1. Introduction

In recent years, photocatalytic oxidation technology has been found capable of effectively removing pollutants^[1]. Photocatalytic technology is the core of photocatalysts. Anatase TiO₂ is in the focus of photocatalytic research because of its strong oxidizing property, high catalytic ability, etc. The anatase TiO₂ surface has an important impact on photocatalytic activity, photovoltaic properties, and gas sensor characteristics. Hence, an intensive study of the anatase TiO₂ surface is very significant. The most stable surface of anatase TiO₂ is the (101) surface, followed by the (100) and (001) surfaces. Hebenstreit et al.^[2] and Hengerer et al.^[3] have conducted a study on the (101) surface of anatase TiO₂ by low-energy electron diffraction and scanning tunneling microscopy (STM), and have obtained the STM images of the anatase surface in a vacuum environment. Some four coordination Ti atoms are found to exist on the surface, containing O vacancies. Ma et al.^{[4, 5]} have studied the anatase TiO₂ (101) surface and superficial native defects. They have concluded that Ti and O point defects easily form on the outer surface. Emanuele et al.^[6] and Chen et al.^{[7, 8]} have studied the N doping anatase TiO₂ (101) surface. N-doping the surfaces with O vacancies is found to be easy. With increased doping concentration, the forbidden bandwidth decreases. However, there is not much research on the optical character of the anatase TiO₂ (101) surface. The current paper provides basic theoretical data for experimental studies on the electronic and optical properties of the anatase TiO₂ (101) surface by first-principle calculations as well as analyses of the geometric structure, surface energy, electronic structure, bond length, electronic populations, and optical character.

2. Computational details

The anatase TiO₂ space group is I41/amd, which has a square crystal structure. Every primitive unit cell contains four Ti atoms and eight O atoms. All calculations based on the density functional theory (DFT) and plane wave ultrasoft pseudopotential method were performed using the VASP package^[9].

The model was built as follows. First, to establish the body model with the different calculation methods for optimization, the optimization of energy convergence criteria was set to 1 × 10$^{-5}$ eV/atom. The cutoff energy selected was 340 eV, and the Monkhorst-Pack k-point sampling was set to 7 × 7 × 3 in the supercells. The results are shown in Table 1. A comparison of the lattice constant values calculated using different methods and the experimental ones showed that the generalized gradient approximation (GGA) and the PBESOL scheme had the minimum errors^[10]. The method-optimized parameters for a = 0.3776 nm, c = 0.9621 nm, 2θ = 155.162$^\circ$. Compared with the experimental value, the errors, all less than 1.2%, were $-0.24$%, 1.12%, and $-0.68$%, respectively.

The PBESOL scheme was chosen as the optimal calculation method in the current paper. Diebold^[17] has proposed that the share of the anatase TiO₂ (101) surface is more than 90%; thus, the (101) surface was established based on the optimized body configuration. Considering that bond breaking situation and atomic positions of the surfaces is different, six different surface atomic termination structures existed on the vertical (101) surface, as shown in Fig. 1. For the six structures, the crystal surface cells were established, the atomic number of unity was 12 and the chemical formula was Ti₄O$_{8}$. Using different methods, the difference energy between the different surface atomic termination structures and bulk structures with same number of atoms were calculated (Table 2). Based on the general principle of minimum energy, the surface that is most likely to appear was selected. The outer layer of the surface was composed of two coordination O atoms, and the second layer was composed of five coordination Ti atoms. Build 18 layers atoms periodic structure from 1 × 2 supercell models. The bottom two atom layers are fixed and a vacuum level thickness of 8 Å was used before geometric structure optimization.

3. Results and discussion

3.1 Geometric structure of the anatase TiO₂ (101) surface

Table 2 shows that after the optimization of the anatase TiO₂ (101) surface, class C and F structures (Fig. 1) have the lowest energies. This finding means that anastase surface atoms terminated by the class C structure are most likely to appear on the surface. The outer layer of the surface is composed of two coordination O atoms, and the second layer is composed of five coordination Ti atoms. This result is consistent with that of Ma et al.^[4]. The discrepancy possibly originates from the different selection models and computational methods. The result is consistent with that of Hebenstreit et al.^[2], who has used STM to observe the anatase TiO₂ (101) surface. Two coordination O ions, five coordination Ti ions, and a small amount of four coordination Ti ions on the anatase TiO₂ (101) surface are found. The small amount of four coordination Ti ions observed in experiments may be due to O vacancy defects.

