have been computed using the FPLAPW density functional theory.³⁸ To study the exchange correlation interaction, the Perdew–Burke–Ernzerhof (PBE) function with generalized gradient approximation (GGA) scheme³⁹ was employed. Ultrasoft pseudopotentials were utilized to determine the valence electronic configuration (Na: 3s¹, K 4s¹, Zn: 3d¹⁰4s², V: 3d³4s², and O: 2S² 2P⁴). Convergence criteria were set to fine quality with kinetic energy, i.e., 500 eV for Na₃VO₄, K₃VO₄ and 700 eV for Zn₃V₂O₈. Before performing electronic calculations, the lattice constants and atomic positions were fully optimized by keeping the residual force on each atom at less than 0.05 eV Å⁻¹. All geometry structures were fully relaxed with 0.002 Å force displacement and an energy value of 2 × 10⁻⁵ eV per atom. The Brillouin zone was sampled by the Monkhorst–Pack scheme with a mesh size of 8 × 8 × 6 for Na₃VO₄, K₃VO₄, and 10 × 5 × 7 for Zn₃V₂O₈. Meanwhile, a k-point grid of 1 × 1 × 1 was used during the calculation of the elastic and thermodynamic properties of all models. The Mulliken population was computed to analyze the bond population and bonding nature. Moreover, the thermodynamic parameters were studied via Troullier–Martin norm-conserving pseudopotentials. The optical properties of the compounds were studied via frequency-dependent dielectric function α(ω).⁴⁰

3. Results and discussion

3.1. Crystal structure

The optimized structures of the orthorhombic (Zn₃V₂O₈) and tetragonal (Na₃VO₄ and K₃VO₄) phases are shown in Fig. 1a. The Zn₃V₂O₈ crystal was constructed in the Cmca space group, having three independent lattice parameters such as a = 6.18 Å, b = 11.76 Å, c = 8.38 Å. The crystal is composed of isolated VO₄ tetrahedra, spreading around the Zn atoms to form the ZnO₈ octahedra. The VO₄³⁻ tetrahedra share edges with the ZnO₈ dodecahedron, forming a chain parallel to the environment of the V-atoms.⁴¹

Fig. 1 Mixed polyhedral and ball-and-stick model of the optimized crystal structures of (a) Zn₃V₂O₈ (orthorhombic), (b) Na₃VO₄ (tetragonal) and (c) K₃VO₄ (tetragonal).

There are two independent lattice parameters, such as ‘a’ and ‘c’, for the tetragonal phase structure. The unit cell parameters are summarized in Table 1. There are two inequivalent coordinated sites in the Na₃VO₄ crystal system shown in Fig. 1b. In the first site, it forms a distorted tetrahedron (NaO₄) that gets bonded with four oxygen atoms, thereby sharing corners with the VO₄ tetrahedron. Secondly, Na coordinates with four oxygen atoms and forms NaO₄, which shares sites with the VO₄ and NaO₄ tetrahedra.⁴² Similar to Na₃VO₄, the tetragonal crystal structure K₃VO₄ crystallizes with the I4̄2m space group, in which V ions are tetrahedrally coordinated. Meanwhile, K¹⁺ ions have two types of chemically distinct environments, which are depicted in Fig. 1. In the first environment, K¹⁺ is bonded to four oxygen atoms, forming a distorted trigonal pyramid that shares edges with the VO₄ tetrahedra. At the second site, the K¹⁺ ions combine with four oxygen atoms to form the KO₄ tetrahedra.¹⁹

Table 1 Optimized lattice parameters, cell volume, angles and lattice energies of Zn₃V₂O₈, Zn_3(1−0.06)V_2(1−0.06)Fe_0.06O₈, Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈, Na₃VO₄, Na₃V_(1−0.06)Fe_0.06O₄, Na₃V_(1−0.06)Co_0.06O₄, K₃VO₄, K₃V_(1−0.06)Fe_0.06O₄, and K₃V_(1−0.06)Co_0.06O₄

Crystals	Lattice constants			Angles			Cell volume (Å)^3	Lattice energy (eV)
	a (Å)	b (Å)	c (Å)	α	β	γ
Zn3V2O8	6.15	11.70	8.34	90	90	90	600.87	−50340.43
Zn3(1−0.06)V2(1−0.06)Fe0.06O8	6.25	11.87	8.13	90	90	90	603.62	−52422.60
Zn3(1−0.06)V2(1−0.06)Co0.06O8	6.20	11.76	8.36	90	90	90	609.85	−52831.13
Na3VO4	5.02	5.02	9.79	90	90	90	247.01	−15288.88
Na3V(1−0.06)Fe0.06O4	5.06	5.06	9.78	90	90	90	250.42	−15072.91
Na3V(1−0.06)Co0.06O4	5.09	5.09	9.80	90	90	90	254.42	−15134.94
K3VO4	5.72	5.72	10.05	90	90	90	329.38	−12140.37
K3V(1−0.06)Fe0.06O4	5.8	5.8	10	90	90	90	338.68	−11985.15
K3V(1−0.06)Co0.06O4	5.95	5.95	9.31	90	90	90	329.59	−11994.68

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When foreign atoms are doped into the host lattice, the lattice distorts as a result of changes in the ionic radii and electronic states. However, the lattice distortions in the Co-doped and Fe-doped Zn₃V₂O₈, K₃VO₄, and Na₃VO₄ are not apparent after placing the dopant atom, which could be explained by the low impurity concentration and similar radii.

