Effect of Fe Concentration on Fe-Doped Anatase TiO 2 fromGGA + U Calculations

To comprehend the photocatalytic mechanisms of anatase Ti1−xFexO2 with various concentrations of Fe, this study performed first principles calculations based on density functional theory with Hubbard U on-site correction to evaluate the crystal structure, impurity formation energy, and electronic structure. We adopted the effective Hubbard U values of 8.47 eV for Ti 3d and 6.4 eV for Fe 3d. The calculations show that higher concentrations of Fe are easily formed in anatase TiO2 due to a reduction in the formation energy. The band gap of Fe-doped TiO2 decreases Fe doping level increases as a result of the overlap among the Fe 3d, Ti 3d, and O 2p states, which enhances photocatalytic activity in the visible light region. Additionally, a broadening of the valence band and Fe impurity states within the band gap might also contribute to the photocatalytic activity.


Introduction
The increase in global pollution has led researchers to search for new techniques and materials to promote environmental protection.Since Fujishima and Honda's report in 1972 [1], the unique properties of TiO 2 have attracted considerable attention in the fields of air and water purification, hydrogen production, and dyesensitized solar cells.Anatase TiO 2 also has a wide band gap capable of only absorbing ultraviolet (UV) light (≤387 nm).UV light accounts for a small fraction (∼5%) of the available solar energy; therefore, the utilization of solar energy is low.To improve the photocatalysis of TiO 2 , determining how to expand the optical absorption into the visible light region (∼45%) has become a topic of considerable interest among researchers.
Considerable research has gone into modifying the band gap of TiO 2 .One of the most effective methods involves doping TiO 2 crystals with impurities including transition metals, such as V [2], Mo [3], Fe [4][5][6], Co [7,8], Pt [9], Au [10], and nonmetals such as N [11,12], F [13], P [14], S [15].Among these impurity elements, the radius of Fe 3+ (0.64 Å) is similar to that of Ti 4+ (0.68 Å) and is therefore easily incorporated into the TiO 2 crystal [16].In addition, the Fe 3+ dopant can serve as a charge trap, impeding the electronhole combination rate and enhancing photocatalysis within a range suitable to the concentration of the dopant [17].Fe is considered an appropriate candidate element and has been widely studied [18][19][20][21][22][23][24][25].Zhang et al. [18] prepared Fedoped mesoporous TiO 2 thin films and suggested that the doped Fe forms Fe 3+ ions, which could play a role as e − or h + traps, thereby reducing the e − /h + pair recombination rate.Wang et al. [19] reported that the band gap of Fedoped TiO 2 thin films decreased from 3.29 to 2.83 eV with an increase in the Fe 3+ content from 0 to 25 wt%.The decrease in unit cell volume indicates that Fe 3+ replaced Ti 4+ in the lattice, forming a solid solution.Some experimental results [20,21] have found that Fe doping in TiO 2 could narrow the band gap of TiO 2 , thereby increasing the efficiency of the photocatalysis in the visible range.Yalc ¸in et al. [22] performed calculations based on density function theory (DFT) to characterize the influence of Fe 3+ doping on the electronic and structural properties of TiO 2 .The results indicate that the visible light activity in Fe 3+ -doped TiO 2 is due to the introduction of additional electronic states within the band gap.Recently, first-principles calculations were conducted for Fe-doped TiO 2 , but these have been restricted to a single concentration of Fe [23][24][25].To the best of our knowledge, few of these studies have focused on the photocatalytic mechanisms of Fe-doped anatase TiO 2 with different Fe concentrations.Additionally, most theoretical calculations have greatly underestimated the band gap of TiO 2 due to the adoption of the conventional DFT method, which is known to include an insufficient description of the on-site Coulomb interaction between electrons occupying the Ti 3d orbitals.This study performed first-principles calculations using the generalized gradient approximation + Hubbard U (GGA + U) approach to investigate the crystal structure, formation energy, and electronic structure of anatase Ti 1−x Fe x O 2 .

