First-principle study for influence of an external electric field on the electronic structure and optical properties of TiO₂

Abstract

The influence of external electric fields on the electronic structure and optical properties of TiO₂ was studied using first-principle calculations. The results showed that the TiO₂ energy gap became gradually narrower as the electric field was increased, and it decreased to 0 eV when the electric field was 0.25 eV. All peaks in the density of states gradually broadened, and extended towards low energy with increasing electric field, and overall the peaks of the density of states were split, and the split became larger. The lattice parameter c increased, whereas a and b decreased with increased electric field in the z direction. At the same time, the bond lengths of Ti–O in the z direction increased, whereas those in other directions decreased. The charges mainly transfer from the O 2p orbital to the Ti 3d orbital under an electric field. The dielectric constant, conductivity, refractive index and extinction coefficient of TiO₂ with a 0.15 eV electric field were larger than those of TiO₂ without an electric field. The absorption edges extended to the visible region, and the absorption in the visible range intensified greatly under the effect of an external electric field, which would enhance the photocatalytic activity of TiO₂ and solar energy utilization.

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1. Introduction

Titanium dioxide (TiO₂) is widely used in photocatalytic environmental remediation and solar energy conversion due to its high photoactivity, good chemical stability, and low cost.^1–4 However, technical challenges still remain in the photocatalytic application of TiO₂: (1) low efficiency solar energy utilization; TiO₂ can only absorb UV light, merely about 3–5% of the total sunlight, and, (2) the fast recombination of e⁻ and h⁺ pairs, which results in low photocatalytic activity. The photocatalytic activity of TiO₂ is due to the production of excited electrons in the conduction band, along with corresponding positive holes in the valence band by the absorption of UV radiation. These energetically excited species are mobile, and capable of initiating many chemical reactions, usually by the production of radical species at the semiconductor TiO₂ surface. However, they are unstable, and recombination of the photogenerated electrons and holes can occur very quickly (around 250 nanoseconds for TiO₂).⁵ Therefore, the time is not long enough for any other chemical reaction to occur.

To reduce recombination of photogenerated electrons and holes, and to extend its light absorption into the visible region, a lot of approaches have been studied extensively, including dye sensitization, doping, coupling and capping of TiO₂.^6–14 These studies showed that dye sensitization and coupling of semiconductors can expand the light response of TiO₂ to the visible region. Excited dyes and small band gap semiconductors can inject electrons into the conduction band of large band gap TiO₂, resulting in efficient charge separation and high photocatalytic efficiency.^6,7,11 Doping of TiO₂ can inhibit charge recombination and expand its photoresponse to the visible region through the formation of impurity energy levels.^8–10 In recent years, a number of studies showed that composite plasmonic-metal/semiconductor photocatalysts achieve significantly higher rates in various photocatalytic reactions compared with those of pure semiconductor counterparts.^15–17

Although the great progress made in TiO₂ photocatalysis has been recognized for a long time, commercial applications of these technologies are scarce. Continuous efforts to improve the photocatalytic properties of TiO₂ will allow the full potential of this photocatalyst to be realized. In several cases, the electric field effect can vary the carrier concentration in a semiconductor device, and consequently change the electric current; the electric field is clean, easily acquirable, and adjustable in both direction and intensity compared with other approaches. Novoselov et al.¹⁸ studied the electric field effect in atomically thin carbon films. The results showed that thin carbon films exhibit a strong ambipolar electric field effect such that electrons and holes were in concentrations up to 10¹³ per square centimeter. Tank et al.¹⁹ reported the effect of electric field on the photocatalytic efficiency of TiO₂ nanoparticles. The results showed that the photocatalytic efficiency increased by 18% at −3.0 V of the negative bias potentials as compared to the zero field value. In addition, the shift and broadening of the band edge absorption is usually known as the Franz–Keldysh effect induced by the electric field, which leads to a red shift of the absorption edge. As a result, the activity of TiO₂ photocatalysts can be improved and the utilization of sunlight can be enhanced in the presence of an external electric field. Therefore, it is very important and valuable to study the relationship between the external electric field and optical properties.