3.2. Band structure

To better characterize the electronic properties, the band structure provides significant information. Around the Fermi level, the electronic structures exhibit separation of the conduction and valence bands.⁴³ Here, the Fermi level is set to zero. A set of certain line segments linking distinctive Brillouin zone (BZ) points, as well as those connecting the distinctive BZ sites with the Γ-points, are chosen as the band pathways (Fig. 2). Fig. 3a, 4a and 5a illustrate the high symmetry points along the selected Brillouin zone path: G(0 0 0) − Z(0 0 0.5) − T(−0.5 0 0.5) Y(−0.5 0 0) − S(−0.5 0.5 0) − X(0 0.5 0) − U(0 0.5 0.5) − R(−0.5 0.5 0.5), Z(0 0 0.5) − A(0.5 0.5 0.5) − M(0.5 0.5 0.0) − G(0 0 0) − Z(0 0 0.5) − R(0 0.5 0.5) − X(0 0.5 0) − G(0 0 0), and Z(0 0 0.5) − A(0.5 0.5 0.5) − M(0.5 0.5 0) − G(0 0 0) − Z(0 0 0.5) − R(0 0.5 0.5) − X(0 0.5 0.0) − G(0 0 0) for Zn₃V₂O₈,⁴⁴ Na₃VO₄ (ref. 45) and K₃VO₄ (ref. 46) photocatalysts, respectively. From Fig. 3a and 4a, the top of the valence band is ostensibly obtained within the G line for Zn₃V₂O₈ and at the M line for Na₃VO₄. Subsequently, the bottom of the conduction band is connected in the Y point and G point between the SIG of the Brillouin zone for Zn₃V₂O₈ and Na₃VO₄; hence, indicating the indirect band gap. In the case of K₃VO₄, the G point specifies the direct band gap between the valence and conduction bands, as shown in Fig. 5a. Concerning the photocatalytic activity, the indirect band gap is more effective than direct ones.⁴⁷ Due to the difference in the momentum between the location of the two high symmetry points, it prevents the recombination of excited electrons with holes.⁴⁸ The estimated theoretical band gaps of Zn₃V₂O₈, Na₃VO₄ and K₃VO₄ are 2.69 eV, 3.80 eV and 3.65 eV, respectively.⁴⁹ The electronic properties of zinc orthovanadate are in good agreement with the reported theoretical study of orthovanadates.⁵⁰

Fig. 2 Brillouin zone showing k points of (a) orthorhombic Zn₃V₂O₈ as well as (b) tetragonal Na₃VO₄ and K₃VO₄.

Fig. 3 Band structure with the indicated dashed line at 0 eV as the Fermi level of (a) Zn₃V₂O₈, (b) Zn_3(1−0.06)V_2(1−0.06)Fe_0.06O₈ and (c) Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈.

Fig. 4 Band structure of (a) Na₃VO₄, (b) Na₃V_(1−0.06)Fe_0.06O₄, and (c) Na₃V_(1−0.06)Co_0.06O₄, in which the Fermi level is referenced as 0 eV as indicated by the horizontal dashed line.

Fig. 5 Band structure of (a) K₃VO₄, (b) K₃V_(1−0.06)Fe_0.06O₄ and (c) K₃V_(1−0.06)Co_0.06O₄, in which the Fermi level is referenced as 0 eV as indicated by the horizontal dashed line.

The band gap varies inversely with the atomic mass. This agrees with the sequence of band gap values for the selected compounds, such as Na₃VO₄ > K₃VO₄ > Zn₃V₂O₈.⁵¹ However, among all compounds, the dramatic decrease in the band gap of Zn₃V₂O₈ is due to the transition metal involving the d-orbital in bonding. Specifically, Zn (3d-orbital) is involved in the excitation of electrons from VB to CB.¹⁹

3.3. Doping/band gap engineering

To construct the band gap covering the whole range of visible light, 6% Fe was doped on the vanadium atom of Na₃VO₄ and K₃VO₄. Similarly, 6% Co was doped on the V-3D sites of the crystal lattices, and their band structures were compared. Meanwhile, in the case of Zn₃V₂O₈, the doping of 6% Fe or Co only on the V-3D sites was not significant. However, doping both Zn and V atoms with 6% Fe or Co magnified the narrowing of the band gap. The Co-doped Zn₃V₂O₈ system shows the narrowest energy gap between the dopant states and the VBM. Fig. 3 shows the reduction in the band gap values as follows: 2.69 eV > 1.72 eV > 1.68 eV for Zn₃V₂O₈ > Zn_3(1−0.06)V_2(1−0.06)Fe_0.06O₈ > Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈, respectively. Meanwhile, in the case of Na₃VO₄, Na₃V_(1−0.06)Fe_0.06O₄ and Na₃V_(1−0.06)Co_0.06O₄, the computed band gaps are 3.80 eV, 2.99 eV and 2.59 eV, respectively (Fig. 4). Similarly, the sequence of reduction in the band gap has been observed as 3.68 eV > 2.76 eV > 2.58 eV for K₃VO₄ > K₃V_(1−0.06)Fe_0.06O₄ > K₃V_(1−0.06)Co_0.06O₄, respectively (Fig. 5). In all three host crystal lattices, cobalt is computed as being a vital dopant for enhancing their photocatalytic activity.

3.4. Density of state

To develop qualitative insight into the nature of orbitals and their splitting, the total and projected density of states are examined. This provided substantial information for the characterization of the electronic properties of the materials. As a central atom, vanadium has a 3-d orbital that splits into two sets. The “e” state (including 3d_x²y² and 3d_z²) and t₂ state (having 3d_xy, 3d_yz, and 3d_xz) are considered as two distinct parts in crystal field theory.⁵² The conduction band is mainly composed of the vanadium 3d atom for tetragonal and orthorhombic phase structures. This band falls in the energy region between 0 and 10 eV. There is also a minor contribution of O-2p states, resulting in p–d hybridization.⁵³

The upper valence band of Zn₃V₂O₈ ranging from 0–3.5 eV is mainly dominated by the O-2p state, which is the major hallmark of oxide-based semiconductor materials, as shown in Fig. 6. The V-3d and Zn-3d orbitals are also involved in the upper valence band. One of the most striking attributes is the involvement of Zn-3d in the entire VB. However, p–d hybridization results in a strong covalent interaction between the Zn-3d and O-2p states.⁵⁰ The calculated DOS and PDOS for the Fe-doped and Co-doped systems with 6% concentration are compared in Fig. 7a and b. The Fermi level of each doped system lies within the dopant states, which means that both dopants inherently have empty states that can accept the active electrons. The dopant states become eminently closer in the case of the Co-dopant, and the energy gap is noted to be significantly narrower for Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈. Moreover, expansion of the VBM can be attributed to the increasing contribution of Co-3d states in the conduction band, as evident from the PDOS plots shown in Fig. 6b. Similarly, Fig. 7a and b show that the contribution and distribution of O-2p states in the valence bands are not significantly altered.

Fig. 6 Calculated total density of states and partial density of states of pure Zn₃V₂O₈ showing the O-2p states and V-3d states below and above the Fermi level (E_g).