3.2 Surface energy

The selected number of atomic layers and layers on the vacuum surface greatly influence the surface energy. Ma et al.^[4] have concluded that the surface begins to converge when surface atomic layers are more than three layers, and the vacuum layer is thicker than 4 Å. To facilitate an accurate calculation of the surface energy, the computer ability and computational accuracy must both be considered. This paper establishes an 18-layer surface atoms model with upper and lower vacuum thicknesses of 8 Å to remove upper and lower surface interactions. The surface energy definition is:

$\begin{equation} E_{\rm sur} =\frac{1}{2A}\left(E_{\rm slab} -\frac{N_{\rm slab} }{N_{\rm bulk} }E_{\rm bulk} \right), \end{equation}$

(1)

$E_{\rm sur}$ is the surface energy of the film surface, A is the surface area of the cell surfaces, $E_{\rm slab}$ is the surface crystal cell total energy, $N_{\rm bulk}$ is the number of molecules in a unit cell, $N_{\rm slab}$ is the surface crystal cell molecule number, and $E_{\rm bulk}$ is the total energy of the unit cell.

By the calculated surface crystal cell, the surface area for A is 5.56 × 7.60 Å$^{2}$. The bulk unit cell contains a molecular number $N_{\rm bulk}$ = 4 (chemical formula, Ti₄O$_{8})$. The number of surface crystal cell molecules $N_{\rm slab}$ = 12 (chemical formula, Ti₁₂O₂₄. The computation for the superficial total energy is 0.580 J/m$^2$. Compared with Ref. [4], this result of 0.12 J/m$^2$ is slightly higher.

3.3 Relaxation of anatase TiO₂ (101) surface atoms

The anatase TiO₂ (101) surface structure is optimized by relaxing surface atoms to eliminated surface atomic tension. The surface atomic position is adjusted to reduce the total energy of the system. Figure 2 shows the structure before and after optimization. After optimization, the atomic surface underwent relaxation, but not reconstruction. The top layer of the surface is still a two-coordinated O (marked as O2c), the sub-layer is a three-coordinated O (marked as O3c), and the third layer is a five-coordinated Ti (marked as Ti5c). After relaxation, the nearest neighbor titanium atoms moved 0.136 Å toward the inner layers and the nearest neighbor oxygen atoms moved 0.164 Å toward the surface, as shown in the arrow symbols.

3.4 Surface electronic structure

Figure 3 shows the band structure and density of states of anatase TiO₂ (101) surface and bulk. Comparing the energy band diagrams in Figs. 3(a) and 3(c) reveal that the bulk energy gap $E_{\rm gap}$ is 2.14 eV. The current article utilizes a scissors operator^[18] to carry on the revision. The modification factor is 1.06 eV (3.2 to 2.14 eV). The energy band gap $E_{\rm gap}$ of the TiO₂ (101) surface is 2.50 eV; relative to the bulk band gap this is larger by 0.36 eV. In the lateral and inner surfaces, the electronic wave function changes in index relations attenuation. This finding indicates that the distribution probability of the electron is biggest in the surface, i.e., the electron is limited near the surface. This kind of electronic state is called the surface state, and the corresponding energy level is called the surface level. Figures 3(c) and 3(d) clearly show that in the surface, no surface level appears near the Fermi level, indicating that no surface states exist. These changes can be mainly attributed to the redundant electronics of the anatase TiO₂ (101) surface transfer to the other atoms. Hence, the anatase TiO₂ (101) surface is relatively stable. The valence band energy in the surface near the Fermi level is from 0 to $-4.4$ eV and the width is 4.4 eV, which is mainly attributed to the O2p orbit and Ti3d orbit hybridization. The conduction band energy is from 2.5 to 4.2 eV and the width is 1.7 eV, mainly constituted by the Ti3d track and a small amount of O2p track. The width of the conduction become smaller and the electrons are locally enhanced near the TiO₂ (101) surface. Compared with the band of the bulk phase, the anatase TiO₂ (101) surface band is smoother, and the corresponding surface state density peak is higher than that of the bulk phase. This result can be attributed to two reasons: first, the system is made up of different numbers of atoms; second, the band gap width changes strengthen the surface electronic density of states and localization.