In this study, the influence of external electric fields on electronic structure and the optical properties of TiO₂ were investigated using first-principles calculations, including energy band, density of states (DOS), bond length, and optical properties. This study can provide the microscopic explanations for the photocatalytic mechanism for TiO₂ with the electric field from the viewpoint of electronic structure.

2. Computational details and models

Based on density functional theory (DFT), all calculations were performed using the CASTEP (Cambridge serial total energy package) program module developed by Payne et al.²⁰ The exchange correlation functional used was the generalized gradient approximation (GGA), developed by Perdew and Wang (PW91).²¹ Calculations used the DFT+U method with a value of 7.0 eV (U) in order to match the calculated band gap with the experimental value. The interactions between valence electrons and the ionic core were represented with ultrasoft pseudopotentials.^21,22 A Monkhorst–Pack k-point sampling density of 2 × 2 × 1 was used for all the absorption calculations.²³ Based on the test results, a plane wave cutoff energy of 340 eV was used in all calculations, which was the most stable state. The convergence tolerances for geometry optimization calculations were set to a maximum displacement of 0.002 Å, maximum force of 0.08 eV Å⁻¹, maximum energy change of 2.0 × 10⁻⁵ eV per atom and maximum stress of 0.1 GPa, and the self-consistent field (SCF) convergence tolerance was set to 2.0 × 10⁻⁶ eV per atom.

TiO₂ belongs to the family of transition metal oxides and exists as three different polymorphs: anatase, rutile and brookite. In all three forms, titanium (Ti⁴⁺) atoms are coordinated to six oxygen (O²⁻) atoms, forming TiO₆ octahedra. Both anatase and rutile TiO₂ have a tetragonal structure. The TiO₆ octahedron is slightly distorted, and the distortion of the TiO₆ octahedron for the anatase TiO₂ is slightly larger than that for rutile TiO₂.^24–28 Brookite TiO₂ belongs to the orthorhombic crystal system. Its unit cell is composed of eight formula units of TiO₂ and is formed by edge-sharing TiO₆ octahedra. Of the three forms, anatase TiO₂ is considered to be the active photocatalytic component based on charge carrier dynamics, chemical properties and the activity of photocatalytic degradation of organic compounds. Therefore, anatase TiO₂ was selected as the research object in this paper. The structure of anatase TiO₂ is from ICSD-76173 (ref. 29) and is shown in Fig. 1.

Fig. 1 TiO₂ unit cell.

3. Results and discussion

Influences of various electric field directions on the electronic structure and optical properties of TiO₂ were investigated, including the x, y and z directions. The changing trends of the electronic structures and optical properties for TiO₂ under the three (x, y, z) directions were found to be similar. However, the influence of electric field along the z direction on TiO₂ through an optimization test of the three directions was relatively the greatest. As a result, we only studied the electronic structures and optical properties of TiO₂ under the influence of electric field along the z direction.

3.1 Band structure and densities of states

Fig. 2 shows the band structure of TiO₂ (a) and the corresponding density of states (b). It is observed from Fig. 2(a) that the valence band maximum (VBM) is located at the M and the conduction band minimum (CBM) is located at the G point of the Brillouin zone, suggesting that TiO₂ is an indirect band gap semiconductor. The calculated band gap of TiO₂ was 3.11 eV, which is close to the experimental value (3.2 eV).³⁰ The character of bands can be evaluated by the DOS. Fig. 2(b) presents the partial densities of states (PDOS) of Ti and O in TiO₂. It was observed that the group between −60 and −55 eV is from the Ti 4s orbital, and the group between −35 and −30 eV is from Ti 4p orbital, which attributes to the hybridization of atomic orbitals. The band between −20 and −15 eV is mainly from O 2s orbital with few contributions from 4s and d–p hybrid orbitals of Ti atom. The band in the range from −5 to 0 eV is from O 2p and s–p–d hybrid orbitals of Ti atoms. The conduction band is mainly from Ti 3d orbital, only with few contributions from O 2s and O 2p states.

Fig. 2 Band structure (a) and PDOS (b) of TiO₂ without the electric field.