Fig. 7 Calculated total density of states and partial density of states of (a) Zn_3(1−0.06)V_2(1−0.06)Fe_0.6O₈ and (b) Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈, showing the O-2p, Fe-3p and Co-3p states in VBM, along with the V-3d, Fe-3d and Co-3d states in CBM.

On the other hand, the VBM of the tetragonal phase system is composed of O 2p states. There is a contribution of V 3d, Na 3s, and K 4s in the CBM of these isostructural crystal systems, as shown in Fig. 8 and 10. Fig. 9a and 11a reveal that the Fe-dopant prominently induces its 3d before the V 3d state in Na₃V_(1−0.06)Fe_0.06O₄ and K₃V_(1−0.06)Fe_0.06O₄, thus placing the CBM near the Fermi level (E_g). Meanwhile, in the case of the Co-doped alkali metal orthovanadates, such as Na₃V_(1−0.06)Co_0.06O₄ and K₃V_(1−0.06)Co_0.06O₄, the CBM has been computed to shift closer to E_g, resulting in the vital reduction of the band gap due to the low-lying Co 3d state (Fig. 9a and b).

Fig. 8 Calculated total density of states (TDOS) and partial density of states of pure Na₃VO₄, showing O-2p and V-3d states below and above the Fermi level (E_g).

Fig. 9 Calculated total density of states (TDOS) and partial density of states of (a) Na₃V_(1−0.06)Fe_0.06O₄ and (b) Na₃V_(1−0.06)Co_0.06O₄, showing the O-2p, Fe-3p and Co-3p states in VBM, along with the V-3d, Fe-3d and Co-3d states in CBM.

Fig. 10 Calculated total density of states (TDOS) and partial density of states of pure K₃VO₄, showing the O-2p states and V-3d states below and above the Fermi level (E_g).

Fig. 11 Calculated total density of states (TDOS) and partial density of states of (a) K₃V_(1−0.06)Fe_0.06O₄ and (b) K₃V_(1−0.06)Co_0.06O₄, showing the O-2p, Fe-3p and Co-3p states in VBM, along with the V-3d, Fe-3d and Co-3d states in CBM.

3.5. Magnetic properties

Since Fe and Co are magnetic elements⁵⁴ as compared to V, the inversion symmetry of the atomic geometry diminishes in the direction normal to the plane of symmetry, while it still persists within the plane. Nevertheless, it is possible for spin polarization to break the in-plane inversion symmetry of spin-resolved electronic states, resulting in energetic spin splitting at the Fermi level.⁵⁵ Therefore, the spin electron density in real space and the spin-dependent orbital-resolved (partial) atomic-projected density of states (DOS) were calculated by collinear spin polarization to investigate the spin asymmetry magnetic order and the atom-orbital contributions to the spin magnetic moment.

In the total electron density, the major α (spin up) and β (spin down) components of the spin-polarized electrons are appreciably distinguished by the positive and negative spin densities of the s, p and d orbitals.⁵⁶ The α and β spin states for the undoped and doped Zn₃V₂O₈ Na₃VO₄ and K₃VO₄ are shown in Fig. 12–14, respectively. In Zn₃V₂O₈, the valence band is attributed mainly to the α-spin states of O 2p with a minor contribution from the 3d α-spin electrons of Zn and V. Particularly, the higher absolute value of the α-dominated spin density of the V-3d electrons in the conduction band compared to that of the β-dominated electrons indicates the presence of more α electrons than β electrons, as manifested in Fig. 12a. However, the α-spin magnetic moment predominantly originates in the conduction band from the spin-polarized components of the V-3d and Co-3d orbitals in Zn_2.82V_1.88Co_0.12O₄, and the V-3d and Fe-3d orbitals in Zn_2.82V_1.88Fe_0.12O₄ with an energetic-preferential spin splitting near the Fermi level, as shown in Fig. 12b and c.

Fig. 12 Spin-up (↑) and spin-down (↓) partial density of states of (a) Zn₃V₂O₈, (b) Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈, and (c) Zn_3(1−0.06)V_2(1−0.06)Fe_0.06O₈ with indicated dashed lines as the Fermi level (E_g).

Fig. 13 Spin-up (↑) and spin-down (↓) partial density of states of (a) Na₃VO₄, (b)- Na₃V_(1−0.06)Co_0.06O₄, and (c) Na₃V_(1−0.06)Fe_0.06O₄ with indicated dashed lines as the Fermi level (E_g).

Fig. 14 Spin-up (↑) and spin-down (↓) partial density of states of (a) K₃VO₄, (b) K₃V_(1−0.06)Co_0.06O₄, and (c) K₃V_(1−0.06)Fe_0.06O₄ with indicated dashed lines as the Fermi level (E_g).

For Na₃VO₄ and K₃VO₄, the entire crystal lattice comprises the α-dominated spin states of O-2p and V-3d near the Fermi level (Fig. 13 and 14). The valence band maximum (VBM) and conduction band minimum (CBM) are mainly from the α-spin states of the O-2p and V-3d orbitals. Meanwhile, the 3d α-spin states of the Fe and Co dopants significantly shifted the CBM close to the Fermi level. However, the β-dominated electrons occur less frequently in VBM and CBM (Fig. 13b, c and 14b, c). It is evident from the spin polarization that Co and Fe are magnetic metals because both up-spin and down-spin DOS cross the Fermi level with a clear polarization between the DOS of the two spin channels.⁵⁷ In Zn₃V₂O₈, Fe and Co established the p-type extrinsic semiconductor with the shifting of the Fermi level towards VB.⁵⁸ Meanwhile, in the case of Na₃VO₄ and K₃VO₄, DOS more prominently crosses the Fermi level, indicating Fe-doped and Co-doped p-type extrinsic semiconductors.

3.6. Mulliken population analysis

Mulliken charge analysis and bond population facilitated the investigation of the bonding behavior of the Zn₃V₂O₈, Na₃VO₄ and K₃VO₄ crystal systems. The bond population and atomic bond lengths are expressed in Table 2.

