Review of First-Principles Studies of TiO 2 : Nanocluster, Bulk, and Material Interface

: TiO 2 has extensive applications in the ﬁelds of renewable energy and environmental protections such as being used as photocatalysts or electron transport layers in solar cells. To achieve highly efﬁcient photocatalytic and photovoltaic applications, ongoing efforts are being devoted to developing novel TiO 2 -based material structures or compositions, in which a ﬁrst-principles computational approach is playing an increasing role. In this review article, we discuss recent computational and theoretical studies of structural, energetic, electronic, and optical properties of TiO 2 -based nanocluster, bulk, and material interface for photocatalytic and photovoltaic applications. We conclude the review with a discussion of future research directions in the ﬁeld. (112)A/(111)(S/B)TO, and (100)R/(111)(S/B)TO—based on the experimental ﬁndings, volumetric formation and strain, and areal substrate, and then calculated their interface energies. Their computational models predict a decrease order of anatase stability from (001) to (011) to (111) perovskite substrates, which is in a general agreement with the experimental epitaxial stability investigations. In contrast, rutile is energetically more favorable than anatase on the (111) perovskite substrate. This work indicates that


Introduction
Titanium dioxide (TiO 2 ) is an ideal semiconductor photocatalyst because of its excellent properties (e.g., high activity, good stability, nontoxicity, and low cost), which has many promising applications in the fields of renewable energy and environmental protections [1][2][3][4][5]. However, the larger bandgap of TiO 2 (~3.2 eV for anatase and~3.0 eV for rutile) makes it inefficient for visible light to excite the electron-hole pairs, which are necessary to initiate a photocatalytic process [2][3][4]. Therefore, the photocatalytic applications of TiO 2 in the visible light range are heavily limited. To improve the spectra response and photocatalytic activity of TiO 2 in the whole solar spectra, numerous efforts have been carried out [2][3][4][6][7][8]. For instance, in recent years, a number of attempts have been made to improve the visible light absorption of TiO 2 by nonmetal doping, which either introduces some impurity states in the bandgap or modifies the fundamental bandgap of TiO 2 , and promote the photocatalytic activity of TiO 2 to some degree [4,[6][7][8][9][10]. To achieve high photocatalytic efficiency, in addition to increasing the number of photoinduced electron-hole pairs, another approach is to increase the separation rate of photoinduced electron-hole pairs in TiO 2 . To do so, increasing efforts are being made to synthesize TiO 2 -based materials heterostructures and/or composites that utilize the interfacial charge transfer mechanism to promote the separation rate of electron-hole pairs [11][12][13][14][15]. A high charge separation rate is beneficial not only for photocatalysts, but also for photovoltaic cells in which TiO 2 is often used as electron transport layers as they both use the separated electrons and holes [16][17][18][19][20][21][22][23][24].
In this review article, we review and discuss recent computational and theoretical progress on the TiO 2 nanoclusters, bulk, and material interfaces from the viewpoints of first-principles calculations. This review is divided into three sections. First, we review the energetic stability and electronic properties of TiO 2 nanoclusters. Second, we focus on the nonmetal doping and co-doping effects on  [26]. Copyright (2020) Elsevier.
In 2017, Gan et al. reported a first-principles computational study on the mechanism of photo-selective catalytic reduction of 4-bromobenzaldehyde (4-BBA) in different solvents using OH-defected TiO 2 cluster model [34]. In this work, a neutral cluster model of Ti 3 O 9 H 6 was used to represent the reaction sites with photocatalytic activity on TiO 2 surfaces. The authors selected the Ti 3 O 9 H 6 cluster based on three considerations: (i) the dangling bonds of TiO 2 clusters can be saturated by replacing the Ti-O bonds with the Ti-OH bonds without altering Ti 4+ oxidation state; (ii) the chemical environment of each Ti atom is equivalent and the small cluster size also saves computational cost; and (iii) the surface species such as OH defects on the TiO 2 surface were already involved in the Ti 3 O 9 H 6 cluster. This work shows that the Ti 3 O 9 H 6 cluster is an ideal molecular model to study photocatalytic reactions on the TiO 2 surface, which servers as an important reference in future work. TiO 2 photocatalysts are usually nanosized materials with highly hydrated surfaces. Hydrogen is thus one of the most common impurities and hydroxyl groups of different nature often appear on TiO 2 surfaces. The size-dependent evolution of the chemical reactivity and site selectivity of (TiO 2 )n nanoclusters (n = 5-10) toward hydrogen peroxide were explored from first-principles computational studies by Mohajeri et al. [25]. The adsorption energy of H 2 O 2 onto the TiO 2 nanoclusters is found to have a converging trend at n ≥ 8. After adsorption, the H 2 O 2 can be easily decomposed into two OH radicals that can be stabilized through forming bonding interactions with Ti and O atoms on the cluster surface.
The O 2 molecule adsorption on the TiO 2 nanoparticles was also modeled from first-principles DFT calculations [35]. The authors studied two possible adsorption modes: The first one is that O 2 was Catalysts 2020, 10, 972 4 of 34 adsorbed on TiO 2 nanoclusters via the interaction between H and O atoms. This reduction process of TiO 2 nanoparticles have three following features: (i) forming stable OH group, (ii) low activation energy, and (iii) introducing Ti 3+ ion related states within the energy gap. Figure 2 shows the structural model, electronic structure diagram, and spin density surface of Ti 8 O 16 cluster. The second one is that O 2 was directly adsorbed on each Ti 3+ ion of the reduced Ti 8 O 16 H cluster. This process shows three features: (i) formation of stable O 2 -species, (ii) no any energetic barrier, and (iii) no Ti 3+ defect states within the energy gap. These calculated results are generally consistent with the experimental phenomena of H plasma reduction and interaction between reduced TiO 2 and O 2 . Figure 2. Illustration of (a) geometrical structure, (b) electronic structure diagram, and (c) spin density surface of Ti 8 O 16 cluster. Reprinted with permission from the authors of [35]. Copyright (2012) American Chemical Society.