It is generally known that only light above the band-gap hits TiO₂ and electrons in the valence band are excited to the conduction band leaving behind holes. A photocatalyst is characterized by its ability to adsorb two reactants simultaneously, which can be reduced and oxidized by electrons and holes. Therefore, pure TiO₂ has low efficiency for solar energy utilization because of the large band gap (3.2 eV). The external electric field can change the band gap, and prevent the recombination of electrons and holes.

Fig. 3 and 4 show the band structure of anatase TiO₂ with different electric field strengths ranging from 0.10 to 0.25 eV (Fig. 3) and the corresponding density of states (Fig. 4). Because the physical processes in solids occur mainly in the vicinity of the Fermi level, the energy ranges from −20 eV to 10 eV of band structure and DOS. Clearly, the band gaps of TiO₂ strongly depend on the applied electrical field. At a lower external field (0.10 eV), TiO₂ still remains an indirect band gap semiconductor. However, as the strength of the electric field increases to 0.15 eV and 0.20 eV, TiO₂ becomes a direct band gap semiconductor. The VBM and CBM are located at the center of G and Z points of the Brillouin zone. At the same time, band gaps of TiO₂ decrease from 2.61 eV to 0.62 eV with an increase of the electric field from 0.10 eV to 0.20 eV. The band gap becomes 0 eV by further increasing the electric field from 0.20 eV to 0.25 eV. Although the DFT method underestimates the band gap due to a self-interaction error, and the actual band gap of TiO₂ should be slightly larger than the calculated values, but the relative order and trend are credible. The bandgaps decrease monotonically with increasing electric field strength, and TiO₂ becomes metallic from semiconducting when the electric field is 0.25 eV. We could see from the band structure that as the external electric field was varied, the conduction band shifted downward relative to the Fermi level for all TiO₂ under electric fields from 0.10 eV to 0.25 eV, and all energy levels became wide (dispersion). The band collapsed when the electric field was further increased to 0.28 eV. The critical fields were estimated to be about 0.28 eV. The band gap reflects the possibility of electronic transitions between the occupied and unoccupied electronic states, and to some extent, it represents the ability of atoms to be involved in chemical interaction. Narrower gap of TiO₂ with the electric fields means that it is easier for electrons to transit from occupied orbitals to unoccupied orbitals, forming holes.

Fig. 3 Band structure of TiO₂ with different electric fields. (a) 0.10 eV, (b) 0.15 eV, (c) 0.20 eV and (d) 0.25 eV.

Fig. 4 DOS results of O atoms and Ti atoms of TiO₂ with different electric fields.

The bandgap modulation may be explained by the well-known Stark effect. The Stark effect can lead to splitting of degenerate energy levels. The stronger the electric field is, the larger the band splitting is, and the smaller the band gap. These band characteristics can be evaluated by the DOS (Fig. 4). It is easily seen that all peaks corresponding to the orbital energy level in the DOS curve gradually broaden, and extend toward low energy with increased electric field strength. When the electric field was 0.25 eV, the DOS of O 2p and Ti 3d orbitals crossed the Fermi level, suggesting the strong metallic and degenerative feature of TiO₂, which is also consistent with the change of the band structure (Fig. 3). The CBM shifted towards lower energy with an increase of electric field, and crossed with VBM in the vicinity of the Fermi level when the electric field was increased to 0.25 eV, which leads to a 0 eV energy gap. In addition, the peak in the range of −20 to −14 eV in the DOS curve was split when the strength of the electric field was 0.15 eV, and the split became larger when the strength of the electric field increased from 0.15 eV to 0.25 eV. The split energy level was also found in the 0 to 5 eV range when the electric field was 0.20 eV, and the split became larger as the electric field was increased to 0.25 eV. All these behaviors that occur in the DOS curve are also observed in the energy-band structures (Fig. 3).

The physical mechanisms of the electrical field effects of the band structure of TiO₂ are attributed to the bond length changes of lattice cell and the charge transfer between Ti and O atoms induced by the electric field. The lattice distortion caused by the mutation of the lattice parameter may lead to a change of the microscopic electric structure, band structure and density of states. In addition, the crystalline natures are closely related to the bonds, band gap and optical properties of TiO₂, and are strongly influenced by the changes of Ti–O bond in TiO₂.