Calculation Models and Methods
Anatase TiO 2 has a tetragonal structure with lattice parameters a = b = 3.776 Å, c = 9.486 Å.To calculate various concentrations of Fe, a 2 × 2 × 1 supercell was constructed with 16 Ti and 32 O atoms, as shown in Figure 1(a).The calculations for the doping system were conducted for the 2 × 2 × 1 supercell containing one, two, and three Fe atoms in the substitutional sites of the Ti atoms, as shown in Figures 1(b)-1(d), which correspond to the Fe concentrations of 2.08, 4.17, and 6.25 at.%, respectively.First-principles calculations were performed using the CASTEP module [26] in Materials Studio 5.0 developed by Accelrys Software Inc. Electron-ion interactions were modeled using ultrasoft pseudopotentials in the Vanderbilt form [27].The valence configurations of the atoms were 3s 2 3p 6 3d 2 4s 2 for Ti, 2s 2 2p 4 for O, and 3d 6 4s 2 for Fe.The wave functions of the valence electrons were expanded through a plane wave basis-set and the cutoff energy was selected as 400 eV.The Monkhorst-Pack scheme [28] Kpoints grid sampling was set at 4 × 4 × 3 (less than 0.04 Å−1 ) in the supercells.The convergence threshold for self-consistent iterations was set at 5 × 10 −6 eV.The lattice parameters and atomic positions for each supercell system were first optimized using the generalized gradient approximation (GGA) together with the method introduced by Wu and Cohen [29].The optimization parameters were set as follows: energy change = 9 × 10 −5 eV/atom, maximum force = 0.09 eV/ Å, maximum stress = 0.09 GPa, and maximum displacement tolerance = 0.009 Å.To describe the electronic structures more accurately, the GGA + U method was adopted with the strong on-site Coulomb repulsion among the localized Ti 3d electrons described according to the following formalism [30,31]: where ρ σ denotes the spin (σ) polarized on-site density matrix.The spherically averaged Hubbard parameter U describes the increase in energy caused by the placement of an additional electron at a particular site, and the parameter J (1 eV) represents the screened exchange energy.The effective Hubbard parameter U eff = U − J, which accounts for the onsite Coulomb repulsion for each affected orbital, is the only external parameter required for this approach.

Results and Discussion
3.1.Structural Optimization.Table 1 summarizes the optimized lattice parameters, average bond lengths, and differences in volume, obtained from structure optimization.The optimized lattice parameters, a = b = 3.778 Å, and c = 9.549 Å, for pure anatase TiO 2 are in good agreement with the experimental values of a = b = 3.782 Å and c = 9.502 Å [32], indicating that our calculations are reliable.In anatase TiO 2 , each Ti atom is bonded to its four nearest and two second nearest oxygen neighbors.Average bond lengths are represented as Ti-O 1st and Ti-O 2nd .With the same concentration of Fe, all of the Fe-O bond lengths are shorter than those of Ti-O.Therefore, the volume decreases with an increase in Fe concentration, which is consistent with the experimental results [33].This indicates that Fe doping causes a contraction in the overall volume, which may be the result of the difference in the radii of the ions: 64 pm for Fe 3+ and 68 pm for Ti 4+ .

Hubbard U Parameter.
The GGA + U approach uses an intra-atomic electron-electron interaction for on-site correction to describe systems with localized d and f electrons, capable of producing a more optimal band gap.Determining an appropriate Hubbard U eff parameter is necessary in GGA + U calculations to correctly interpret the intra-atomic electron correlation.As shown in Figure 2, for anatase TiO 2 , the band gap widens with an increase in the effective Hubbard U eff of Ti 3d.The band gap was widened by increasing U eff from 2 to 8 eV.Here, the on-site Coulomb interaction was U eff = 8.47 eV for Ti 3d using the GGA + U approach and the calculated band gap of pure anatase is 3.21 eV, which is close to the experimental value of 3.2 eV.Using the same method, the U eff = 6.4 eV of Fe 3d was determined by fitting the band gap of Fe 2 O 3 (2.2 eV).