Table 2 Calculated bond lengths (Å) and bond populations of the Zn₃V₂O₈, Na₃VO₄ and K₃VO₄ crystal systems

Compounds	Population R–O	Bond length R–O/Å	Bond length V–O/Å	Population V–O
Zn3V2O8	0.33, 0.20, 0.17	2.04, 2.23, 2.18	1.71, 1.74, 1.81	0.76, 0.67, 0.64
Na3VO4	0.10, 0.08	2.32, 2.36	1.73	0.74
K3VO4	0.04, −0.05	2.60, 2.8	1.74	0.76

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The R and V atoms of the photocatalytic systems have higher positive charge, while oxygen has a negative charge. The data sets in Table 3 demonstrate that charge transfer is carried out from R and V to O atoms. Moreover, the higher positive charge transfer values demonstrated the greater overlap between the atoms of the bonding atoms.⁴⁴

Table 3 Mulliken atomic population, Mulliken charge (e) and Hirshfeld charge (e) of Zn₃V₂O₈, Na₃VO₄ and K₃VO₄

Compounds	Species	Mulliken atomic population				Mulliken charge/e	Hirshfeld charge/e
		s	p	d	Total
Zn3V2O8	Zn	0.34, 0.31	0.65, 0.67	9.97, 9.98	10.97, 10.96	1.03, 1.04	0.46, 0.45
	V	2.25	6.43	3.35	12.03	0.97	0.46
	O	1.86, 1.85	4.73, 4.79, 4.84	0.00	6.59, 6.70, 6.64	−0.59, −0.70, −0.64	−0.28, −0.29
Na3VO4	Na	2.14	6.09, 6.10	0.00	8.24	0.76	0.34, 0.32
	V	2.25	6.82	3.42	12.48	0.52	0.49
	O	1.87	4.83	0.00	6.70	−0.70	−0.37
K3VO4	K	2.18, 2.11	5.81, 5.96	0.00, 0.00	7.99, 8.07	1.01, 0.93	0.36, 0.45
	V	2.27	6.87	3.44	12.59	0.41	0.45
	O	1.90	4.92	0.00	6.82	−0.82	−0.40

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The characteristic of a bond as either covalent or ionic can be evaluated by the overlap population of the orbitals. A complete ionic bond has a value of zero in the overlap bond population. Conversely, bonds having values greater than zero have a higher covalent character.⁵⁹ So, the stronger covalency is implied by the greater bond population, whereas the opposite is true for the ionic character. The V–O bond populations for these three systems (0.76, 0.74, 0.64, 0.67) are much higher than the R–O bonds (0.04, −0.05, 0.10, 0.08, 0.33, 0.20, 0.17). As compared to the R–O bond, the covalent character is stronger in the V–O bond. Na₃VO₄ and Zn₃V₂O₈ exhibit more covalent character, as listed in Table 2. Meanwhile, the bonding of K₃VO₄ is actually a mixture of covalent–ionic properties. Having greater bond population and shorter bond lengths, the V–O bonds also have a considerable impact on the hardness, instead of the R–O bonds, in the R₃VO₄ and R₃V₂O₈ crystal systems.

The crystal structures and electronic band structures have led to the conclusion that vanadium atoms develop covalent and ionic interactions with O to produce VO₄³⁺, which confirmed the lack of a lone pair electron in the V atoms. Therefore, the interaction forces between R²⁺ and V⁵⁺ are directly related to the V–O bonds. The distortion in the Zn–O bond lengths in Zn₃V₂O₈ is due to the 4s² or 6s² lone pair of the transition metal (Zn⁺²). The strong photocatalytic activity of visible light is greatly influenced by this attribute,⁴⁴ which seems favorable for Zn₃V₂O₈.

3.7. Elastic and mechanical properties

The Debye temperature, dynamic properties, and stress strain behavior are all influenced by a solid's mechanical properties and elastic constants, which are crucial parameters. Moreover, the structural stability and binding energy are associated with the elastic properties. As the elastic constants vary depending on the crystal symmetry,⁶⁰ there are six distinct elastic constants for the tetragonal systems: C₁₁, C₃₃, C₄₄, C₆₆, C₁₂, C₁₃.⁶¹ Meanwhile, the orthorhombic system has nine independent elastic parameters; namely, C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, C₂₃.⁶² The positively defined quadratic form of the energy density, in which the coefficients are determined from the DFT calculations (elastic stiffness), fulfill the following fundamental criteria of the tetragonal and orthorhombic structures that explore the mechanical stability of the compounds.⁶³

• Born stability standard for the tetragonal system (Na₃VO₄ and K₃VO₄):

C₁₁ > |C₁₂|; 2C₁₃² < C₃₃ (C₁₁ + C₁₂), C₄₄ > 0; C₆₆ > 0

• Criteria for the mechanical stability of the orthorhombic system (Zn₃V₂O₈) is:

(C₁₁ > 0; C₁₁C₁₂ > C₁₂²

C₁₁C₂₂C₃₃ + 2C₁₂C₁₃C₂₃ − C₁₁C₂₃² –C₂₂C₁₃² − C₃₃C₁₂² > 0

C₄₄ > 0; C₅₅ > 0; C₆₆ > 0

Table 4 shows that the values of all elastic parameters are positive, and meet all aforementioned requirements. So, the crystal systems are mechanically stable.

Table 4 Elastic constants C_ij (in GPa) of Zn₃V₂O₈, Na₃VO₄ and K₃VO₄

Compounds	C11	C33	C44	C66	C12	C13	C16	C22	C23	C55
Zn3V2O8	173.98	164.66	53.86	19.44	52.25	87.42	—	182.03	97.04	61.23
Na3VO4	71.70	52.97	26.51	20.75	31.70	32.84	0.00	—	—	—
K3VO4	50.40	32.07	18.80	11.45	19.19	29.58	0.00	—	—	—

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To assess the aggregate average elastic characteristics of the crystal structure, additional approximations named as isostress (Reuss state) and isostrain (Voigt state) were used. Hill demonstrated that the Voigt and Reuss approximations accurately exhibit the upper and lower limits of the elastic constants. The Voigt (B_v) and Reuss (B_R) bulk modulus is given as:⁶⁴

For the tetragonal system, the bulk modulus can be calculated as:⁶⁵

whereas, C² = (C₁₁ + C₁₂) C₃₃ − 2C₁₃², M = C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃

The Voigt–Reuss shear modulus is then derived as:

The VRH averages for the shear modulus (G) and bulk modulus (B) are given as:

