Dated: Received on November 15, 2017; Accepted on December 15, 2017
School of Chemistry and Chemical Engineering, Shandong University, Jinan 250100, China
*Author to whom correspondence should be addressed.
Yu-chen Ma, E-mail:myc@sdu.edu.cn
^{†}Part of the special issue for "the Chinese Chemical Society's 15th National Chemical Dynamics Symposium"
Ⅰ. INTRODUCTION
Anatase TiO$_2$ is more favorite than its rutile isomer in photovoltaic and photocatalytic applications [1, 2]. However, anatase has lower solar energy absorption ability than rutile because its optical band gap is reported to be 3.2 eV, being 0.2 eV larger than rutile [2-4]. The higher activity of anatase, theoretically, is usually assigned to its indirect electronic band gap which prevents recombination of photogenerated electron and hole. Both density functional theory (DFT) and the more advanced GW method within the many-body Green's function theory have presented that anatase has an indirect electronic band gap, while rutile is a direct-band-gap solid [5-7]. Sun et al. put forward another mechanism to account for the better water splitting quantum efficiency of anatase than rutile [8]. They find that the O2p state belonging to water molecule is closer to the valence band maximum (VBM) of anatase than rutile, making hole trapping more favorable in anatase surface than rutile. Defects are generally regarded as another crucial factor to affect the photocatalytic behaviour of TiO$_2$ [9]. Although there have been intense researches on defects in TiO$_2$, many fundamental problems related to defects in TiO$_2$ remain to be solved.
In the past two decades, a great deal of theoretical effort has been made on TiO$_2$. DFT with the local or semi-local exchange-correlation functionals underestimates the band gap greatly. Hybrid functional methods, such as B3LYP [10, 11] and HSE [12], or DFT+$U$ are usually applied in calculations on TiO$_2$. For example, HSE predicts a 3.6 eV band gap for the defect-free anatase TiO$_2$ bulk using suitable parameters, which seems at the level of GW method [5]. For the defective TiO$_2$, such as hydroxylated anatase (Hy-ana) and anatase with oxygen vacancy (Ov-ana), B3LYP, HSE and DFT+$U$ yield localized Ti3d defect states 0.7$-$1.1 eV below the conduction band minimum (CBM) [13-16]. However, these results on defects are sensitive to the $U$ value or the percentage of exact Hartree-Fock exchange interaction in the hybrid functionals. First principle investigation on the defective TiO$_2$ by the parameter-free GW method is still rare due to its large computational cost [17-19]. Ataei et al. studied electronic and excitonic properties of the hydrogen-doped anatase TiO$_2$ bulk with the GW method and Bethe-Salpeter equation [17]. Malashevich et al. examined the stability of different charge states of oxygen vacancy in the rutile TiO$_2$ bulk by GW [18]. Zhang et al. calculated the quasiparticle band structure of the rutile TiO$_2$ bulk with Nb impurity by the all-electron GW method [19]. There is still no report on GW studies related to oxygen vacancy and Ti interstitial in anatase.
Another issue concerning theoretical studies on TiO$_2$ is the notable difference between the calculated band gap and the experimental spectroscopies, even for the state-of-the-art GW method. The direct (indirect) band gap of anatase is estimated to be 4.14 eV (3.56 eV) and 4.29 eV (3.83 eV) within GW by Kang et al. and Chiodo et al. [6, 7]. However, the optical gap of anatase is measured to be 3.2 eV in experiments [2-4]. Origin of this 1 eV disparity is still unclear. Some researchers attribute it to the electron-phonon coupling effects [5-7]. But the phonon energy at room temperature should not be large enough to fill this gap. The electron-hole binding energy, which is reported to be several meV by experiments for TiO$_2$ [6], is not the possible cause, either. Performance of the GW method on TiO$_2$ is therefore doubted [20]. Further studies are required to explain the gap between GW method and experiments.
In this work, we systematically investigate the electronic properties of the defective anatase TiO$_2$, including Hy-ana, Ov-ana and Ti interstitial (Ti-int-ana), using the GW method. Our calculations could explain very well the difference in the band gap of anatase between the GW method and experiments. This can not only help us understand TiO$_2$ in depth, but also make us more confident in the GW method.