Formation Energy.
To examine the relative stability of TiO 2 doped with various concentrations of Fe, the defect formation energies were calculated according to the following formula: Here, E tot (Fe-doped) and E tot (pure) are the total energies of Fe-doped TiO 2 and pure TiO 2 ; n is the number of substitutional Fe atoms; μ Fe and μ Ti represent the chemical potentials of the Fe and Ti atoms, respectively.The formation energy depends on the growth conditions, which can be Tirich or O-rich.For TiO   the Ti-rich conditions, indicating that the incorporation of Fe into TiO 2 at the site of the Ti atom is favorable.In addition, the formation energies decrease with an increase in Fe concentration under O-rich conditions, suggesting that higher concentrations of Fe facilitate the synthesis of Fe-doped anatase TiO 2 .The electrons in the VB can be excited to localized impurity states within the band gap and subsequently to the CB through the absorption of visible light.In addition, the substitutional Fe atom only transfers 3 electrons to the surrounding O atoms, which leads to unfilled O 2p orbitals and an electron remaining in the Fe 3+ ion as shown in Figure 4.At 4.17 at.% (Figure 3(c)), the overlap of Fe 3d, Ti 3d, and O 2p bands near the CB results in a decrease in the CBM.Therefore, the band gap of Fe-doped TiO 2 at 4.17 at.% narrows to 2.70.From 4.17 at.% to 6.25 at.% (Figure 3(d)), the CBM continuously moves toward the Fermi level and hybridization among Fe 3d, Ti 3d, and O 2p near the Fermi energy level also occur, resulting in a reduced band gap. Figure 5 shows the relationships between E g and W VB and Fe concentration.E g decreases with an increase in the Fe doping level, which is similar to other experimental results [18,19].It also reveals that the decrease in E g from 2.08 at.% to 6.25 at.% is more obvious than that from 0 to 2.08 at.%. W VB increases with an increase in Fe concentration due to the contribution from the lower Fe 3d band, which benefits the hole mobility in VB.As a result, the electron transition energy from the valence band to the conduction band decreases as a result of Fe-doping, which may induce a red shift at the edge of the optical absorption range.In addition, the valence band was found to broaden after Fe was incorporated into TiO 2 due to the contribution from the lower Fe 3d states.W VB broadens with an increase in Fe concentration.The wider valence band results in an increase in the mobility of the photogenerated electron-hole pair.In this manner, both the narrowing of the band gap and the increased mobility of the photogenerated carriers can improve the photocatalytic activity under visible light.

Conclusions
This study used the GGA + U method to investigate the influence of doping concentration on the crystal structure, impurity formation energy, and electronic properties of Fedoped anatase TiO 2 .We adopted the effective Hubbard U values, 8.47 eV for Ti 3d and 6.4 eV for Fe 3d, to accurately determine the band gap in the experiments.The calculated results imply that higher concentrations of Fe facilitate the formation of anatase TiO 2 .In addition, doping anatase TiO 2 with Fe can effectively narrow the band gap, thereby increasing photocatalytic activity in the visible light region and the trend of band gap decreases with an increase in Fe concentration.Both the broadening of the valence band and Fe impurity states within the band gap might also enhance photocatalytic activity.

2 Figure 2 :
Figure 2: Relationship between the effective Hubbard parameter (U eff ) and the band gap (E g ) for anatase TiO 2 and Fe 2 O 3 .

Figure 3
indicates the total density of states (TDOS), and the projected density of states (PDOS) to investigate the electronic properties of Fe-doped anatase TiO 2 .The zero-point energy is taken as the Fermi level.The band gap (E g ) of pure anatase TiO 2 is 3.21 eV, as shown in Figure3(a), and is consistent with the experimental value of 3.2 eV.In pure anatase TiO 2 , the valence band (VB) mainly comprises O 2p states with a small number of Ti 3d states, while the conduction band (CB) comprises Ti 3d states with a small number of O 2p states.This indicates that there exists a little covalence bond character between Ti and O atoms.The valence band of TiO 2 has a large bandwidth (W VB ) of approximately 4.63 eV, showing among the O 2p electrons.At 2.08 at.% (Fe concentration) from Figure 3(b), we observe Fe 3d impurity states in the band gap ranging from 1.74 eV above the valence band maximum (VBM) to 0.52 eV below the conduction band minimum (CBM).

Table 1 :
2 , μ Ti and μ O satisfy the relationship μ Ti + 2 μ O = μ TiO2 .Under the O-rich growth condition, μ O is determined by the total energy of an O 2 molecule (μ O = μ O2 /2) and μ Ti is determined by the formula μ Ti = μ TiO2 − 2 μ O .Under the Ti-rich growth condition, μ Ti is the energy of one Ti atom in bulk Ti and μ O is determined by μ O = (μ TiO2 − μ Ti )/2.The values of μ Ti are −1601.80and−1592.61eVunder the O-rich and Ti-rich, respectively.μFe is the energy of one Fe atom in bulk Fe, and the calculated value is −856.33 eV.Table 2 summarizes the calculated formation energies for TiO 2 doped with various concentrations of Fe.It should be noted that the smaller the E Optimized lattice parameters, average bond lengths, and volume difference (ΔV ) of TiO 2 doped with various concentrations of Fe.
f value is, the easier it is to incorporate impurities into the TiO 2 supercell.The formation energy of Fe-doped TiO 2 is reduced to a greater degree under O-rich conditions than under

Table 2 :
Formation energy of Ti 1−x Fe x O 2 for different Fe concentrations.