TiO₂ crystals displayed the structural changes under an external electric field. The changes of lattice constants are shown in Table 1. It is clear that the lattice parameter c increased, whereas a and b decreased with the increase of the electric field along the z direction from 0 eV to 0.25 eV. Table 2 shows the changes of the bond lengths of TiO₂ with different electric fields. It is clearly observed that the bond lengths of Ti–O1 in the z direction increased, whereas those in other directions (Ti–O2) decreased with increasing the electric field from 0 to 0.20 eV. The calculated results also showed that the Ti–O bonds in the z direction were broken when the electric field was increased up to 0.28 eV. The changes of the bond lengths are one of the major reasons for the serious lattice distortion.

Table 1 Lattice constants of TiO₂ with different electric fields

Electric field	Lattice constants (Å)
	a	b	c
0	3.901	3.901	9.832
0.10	3.897	3.897	9.896
0.15	3.818	3.818	10.953
0.20	3.791	3.791	11.842
0.25	3.735	3.735	12.471

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Table 2 Bond length of TiO₂ with different electric fields

Bond	Electric field (eV)	Length (Å)
Ti–O1	0	2.026
	0.10	2.110
	0.15	2.186
	0.20	2.287
Ti–O2	0	1.999
	0.10	1.979
	0.15	1.964
	0.20	1.950

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Although the bond length of Ti–O2 decreases with the increase of electric filed, the increase in the bond length of Ti–O1 is much bigger, which leads to the weak interaction between O and Ti (Fig. 4). According to Fig. 4, the distance between the bonding of O 2p and Ti 3d in the −5 to 0 eV range and the anti-bonding in the 0–5 eV range gradually became smaller with increased electric field, suggesting weak bonding between O and Ti. As a result, the bondage of Ti and O atoms from the surrounding atoms decreased, leading to an increase in activity of the Ti and O atoms. When the electric field increased to 0.25 eV, the DOS curves of O 2p and Ti 3d orbitals crossed the Fermi level.

Table 3 shows the Mulliken charge populations of O1(O2) and Ti atoms before and after applying the electric field. It was found that oxygen atom loses electrons, whereas titanium atom gains electrons under the influence of an external electric field. The loss of electrons for oxygen atom comes from O 2p orbitals. The gain of electrons for titanium is attributed to the Ti 3d orbitals. In other words, the charges mainly transfer from O 2p orbitals to Ti 3d orbitals under an electric field. The electrons lost for oxygen atoms or gained for titanium atoms increased when the electric field was increased from 0 to 0.25 eV, which shows the strong effect of the electric field. The electron transfer behavior can be characterized by the electron density difference. Fig. 5 shows the electron density difference of TiO₂ with a 0.15 V electric field. The blue colours represent electron density depletion, and the red colours correspond to increased electron density. It is evident that the red region near Ti in Fig. 5(b) decreased compared with Fig. 5(a), indicating a decrease of electron density, whereas the red region near the O atom increased, implying an increase of electron density, which is in good agreement with the Mulliken charge population results.

Table 3 Mulliken charge populations of TiO₂ before and after applying the electric field

Atomic label	Electric field	s	p	d	Charge/e
O1(O2)	0	1.85	4.88		−0.73
	0.10	1.85	4.87		−0.72
	0.15	1.85	4.85		−0.70
	0.20	1.85	4.85		−0.69
	0.25	1.85	4.84		−0.68
Ti	0	2.34	6.34	1.85	1.47
	0.10	2.33	6.34	1.87	1.46
	0.15	2.32	6.34	1.89	1.45
	0.20	2.31	6.35	1.90	1.44
	0.25	2.29	6.36	1.92	1.43

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Fig. 5 Electron density difference map of TiO₂ with 0 eV (a) and 0.15 eV (b) electric fields.