Ⅱ. COMPUTATIONAL DETAILS
Anatase TiO$_2$ belongs to the I4amd spatial group. Its experimental lattice parameters are $a$=$b$=3.784 Å, $c$=9.515 Å [21], which are taken in the calculations of this study. Its conventional cell is in a tetragonal structure, including 4 Ti atoms and 8 O atoms, as displayed in FIG. 1(b). The primitive cell, however, is in a triclinic structure (FIG. 1(a)), containing 6 atoms (2 Ti atoms and 4 O atoms). We employ the primitive cell for the calculations on the intact bulk. Studies on the defective anatase are carried out in a 3$\times$3$\times$1 supercell constructed based on the conventional cell. Two H atoms (one Ti atom) are added into the supercell to build Hy-ana (Ti-int-ana), as shown in FIG. 1 (c) and (d). Ov-ana is modeled by removing an oxygen atom from the supercell (FIG. 1(e)). Structural relaxations are performed by the conjugate gradient method using the Vienna ab initio simulation package (VASP) [22, 23] within the local density approximation (LDA) [24]. A 2$\times$2$\times$2 $k$-mesh is sampled in the first Brillouin zone. Energy cutoff for the wave function is set to 400 eV. Relaxations are stopped when the change in total energy between successive steps is less than 1 meV.
GW calculations are executed at the level of G$_0$W$_0$ with DFT-LDA as the starting point using a Gaussian-orbital based code [25, 26]. Construction of the matrix elements for polarizability and self-energy, which involves band summation over a large number of unoccupied bands, is the most time-consuming part of GW. For calculations on the primitive cell, all the unoccupied bands are taken into account to get the most accurate band structure for bulk anatase. For studies on the larger defective systems, summation over the unoccupied bands is truncated at certain band, whose value has been well tested, in order to save the computational cost. This truncation leads to 0.25 eV underestimation of the VB-CB band gap. However, the error caused by this truncation on the gap between the band-gap defect state and CB, which we are most interested in, is little, much less than 0.1 eV. The dynamical electronic screening is described by the random phase approximation and plasmon-pole model. The $k$-point meshes of 5$\times$5$\times$5 and 3$\times$3$\times$3 are applied to the bulk primitive cell and defective systems in GW, respectively.
Ⅲ. RESULTS AND DISCUSSION
FIG. 2 illustrates the band structure of the anatase bulk primitive cell. In both LDA and GW, anatase bulk primitive cell exhibits an indirect band gap, with CBM at $\Gamma$ point and VBM lying between $\Gamma$ and X points. In LDA, the direct and indirect band gaps are 2.30 and 1.86 eV, respectively. In G$_0$W$_0$, these two gaps are enlarged to 4.28 and 3.84 eV, consistent with the previous GW results by Kang and Hybertsen (4.14 and 3.56 eV) and Chiodo et al. (4.29 and 3.83 eV) [6, 7].
FIG. 3 shows the GW band structures of the defective systems, Hy-ana, Ti-int-ana and Ov-ana based on the LDA-optimized geometries. As mentioned above, band summation truncation in GW leads to 0.25 eV underestimation of the VB-CB band gap. In the band structures of FIG. 3, this error is complemented to have a reasonable comparison with experiments. The most pronounced effect of these defects on the band structure of anatase is that the VB-CB band gap is narrowed by a large amount. The direct band gap at $\Gamma$ point reduces from 4.28 eV for the primitive cell to 3.2 eV for the defective system, while the indirect band gap decreases from 3.84 eV to 3.0 eV. The 3.2 eV direct band gap coincides with the experimental optical absorption edge of anatase. Optical absorption edge corresponds to the momentum-conserving transition at $\Gamma$ point. It has been long discussed that both GW and HSE overestimate the band gap of anatase. In our work, the GW direct band gap of bulk anatase at $\Gamma$ point deviates from the optical absorption edge of real anatase specimens in experiments by 1 eV. Defects, including oxygen vacancy and Ti interstitial, inevitably exist in the real anatase specimen. It seems that the disparity between theoretical calculations and experiments is due to the difference in the objects investigated, i.e. perfect bulk in theory and defective one in experiment. This remarkable difference in the band gap between the perfect and defected anatase bulk can also be discerned in the GW study on Hy-ana by Ataei et al. [17]. In their work, the indirect band gap of perfect anatase bulk is 3.83 eV, the same as ours, while the hydrogen-doped anatase bulk is about 3 eV. Our observation on this respect proves that GW method can still exhibit high accuracy in TiO$_2$. This is helpful for the method development in GW.