3.5 Bond lengths and charge population

The bond lengths and charge population of ideal and relaxation surfaces are shown in Table 3.

Table 3 shows that the bond lengths significantly change after surface relaxation. The most obvious change is in the Ti5c-O2c bond length, which reduces 0.171 Å. The Ti6c-O2c bond length reduces 0.099 Å, which can mainly be attributed to the O2c ion losing the linked Ti ion above it. The outermost layer of the surface is O2c, which indicates that this surface very easily forms O vacancy defects.

Table 4 shows that the TiO₂ (101) surface atomic charge distribution has also changed. Extranuclear electron transfer in O2c and Ti5c is relatively larger. The charge population near O2c increase 0.05e, the charge population near Ti5c increase 0.03e, and the charge in the vicinity of Ti6c reduce 0.03e. The charge transfer of other nearby atoms is relatively small. The general trend of charge transfer is from surface to inner atoms.

3.6 Optical properties

After the computational optimization of the TiO₂ (101) surface, the real and virtual parts of the dielectric function versus. photon energy change curve are shown in Fig. 4. There are three peak positions in the imaginary part. The first high and sharp peak appears at 2.9 eV because the valence band O2p states jump to the conduction band Ti3d states, according to analyses of the energy band diagram and density of states. The second and third peaks are at 19.5 eV and 35.8 eV mainly owing to the electron transition from the O2s state to the Ti3d state.

Figure 5 shows no absorption in the low-energy region. An absorption edge, which gradually appears near 2 eV, is observed near the ultraviolet region. This finding illustrates that the TiO₂ (101) surface very weakly absorbs long-wavelength energy, such as visible light.

Considering that the correction factor is 1.06 eV, the corresponding energy is 3.06 eV after correction of the absorption peaks. The corresponding absorption wavelength is calculated as: $\lambda$ = $hc/E$ = 405 nm. The result suggests that the material can stably absorb wavelengths below 405 nm, consistent with the results of the band structure and density of state analyses. To produce an absorption red-shift, the response to visible light needs to be enhanced using surface doping methods. This conjecture is consistent with Chen and Tang^[7], who have performed theoretical calculations of the N-doped TiO₂ (101) surface to achieve a visible light response.

The TiO₂ (101) surface reflection spectrum is shown in Fig. 6. There are two reflection peaks at 4.5 and 36.1 eV. Figure 7 is the energy loss spectra. The energy loss spectrum describes the energy loss of electrons when passing through a uniform dielectric, and the energy loss peak describes the plasma resonance frequencies. The main energy loss is at about 7.6 eV. Figures 6 and 7 reveal that the energy loss peaks correspond to the sharp decline in the reflection spectrum.

4. Conclusion

In the current paper, analyses of the geometric structure, surface energy, electronic structure, bond length, charge population, and optical properties of the anatase TiO₂ (101) surface were performed by the first-principle method based on the DFT with the plane-wave ultrasoft pseudopotential method. The calculation results show that:

(1) The most likely structure of the anatase phase TiO₂ (101) surface is terminated by two-coordination O atoms, and the sub-layer is terminated by five-coordinated Ti atoms. This superficial termination is the most stable and has the lowest energy.

(2) The energy of an 18-layer surface model is 0.580 J/m$^{2}$.

(3) The surface electronic structure is similar to that of the bulk, and the surface state does not appear. The width of the surface band gap increases 0.36 eV compared with the bulk phase.

(4) Changes in surface bond lengths are obvious after relaxation, with the Ti5c-O2c bond length reducing about 0.171 Å. Surface electrons transfer to the body. Surface atomic relaxation occurs without reconstruction. After relaxation, the top layer of surface atoms is still two-coordination O, the sub-layer is a three-coordination O, and the third layer is a five-coordination Ti.

(5) A series of optical parameters such as real and virtual parts of the dielectric function, absorption coefficients, reflex spectra, and energy loss spectra have been analyzed in the paper. There is no energy absorption in the low-energy region. An absorption edge appears near the ultraviolet region. The corresponding energy is 3.06 eV and the corresponding absorption wavelength is below 405 nm by calculation.