3.2 Optical properties

3.2.1 Dielectric function. Fig. 6 shows the dielectric functions of TiO₂ with 0 eV (a) and 0.15 eV (b) electric fields. The dielectric function can be expressed by the following formula:

ε(ω) = ε₁(ω) + iε₂(ω) (1)

where ε₁(ω) = Re(ω), ε₂(ω) = Imε(ω), and Reε(ω)and Imε(ω)are the real and imaginary parts of the dielectric function, respectively, and ω is the frequency. It is observed from Fig. 6 that the peak intensity of the imaginary part for TiO₂ with a 0.15 eV electric field was larger than that of TiO₂ with zero electric field, whereas the peak intensity of the real part for TiO₂ with 0.15 eV electric field was smaller than that of TiO₂ with zero electric field. The dielectric peak position of the imaginary part for TiO₂ with 0.15 eV changed slightly, whereas the dielectric peak of the real part shifted to low energy related to TiO₂ with zero electric field.

Fig. 6 Dielectric functions of TiO₂ with 0 eV (a) and 0.15 eV (b) electric fields.

3.2.2 Conductivity. Fig. 7 shows the conductivities of TiO₂ with 0 eV and 0.15 eV electric fields. It was found that the conductivity of TiO₂ with a 0.15 eV strength of electric field was larger than that of TiO₂ with zero electric field, though the peak of the imaginary part for TiO₂ with 0.15 eV electric field was slightly lower than that of TiO₂ with zero electric field. The valley value of the imaginary part for TiO₂ with 0.15 eV electric field was much larger. The conductivity of polarization current σ_p and the dielectric function ω(ω) are related through eqn (2) as follows:

σ_p = ωε₂(ω) (2)

Fig. 7 Conductivities TiO₂ with 0 eV (a) and 0.15 eV (b) electric fields.

Therefore, the conductivity is proportional to the imaginary part of dielectric function,³¹ which is in good agreement with our results (Fig. 6) and also with that of band structure, in which the band gap of TiO₂ with the electric field of 0.15 eV is smaller (Fig. 3). As a semiconductor, the conductivity of TiO₂ increases with a decrease in band gap. When light is irradiated to a semiconductor material, the photon energy larger than the band gap can excite electron–hole pairs and cause an increase of carrier concentration, resulting in increasing conductivity.

3.2.3 Refractive index. Fig. 8 shows the refractive indexes and extinction coefficients of TiO₂ with 0 eV and 0.15 eV electric fields. It reveals that the refractive index and extinction coefficient peaks of TiO₂ with the 0.15 eV electric field were higher than those of TiO₂ with zero electric field.

Fig. 8 Refractive indexes (n) and extinction coefficients (k) of TiO₂ with 0 eV (a) and 0.15 eV (b) electric fields.

The refractive index of TiO₂ can be described with a complex refractive index as follows:^31,32

n = n(ω) + ik(ω) (3)

where the real part n(ω) is the refractive index, and the imaginary part is the extinction coefficient that is related to optical absorption. The relation of complex refractive index and dielectric function is represented with the following equations:

[n(ω) + ik(ω)]² = ε₁(ω) + iε₂(ω) (4)

n(ω)² − k(ω)² = ε₁(ω) (5)

2n(ω)k(ω) = ε₂(ω) (6)

The above equations show that the intensity of the imaginary part of the dielectric function increases with an increase in intensity of the refractive index and extinction coefficient. This behavior is in good agreement with our results. Greater values of refractive index and extinction coefficient for TiO₂ with the electric field of 0.15 eV (Fig. 8) increase the peak intensity of the imaginary part of the dielectric function (Fig. 6).

3.2.4 Optical absorption. Fig. 9 shows the optical absorption spectra of TiO₂ with different strength of electric fields. It reveals that the electric field prominently affects optical absorption characteristics of TiO₂. The strong absorption at wavelengths shorter than 400 nm can be assigned to the intrinsic band gap absorption of anatase TiO₂. There exist some small differences between zero electric field and low electric field (0.10 eV) in optical absorption curves of TiO₂. The absorption peak of the latter slightly shifts toward long wavelength, and their absorptions at wavelengths longer than 400 nm are very weak. The absorption of TiO₂ intensifies greatly, and the absorption edges show a very large redshift with increasing the electric field from 0.10 eV to 0.20 eV. It was clear that the absorption edges extend to the range of visible light and the absorption in visible range intensifies greatly. The absorption of TiO₂ with a 0.25 eV electric field makes a big difference, which may be due to semiconducting TiO₂ becoming a conductor (Fig. 3).