Hy-ana and Ov-ana create two Ti3d electrons. They occupy an orbital which is very close to the CB bottom in GW (FIG. 3 (a) and (c)). Ti-int-ana produces two filled defect states in the vicinity of CB edge in GW (FIG. 3(b)). From their band structures, the Fermi level is near the bottom of CB, which is consistent with the experiments and explains the n-type semiconducting character of the reduced TiO$_2$. Position of the defect state for Hy-ana in our calculations agrees with the GW result from Ataei et al. Our GW results on Ov-ana differ from previous DFT+$U$ and B3LYP spin-polarization calculations. Various $U$ values have been applied in investigations of defective anatase. The optimal $U$ value for DFT within the generalized-gradient approximation (GGA) is suggested to be 4.1 eV by Setvin et al. from first principles using the constrained random phase approximation [14]. With GGA+$U$ ($U$=4.0 eV), Finazzi et al. predicted two very deep band-gap states, one 1.07 eV and the other 1.46 eV below CBM [15]. When using $U$=3.0 eV, Finazzi et al. got two sets of vacancy configurations, the stable one contains two defect levels at 0.57 and 0.95 eV below CBM, while the less stable one positions the two defect levels 0.15 and 0.94 eV below CBM. By GGA+$U$ with $U$=3.4 eV, Mattioli et al. suggested that Ov-ana creates two band-gap states at 0.1 and 1.0 eV below CBM [13]. With B3LYP, excess electrons of the most stable Ov-ana are found to occupy two levels which are 1.16 and 1.28 eV below CBM. Overall, GGA+$U$ and B3LYP approaches prefer to put the two excess electrons of Ov-ana into deep band-gap states. Although this seems to be consistent with experimental observations on the band-gap state, it cannot account for the n-type semiconducting behaviour and the infrared absorption of anatase [27].
LDA has been thought to give too delocalized defect states and too less lattice distortion for defects in TiO$_2$ [13, 15]. DFT+$U$, B3LYP and HSE predict larger lattice distortion induced by hydroxyl groups and oxygen vacancy than LDA, which has considered to be one of the key factors why LDA cannot reproduce the experimental deep band-gap state in TiO$_2$ [28]. In LDA, the Ti3d defect state of Ov-ana is indeed delocalized over the entire supercell. We go one further step beyond G$_0$W$_0$. We setup the full matrix, including both the diagonal and off-diagonal elements, of the G$_0$W$_0$ Hamiltonian which is then diagonalized to yield more accurate quasiparticle energies and wave functions. This can be regarded as a sort of one-step self-consistent GW procedure (osscGW) to some extent. If the LDA wave function is not a suitable approximation to the QP wave function, this could be manifested after osscGW since off-diagonal matrix elements of the GW Hamiltonian control the distinction between the starting Kohn-Sham wave function and the QP one. In our recent work on defects in the rutile TiO$_2$, we do find that osscGW could reach converged results very perfectly in comparision with more advanced self-consistent GW scheme. After employing osscGW in Ov-ana, a delocalized QP distribution for the Ti3d defect state is obtained (FIG. 4(d)), analogous to the LDA one. Similarly, QP distributions of the defect states for Ti-int-ana and Hy-ana are delocalized (FIG. 4(a)-(c)).
To examine how much the structural factor impacts the defect level, we optimize Ov-ana with LDA+$U$ ($U$=5.0 eV) additionally. At the LDA+$U$ Ov-ana structure, the defect state is localized in both LDA+$U$, LDA and osscGW (FIG. 4(e)) with the excess electrons trapped at the vacancy site. This kind of state is also called small polaron which has attracted great attention in the community of TiO$_2$ [14, 16, 29-34]. This defect state is 1.0 eV below CBM and 1.9 eV above VBM in LDA+$U$. However, in GW it is within 0.1 eV to CBM (FIG. 3(d)), resembling that of the LDA-optimized Ov-ana. The VB-CB gap at $\Gamma$ point for this polaron structure is 3.1 eV, in line with that for Hy-ana, Ov-ana and Ti-int-ana. These demonstrate that structure is not an important factor to determine the energy of defect state as expected before. Please note that the small polaron discussed here is distinct from the large delocalized polaron which is generated by doping an extra electron into the defect-free TiO$_2$ bulk as discussed in Ref.[30].
From the above discussions on GW results for Hy-ana, Ti-int-ana, Ov-ana and polaron in bulk anatase, none of them could be responsible for the deep band-gap state as observed in experiments. This is distinct from previous calculations by DFT+$U$, B3LYP, and HSE. We guess that the band-gap state must be caused by some surface defects, since the experimental tools including the photoelectron spectroscopy and scanning tunneling microscopy can only detect near-surface region. Future study on the defects in the anatase surface by the GW method is required and we are making such effort on the way.
Ⅳ. CONCLUSION
In summary, we investigate the quasiparticle band structures and wave functions of typical defects in bulk anatase by the first-principle GW method. We discover that these defects result in a huge downshift of the conduction band edge of anatase, which makes the calculated band gap agree with the experimental absorption edge perfectly. This may help to resolve the long-standing argument on the accuracy of GW method for anatase. The occupied Ti3d states arising from these defects situate in the proximity of conduction band, accounting for the pinning of Fermi level to the conduction band bottom.
Ⅴ. ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China (No.21573131, No.21433006 and No.21173130), the Natural Science Foundation of Shandong Province (No.JQ201603). Computational resources have been provided by the National Supercomputing Centers in Jinan.