While the Young's modulus (E) and Poisson ratio (v) are computed as:

The bulk and shear modulus for the orthorhombic system can be calculated by the following equations:⁶⁶

B_R = χ(C₁₁(C₂₂ + C₃₃ − 2C₃₃) + C₂₂(C₃₃ − 2C₁₃) − 2C₃₃C₁₂ + C₁₂(2C₂₃ − C₁₂) + C₁₃(2C₁₂ − C₁₃) + C₂₃(2C₁₃ − C₂₃))⁻¹

The way to calculate the Young's modulus and Poisson ratio is the same as stated for the tetragonal systems. The Voigt and Reuss equations define the upper and lower limits of polycrystalline elastic constants, respectively, and the arithmetic mean of these two limits is used to estimate the realistic bulk and shear modulus. The bulk modulus (B) measures the resistance to volume change, along with the bond length caused by applied pressure. The significant strength of the solid bonds is an indication of the higher bulk modulus.⁶⁷ The condition of the bulk modulus stated as C₁₂ < B < C₁₁ seems to satisfy all these crystal systems.

Panda & Chandran⁶⁸ developed equations to investigate the Young's modulus and Poisson ratio. The resistance to reversible deformation under shear stress was investigated by the shear modulus (G), which is known as the stiffness of a material. A greater shear modulus suggests a stronger degree of bonding between atoms.⁶⁹ As a result, the Zn₃V₂O₈ system is stiffer and exhibits stronger bonding than the other two systems.

Pugh's ratio defines the nature of bonding, which is either ionic or covalent, on the basis of the ductile and brittle character of the fabricated material. G/B < 6 indicates the ductility (ionic bonding) of the material, and the reverse is for the brittle (covalent bonding) nature.⁷⁰ Therefore, K₃VO₄ and Zn₃V₂O₈ have higher ductile nature.

The directionality of the covalent bonds is quantified by the Poisson ratio. In contrast to ionic materials, which have a Poisson ratio ≥0.25, covalent materials have a smaller value (v = 0.1). The lower and higher boundaries of the central force in solids are 0.25 and 0.5, respectively.⁷¹ Our results indicate that all compounds have ionic contribution and central interatomic forces.

Another crucial factor used to assess the constant nature of the structural qualities in all directions is the anisotropy of materials. A suitable parameter is required to describe the degree of anisotropy, as all single crystals are actually anisotropic. Anisotropy factors for the tetragonal structure are as follows:⁴⁴

Using elastic constants, calculations can also be done for the orthorhombic system:

The calculated values of all these parameters are listed in Table 5. As the index for these materials is not zero, all are considered to be anisotropic in nature. Moreover, the elastic anisotropy is associated with the inter-atomic bond strength.

Table 5 Calculated bulk modulus (B), shear modulus (G), Young's modulus (Y), Poisson ratio (ν), Pugh value (G/B), anisotropy factors (A₁, A₂, and A₃), and Kleemann parameter (ξ) of Zn₃V₂O₈, Na₃VO₄ and K₃VO₄

Compounds	Bv	BR	BH	GV	GR	GH	Y	ν	G/B	A1	A2	A3	ξ
Zn3V2O8	110.45	109.5	109.98	45.83	37.3	41.57	110.69	0.33	0.378	1.31	1.60	0.31	0.44
Na3VO4	43.46	42.57	43.02	21.35	19.83	20.59	53.26	0.29	2.86	1.79	—	0.14	0.57
K3VO4	32.18	31.27	31.72	13.44	7.59	10.52	28.40	0.35	0.33	3.24	—	0.73	0.52

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To define the relative positions of the cation and anion sub-lattices under the maintained volume-strain distortions (for which the positions are variable according to symmetry), Kleemann established a significant factor known as the internal strain parameter (ξ). It describes the coloration of the bond stretching and bending.⁴⁴

The minimum bond bending is expressed when ξ = 0. In the case of ξ = 1, there will be minimum bond stretching. Table 5 illustrates that bond-bending is prominent in these photocatalytic materials.

3.8. Optical properties

Optical properties play an important role in determining the photocatalytic activity of photocatalysts, defining their mode of interaction with light. Herein, the absorption, reflection, refractive index, dielectric constant, and loss function are the optical properties calculated in the photon range of 0–60 eV.

3.8.1. Absorption. With electric fields over all directions, the polycrystalline function was used to optimize the optical absorption spectra. A small smearing of 0.5 eV was applied in this simulation to easily detect the absorption peaks. The absorption of incident light by any material can be expressed in terms of the real and imaginary parts of the dielectric function, as described by eqn (1):⁴⁰

The most prominent estimated absorption apexes are 252 269 cm⁻¹, 177 245 cm⁻¹, and 152 260 cm⁻¹ for Zn₃V₂O₈, Na₃VO₄ and K₃VO₄, respectively. The absorption values fluctuate with increasing photon energy. Meanwhile, optimum values were obtained at about 15.8 eV, 7.39 eV, and 26.9 eV for Zn₃V₂O₈, Na₃VO₄ and K₃VO₄, respectively. The absorption cut-off band edge lies at 2.94 eV. This result indicates that the most likely electron transition to occur is VB to CB, as it has been studied for the BiVO₄ photocatalyst.⁷² All these materials have almost zero absorption in the area of low-frequency photons. The photocatalytic degradation of pollutants is directly related to absorption.⁷³ Photo-induced charge carriers (electron and hole pairs) are produced when light interacts and activates the photocatalytic material, causing the migration of photo-sensitive electrons from VB to CB. When these electron and hole pairs interact with water, it results in highly reactive hydroxyl and superoxide radicals. Therefore, these radicals interact with pollutants, and partially or completely degrade them.¹⁴ A lower band gap has a synergetic effect on enhancing the photocatalytic process by allowing the photocatalyst to absorb more photons, and become more sensitive to absorbing incident light.⁷⁴ Hence, from Fig. 15a, Zn₃V₂O₈ exhibits higher absorption than the other two photocatalyst materials.

Fig. 15 Optical properties of Zn₃V₂O₈, Na₃VO₄ and K₃VO₄: (a) absorption, (b) reflectivity, (c) loss function, (d and e) imaginary & real dielectric function, (f and g) imaginary and real refractive indices.