Fig. 9 Absorption spectra of TiO₂ with different electric fields.

Tank et al.¹⁹ reported the effect of electric field on photocatalytic efficiency of TiO₂ nanoparticles. In their study, porous silicon was used as a template to immobilize the nano TiO₂ and to facilitate the application of an electric field. The target of photocatalytic degradation was methylene blue. Tank et al. drew the conclusion from a large number of trials that the photocatalytic efficiency increased by 18% at −3.0 V of the negative bias potentials as compared to a zero field value, showing that the electric field indeed increased the photocatalytic property of TiO₂. This behavior is consistent with our calculation results.

The optimal electric field for the optical properties of TiO₂ semiconductor in our calculation is 2.0 eV equivalent to the kinetic energy obtained for one electron to be accelerated by a 2.0 V potential difference. In Tank's study, the best bias potential was −3.0 V, which is larger than the result of our calculation. The main reason may be that the experimental results are affected by other factors, such as the template porosity, electrophoretic potential and time of electrophoretic deposition, whereas theoretical calculation does not consider the influence of other factors.

The rate of the photocatalytic reaction is proportional to (I_αϕ)ⁿ, where I_α is the photo numbers absorbed by the photocatalyst per second, ϕ is the efficiency of the band gap transition, n = 1 for low light intensity and n = 1/2 for high light intensity. High photocatalytic activity of TiO₂ can be due to large I_αϕ value resulting from intensive absorbance in the UV region.³³

Generally, under the application of an electric field, the electron and hole move in opposite directions at different velocities, which increases the number of photogenerated electrons and holes to participate in the photocatalytic reaction. This would enhance the photocatalytic activity of TiO₂ under the influence of electric field. In our study, the absorption curve of TiO₂ with a 0.10 eV external electric field changes very little relative to zero electric field, which may be due to a small number of photogenerated electrons and holes. The absorption edges shifted toward longer wavelengths with increasing the electric field from 0.15 to 0.20 eV, and the absorptions in the visible range intensified greatly, which was due to a relatively large number of photogenerated electrons and holes. In addition, the redshift of the absorption edges for TiO₂ with the role of the electric field also clearly indicated a decrease in the band gap energy caused by the electric field, which is in agreement with the energy band structure (Fig. 3). In short, TiO₂ with the electric field of 0.15 to 0.20 eV expands the wavelength response range of TiO₂ to the visible region, and increases the number of photogenerated electrons and holes, which can participate in the photocatalytic reactions. All this would enhance the photocatalytic activity of TiO₂.

4. Conclusion

The influence of external electric fields on electronic structure and optical properties of TiO₂ was studied using first-principle calculations. The energy gap gradually becomes smaller on increasing the electric fields from 0 eV to 0.20 eV, until eventually the energy gap becomes 0 eV when the electric field increases to 0.25 eV. All peaks of the density of states gradually broaden and extend toward low energy with increased electric field strength. At the same time, the peaks in the density of states generally are split, and the split becomes larger on increasing the electric field. TiO₂ yields distinct lattice-constant changes and the structural changes under an external electric field. The bond lengths of Ti–O in the z direction increase with the increasing the electric field from 0 to 0.20 eV, and are broken when the electric field is increased up to 0.28 eV. The charges mainly transfer from O 2p orbital to Ti 3d orbital under an electric field. The optical properties of TiO₂ in the near ultraviolet region (100 nm to 380 nm) are better, whereas they are relatively poor in the visible region (380 nm to 650 nm) without the electric field. However, the absorption edge expands to the visible region, and the strength of the optical absorption in visible region increases under the effect of an external electric field.

Acknowledgements

This research was supported by the Guangxi Natural Science Foundation (No. 2014GXNSFAA118342) and the Open Foundation of the Guangxi Key Laboratory for Advanced Materials and Manufacturing Technology.

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