3.8.2. Reflectivity. A series of periodic investigations into the optical phenomena initially focused on the reflectivity of the semiconductor material. The reflectivity can also estimate the extent of incident light directed on the surface of a photocatalyst. This estimation correlates with the absorbance character of a material. Earlier studies revealed that a greater UV or visible absorption corresponds to lower reflectivity.⁷⁵ Fig. 15b shows a dramatic zigzag pattern in the reflectivity curves. Both Na₃VO₄ and Zn₃V₂O₈ have the same reflectance of 0.30 at 6.15 eV and 7.61 eV, respectively. Meanwhile, in the case of K₃VO₄, the optimum R(ω) is 0.31 at 7.61 eV. The lower reflectivity and higher absorption indicate that Zn₃V₂O₈ is a better photocatalytic material.

3.8.3. Loss function. Plasmonic excitations from spectra L(ω) demonstrate the possible loss of energy per unit length of a transversing electron when light penetrates its surface. Within the validity of dielectric theory, the energy loss function for optical properties is correlated with the dielectric function, as expressed in eqn (2):⁷⁶

The high energy region and low energy region are two zones for describing the macroscopic and microscopic parameters of the materials.⁶ The critical onset values for the high energy region are 25 eV, 21 eV, and 27 eV for Zn₃V₂O₈, Na₃VO₄ and K₃VO₄, respectively. As in the visible region, photocatalysts cannot attenuate incident light, which results in less plasmonic excitation. However, the composition and electronic structure can be designed via a low energy function region that is almost less than 1 eV, as shown in Fig. 15c.

3.8.4. Dielectric function. Considering the optical response in the linear range, the complex dielectric function determines all parameters of photocatalyst materials. This function expresses the coupled electron and photon interaction, as given in eqn (3):⁷⁶

ε(ω) = ε_Re(ω) + iε_Im(ω) (3)

Here, the frequency of incident light is described by (ω). The real and imaginary parts of the dielectric constants are denoted by ε_Re(ω) and iε_Im(ω), respectively. The real component defines the electronic polarizability and interaction of incident electromagnetic radiation on materials. The energy storage capability and imaginary number of energy dissipation are also all depicted via the real component of this complex function.⁷⁷ The maximum real dielectric function is almost the same for all photocatalytic materials (6.31), as shown in Fig. 15e. This optimum value lies at an energy of 3.61 eV. There is a regular variation in the curves with increasing photon energy. After 45 eV of energy, the curve for all materials demonstrates constant behavior.

The initial threshold of the dielectric function's imaginary part is basically referred to as the first absorption peak. Such peaks correlate with the electronic transition between the occupied and unoccupied states near the Fermi level. The electron transition takes place between the V-3d and O-2p states. In addition to some other peaks, the critical onset points for the orthorhombic and tetragonal phases are 5.26 eV and 6.20 eV, respectively. By comparing the function of these three materials, the highest dielectric function of Zn₃V₂O₈ confirms its excellent dielectric properties (Fig. 15d).

3.8.5. Refractive index. The refractive index is an important tool for investigating the propagation of incident light throughout the photocatalytic degradation process. N(ω) is determined via the Kramers–Kronig relationship (eqn (4)):⁷⁸

N(ω) = n(ω) + ik(ω) (4)

Here, n(ω) is an absolute refractive index, while k(ω) is the extinction coefficient. Al Mamum reported that a greater density of the medium is associated with a large value of the refractive index.⁶ The refractive index (n) and extinction coefficient (k) are known as “fingerprints of the material” because components of this complex function define the interaction of the material with incident light.⁷⁹ k(ω) represents the attenuation coefficient (imaginary part), which signifies the intensity of dissipated radiation from the material when radiated with incident light. So, the imaginary part of N(ω) is initiated from zero frequency and increases to a maximum value of 1.38 for Na₃VO₄, 1.52 for Zn₃V₂O₈ and K₃VO₄, followed by some fluctuations in curves, as depicted in Fig. 15f.

The variable n(ω) determines the extent of the electromagnetic wave slowdown in comparison to the vacuum speed. The real part of the complex function of such photocatalyst materials shows an inverse pattern. As depicted in Fig. 15g, n(ω) has a larger refractive index in the early stage of photon energy than k(ω), which is virtually zero. Therefore, the real and imaginary parts of the complex N(ω) function are slightly greater for Zn₃V₂O₈ and K₃VO₄, as compared to Na₃VO₄.

3.9. Thermodynamic properties

To analyze the entropy, enthalpy, Gibbs free energy, and specific heat capacity at constant volume, the phonon density of states was calculated (Fig. 16). These parameters have been investigated in temperatures ranging from 0 to 1100 K, where the quasi-harmonic model is valid.⁸⁰ Therefore, the phonon contribution to all thermodynamic properties at elevated temperatures is illustrated in this work.

Fig. 16 Calculated thermodynamic parameters, (a) Gibb's free energy, (b) enthalpy, (c) entropy, (d) heat capacity and (e) Debye temperature of Na₃VO₄, K₃VO₄ and Zn₃V₂O₈.

The Gibbs free energy (G) effectively defines the stability of a compound.⁸¹ Fig. 16a shows the variation in the Gibbs free energy of these systems with temperature. The decreasing trend of the Gibbs free energy for the systems is as follows: K₃VO₄ > Na₃VO₄ > Zn₃V₂O₈. The Gibbs free energy corresponds to the amount of energy of the system that is required for work to be done. Thus, a system having a lower Gibbs energy (G) is thought to be thermodynamically favorable.⁸² The rising trend in the enthalpy and entropy curves with increasing temperature can also be observed in Fig. 16b and c, respectively. The behavior of these two factors is the opposite of that of the Gibbs free energy.⁸³ Because it possesses the highest entropy, enthalpy, and lowest Gibbs energy, the Zn₃V₂O₈ photocatalyst is thermodynamically more stable.

Solid-state physics also involves the investigation of a crystal's specific heat capacity at constant volume.⁸⁴ The phonon thermal softening at high temperature is highlighted by the increasing trend of the C_v curves. When the temperature is low, C_v is proportional to T³. However, at elevated temperature, the heat capacity complies with the Dulong–Petit law, and reaches the asymptotic limit, i.e., 3nR. Here, n is the atom's number in the primitive cell, while R is the general gas constant.⁴¹ The highest heat capacity of Zn₃V₂O₈ is 277.29 cal K⁻¹ at 1000 K (Fig. 16d).

For qualitative insight into the atomic thermal vibrations, the Debye temperature (θ_D) was studied, as these two parameters are parallel (Fig. 16e). According to the Debye theory, the Debye temperature is compatible with the phonon frequency.⁸⁵ Additionally, it represents the stability of the structure and the strength of the bonds, which directly corresponds to several physical characteristics, like the melting temperature and specific heat. The Debye temperature is derived from elastic constants, and observed heat measurements are equivalent at low temperatures. Computing θ_D via elastic constants is one of the standard methods of calculation. Thus, the Debye temperature (θ_D) can be estimated from the average sound velocity (ν_m), as described in the following equation:⁸⁶

Here, h is the Planck's constant. The atomic volume is represented by V_a, and the Boltzmann constant is indicated by K_B. The average sound velocity (ν_m) of a crystalline material can be defined as follows:⁴⁴

The transverse and longitudinal sound velocities (ν_t and ν_l) of a solid material can be determined from the Navier equation using the values of bulk and shear modulus.

The values of all these parameters are summarized in Table 6. To the best of our knowledge, there is no comparable experimental or theoretical data. Therefore, the stated findings are considered as the first-ever prediction. Thus, these results still need an experimental confirmation.

Table 6 Calculated cell volumes, density (ρ), transverse (ν_l) and longitudinal sound velocities (ν_t), average sound velocity and Debye temperature (θ_D) of Zn₃V₂O₈, Na₃VO₄ and K₃VO₄

Crystal systems	V (Å^3)	ρ (kg m^−3)	νl (m s^−1)	νt (m s^−1)	νm (ms^−1)	θD
Zv3V2O8	600.83	46 500	6446	15 548	7295.71	1520.52
Na3VO4	329.38	24 600	2803	5348	3229.71	449.90
K3VO4	247.01	26 100	2004	3891	2244.78	344.72

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3.10. Photocatalytic activity

The electronic properties can be used to quantitatively explore the systematic framework of narrowing band gap, separation, and migration of photoinduced charge carriers that will consequently enhance the photoactivity of a substance. However, this improvement might not be enough to achieve the entire photoactivity required for the degradation and water splitting processes.⁸⁷ It is obvious that the band structure requires a widely applicable design strategy, including band-edge placements and band gaps of newly produced materials.⁸⁸ An efficient tool to examine the enhanced photocatalytic activity of semiconductors for photoelectrochemical water splitting and pollutant degradation is provided by this theoretical calculation method.⁸⁹ Inter-particle electron transfer occurs when the optical incident light is equivalent to or greater than the band gap in semiconductor photocatalysts.⁹⁰ After the absorption of light, electrons are excited from VB to CB, producing e⁻ and h⁺ charge carriers with oxidative and reductive capabilities.⁹¹ The band edge potentials can be calculated as follows:

E_VB = E_VB + E_g

Where, the CB and VB edge potentials are denoted by E_CB and E_VB, respectively, χ is the electronegativity of the semiconductor, and the standard reduction potential (E_e) is estimated as 4.5 eV. The calculated band gap of the photocatalytic material is expressed by E_g. Table 8 and Fig. 17 illustrate the band edge positions of the un-doped and doped Zn₃V₂O₈, Na₃VO₄ and K₃VO_4.

Table 7 First ionization energy (eV), atomic electron affinity (eV) and atomic electronegativity of the individual atoms and dopants

Species	EIE (eV)	EEA (eV)
Na	5.13	0.55	2.84
K	4.34	0.50	2.42
Zn	9.39	0	4.70
V	6.74	0.52	3.64
O	13.68	1.46	7.54
Co-dopant	7.88	0.66	4.27
Fe-dopant	7.9	0.16	4.03

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Table 8 Calculated electronegativity, band gap (eV), conduction and valence band edge positions (eV) of Na₃VO₄, Na₃V_(1−0.06)Fe_0.06O4, Na₃V_(1−0.06)Co_0.06O₄, K₃VO₄, K₃V_(1−0.06)Fe_0.06O₄, K₃V_(1−0.06)Co_0.06O₄, Zn₃V₂O₈, Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈ and Zn_3(1−0.06)V_2(1−0.06)Fe_0.06O₈

Compounds	χ	Eg (eV)	CB (eV)	VB (eV)
Zn3V2O8	5.80	2.69	−0.63	2.06
Zn3(1−0.06)V2(1−0.06)Fe0.06O8	5.90	1.72	−0.36	1.36
Zn3(1−0.06)V2(1−0.06)Co0.06O8	5.93	1.68	−0.31	1.37
Na3VO4	4.77	3.80	−1.63	2.17
Na3V(1−0.06)Fe0.06O4	4.78	2.99	−1.23	1.76
Na3V(1−0.06)Co0.06O4	4.78	2.59	−1.02	1.57
K3VO4	4.50	3.65	−1.83	1.82
K3V(1−0.06)Fe0.06O4	4.50	2.76	−1.38	1.38
K3V(1−0.06)Co0.06O4	4.53	2.58	−1.26	1.32

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Fig. 17 Schematic with band edge positions of pure and doped Na₃VO₄, K₃VO₄, and Zn₃V₂O₈ in the photocatalytic water splitting process.

The following equation is used to calculate the electronegativity of a material:⁸⁷

Here, a, b, and c represent the number of R (Na, K and Zn), V and O atoms in the compound, respectively. The arithmetic mean of the ionization energy and electron affinity defines the atomic electronegativity of individual atoms (χ^a(R) × χ^b(V) × χ^c(O), as discussed in Table 7.

According to the equation, the generation of superoxide acid (⁻HO₂) and superoxide radical (O₂⁻˙) is an essential step for the solar degradation of pollutants. However, photogenerated holes can oxidize the OH⁻ ion and fabricate the reactive hydroxyl radical (OH⁻), as shown in the chemical equations below:⁹²

O₂ + H⁺ + e⁻ → HO₂

O₂ + e⁻ → O₂⁻˙

OH⁻ + h⁺ → OH⁻˙

As discussed earlier, the conduction band potential for all of these systems is more negative as compared to the redox potential of O₂/O⁻₂˙ (−0.33 eV), indicating the reduction of O₂ to the O₂⁻˙ radical anion. Although the conduction band edge potential of the photocatalyst systems is more negative than that of O₂/H₂O₂ (0.685 eV potential), this suggests that H₂O₂ is formed when electrons react with adsorbed oxygen molecules. This H₂O₂ combines with electrons to produce hydroxyl radicals. Furthermore, the valence band potential is more positive than that of HO⁻/HO˙ potential (1.99 eV), indicating that photogenerated holes in these systems oxidize hydroxyl ions to the reactive hydroxyl radicals required in the photocatalytic degradation of pollutants.⁹³

All of the three host systems, either pure, Fe-doped or Co-doped photocatalysts, have a higher positive valence band potential than that of O₂/H₂O (1.23 eV), which is consistent with all of the aforementioned observations, and indicates a strong oxidation potential for the evolution of O₂. Highly reactive radicals (OH⁻˙ and O₂⁻˙) will oxidize both organic and inorganic substances, resulting in water and other non-toxic molecules.⁹⁴ Interestingly, the CB positions of Na₃VO₄, Na₃V_0.94Fe_0.06O₄, Na₃V_0.94Co_0.06O₄, K₃VO₄, K₃V_0.94Fe_0.06O₄, K₃V0_.94Co_0.06O₄, Zn₃V₂O₈, Zn_2.82V_1.88Fe_0.12O₄, and Zn_2.82V_1.88Co_0.12O₄ are negative as compared to the H₂/H⁺ (0 eV) redox potential. A more positive potential provides the strong oxidizing potential for oxygen evolution in these photocatalysts, despite the fact that they are efficient at producing hydrogen gas. Higher positive VB and more negative CB potentials are among the best benchmarks for photocatalytic processes.⁹⁵ The peculiar higher photocatalytic activity of Zn₃V₂O₈ compared to other compounds can be ascribed to the location of the higher V-3d and Fe-3d or Co-3d states, compared with those of Na₃VO₄ and K₃VO₄. For efficient photocatalysis, electron–hole pairs should be energetically and geometrically separated.

Solubility is a key parameter in determining the photocatalytic efficiency. At present, most of the photocatalytic materials usually comprised insoluble semiconductors, while soluble semiconductor materials have restricted validity in photocatalysis.⁹⁶ Soluble inorganics can dissolve in water due to the attraction of positive and negative charges, resulting in the disintegration of crystalline structures and the loss of energy band structures for light absorption.⁹⁷ Meanwhile, the addition of an insoluble solute to its saturated solution will remain as crystals, and these undissolved photocatalytic materials will retain the geometrical shapes required for light absorption and photogenerated carrier transfer.⁹⁶ Higher lattice energies are typically associated with less solubility because they demand more energy to break the ionic bonds within the lattice.⁹⁸ Lattice energy minimization techniques have been used in attempt to enhance the effectiveness of water-soluble photocatalysts.⁹⁹ The lattice energy of Zn₃V₂O₈ (−50340.43 eV) is highest among all three studied orthovanadates, indicating its lowest solubility as compared to K₃VO₄ and Na₃VO₄ (Table 1). Moreover, the smaller size and greater charge on the Zn²⁺ ion than Na⁺ and K⁺ ions enables it to develop the strongest forces of attraction within the lattice, which are difficult to overcome, hence making it significantly less soluble in water. Furthermore, neither distortion of the crystal lattice nor significant reduction in the lattice energy was observed after doping. The solubility trend for all un-doped and doped orthovanadates is as follows: Zn₃V₂O₈ > Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈ > Zn_3(1−0.06)V_2(1−0.06)Fe_0.06O₈ > Na₃VO₄ > Na₃V_(1−0.06)Co_0.06O₄ > Na₃V_(1−0.06)Fe_0.06O₄ > K₃VO₄ > K₃V_(1−0.06)Co_0.06O₄ > K₃V_(1−0.06)Fe_0.06O₄.

4. Conclusion

This study explored the first-principles calculations of vanadium-based Na₃VO₄, K₃VO₄, and Zn₃V₂O₈ photocatalysts. The covalent character is associated with a higher degree of positive charge (for the V–O and R–O bonds). Moreover, the shorter bond length and high electronegativity of the V–O bond introduce hardness in all of these materials. The elastic constants and aggregate elastic modulus (B, G, Y, G/B, Poisson ratio) are also discussed, which signify the stiffness, ductility, anisotropy, and mechanical stability of these compounds. Using a quasi-harmonic approach, the thermodynamic parameters illustrated that a sharp decline in the Gibbs energy can be observed with increased temperature. The increasing heat capacity approaches the Dulong–Petit limit. Hence, K₃VO₄, Na₃VO₄, and Zn₃V₂O₈ are thermodynamically stable photocatalysts. As optical properties, materials showing lower reflectivity and higher absorption are associated with a high dielectric constant, so these parameters actually enhance the photocatalytic activity. A more negative CB value and a more positive VB value are appropriate criteria for photocatalytic water splitting and pollutant degradation. Na₃VO₄ is photocatalytically more active despite its prominent flaw, i.e., band gap, which can be resolved in the future. The electronic structure of a narrow band gap is significantly influenced by cations without a noble gas configuration. The estimated indirect band gap of Zn₃V₂O₈ is 2.69 eV, which is less than the values of 3.80 eV and 3.65 eV for Na₃VO₄ and K₃VO₄, respectively. However, doping with 6% iron and 6% cobalt turned the indirect band gap into a direct one with a prominent reduction in the band gap, especially with 6% Co as compared to 6% Fe. When the electronegativity of the doping species is smaller than that of oxygen, the impurity states will lie above the V. B. M. As a result, interactions between Co 3d and O 2p, as demonstrated by PDOS and the spin up (α) and spin down (β) magnetic moments, destabilize the VB and dramatically reduce the band gap, leading to the most effective photoresponse of Zn_3(1−0.06)V_2(1−0.06)Co_0.06O₈ under visible light irradiation. Furthermore, based on the definition of the semiconductor electronegativity, the CB and VB edge positions demonstrated that the Fe or Co-doped Zn₃V₂O₈ possesses favorable redox potentials under visible light, making it a promising material for further applications.

Data availability

All data are present in the manuscript.

Conflicts of interest

There are no conflicts to declare.

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