Bulk
In 2001, Asahi et al. reported the visible light photocatalytic activity of nitrogen-doped TiO 2 [9,36]. Soon after, various nonmetal-doped (including B, C, Si, P, S, and halogen elements) TiO 2 were studied to explore their photocatalytic performance under visible light. Although most of these nonmetal-doped TiO 2 show the visible light optical absorption and photocatalytic activity in some degree, there are some "tricky" fundamental issues in their experimental optical absorption spectra and visible light photocatalytic mechanism. For example, what leads to the controversy on the origin of the visible light absorption in N-doped TiO 2 ? Why can B-doped TiO 2 show both redshift and blueshift of the optical absorption edge? What is responsible for the different optical absorption thresholds in C-doped TiO 2 ? Which form (anion or cation) of the Si (P and S) ion in TiO 2 is more effective to promote the visible light absorption? Can F-doping lead to the visible light absorption of TiO 2 ? Why can Cland Br-doped TiO 2 show the stronger photocatalytic ability than that of undoped TiO 2 ? What is the difference on the mechanism of the visible light absorption between the I-anion and I-cation doped TiO 2 ? To answer these questions, first-principle theoretical calculation is an effective approach and it has successfully explained many experimental phenomena, including the issues mentioned above. In this section, we review the recent theoretical progress in understanding the electronic structure and optical absorption of nonmetal-doped TiO 2 from first-principles calculations. A brief summary of commonly used nonmetal dopants and the resulting properties of doped TiO 2 is shown in Table 1.

Formation Energy
First-principles DFT calculations for the relative stability of nonmetal-doped TiO 2 can help us understand the formation of the doped structures and provide useful guidance to prepare samples [37][38][39][40][41][42][43][44][45]. First, let us focus on the structural stability of substitutional doping models. In principle, there are two possible substitutional doping ways for X-doped TiO 2 , i.e., an X anion at an O site (X@O) or an X cation at a Ti site (X@Ti). The defect formation energy required for X substituting for either O or Ti in TiO 2 could be calculated from the following formulas, respectively [43][44][45].
E X−doped is the total energy of X-doped TiO 2 and E undoped is the total energy of undoped TiO 2 . µ X is the chemical potential of dopant X, and µ O (µ Ti ) is the chemical potential of the O (Ti). The chemical potentials of Ti and O depend on whether TiO 2 is grown under an O-rich or Ti-rich growth condition. Under the Ti-rich condition, the Ti chemical potential can be assumed as the energy of bulk Ti, while the O chemical potential can be obtained by the growth condition: Under the O-rich condition, the chemical potential of O can be calculated from the ground-state energy of O 2 molecule, while the chemical potential of Ti is then fixed by condition (3). Therefore, a link between the defect formation energy and the external growth condition of doped TiO 2 can be created. The relationship between the formation energies of substitutional nonmetal-doped TiO 2 and its growth condition has been studied systematically [38,[42][43][44][45]. For Si-doped TiO 2 , the formation energy of the substitutional Si-cation-doped model is much less than that of the substitutional Si-anion-doped model under both Ti-rich and O-rich growth conditions [44,46,47]. This indicates that Si is energetically more favorable to substitute Ti than O under both Ti-rich and O-rich growth conditions. For S-and P-doped TiO 2 , the doping sites of S and P strongly depend on the preparing method and growth condition of the doped TiO 2 [43,48,49]. Under O-rich growth condition, S (P) prefers to replace Ti and form substitutional S (P)-cation doped structure. On the contrary, under the Ti-rich growth condition, S (P) prefers to replace O and form S (P)-anion doped structure. This is consistent with the experiments reported by several independent groups [50][51][52][53]. Yu et al. and Ohno et al. used the titanium isopropoxide and thiourea as the titanium and sulfur original materials, respectively, which corresponds to the O-rich growth condition, and prepared the S-cation -doped TiO 2 [50][51][52]. Moreover, it is further confirmed that replacing Ti by S is energetically more favorable than replacing O under the O-rich growth condition [50]. In contrast, Umebayashi et al. used TiS 2 as the starting material, which corresponds to the Ti-rich growth condition, and prepared S-anion -doped TiO 2 [53,54]. Therefore, the first-principles theoretical calculations demonstrated a basic experimental fact that the ionic form and site of S (P) dopant in TiO 2 can be controlled by the growth condition and preparation method of doped TiO 2 sample [43,48,49]. For C-doped TiO 2 , Di Valentin et al.'s theoretical calculations also gave a similar conclusion [38]. In addition, the first-principles theoretical calculations for halogen-doped TiO 2 further revealed the relationship between the doping sites of halogen atoms and their electronegativities as well as the growth condition of doped TiO 2 [45,55]. The following conclusions were drawn [45].
(i) Substitutional X-anion-doped TiO 2 (X = F, Cl, Br, and I) is energetically preferred to form under the Ti-rich rather than under the O-rich growth condition, and the formation energy increases in the order F < Cl < Br < I. This indicates that it is more difficult to replace an O atom using a larger and less electronegative X atom.
(ii) Substitutional X-cation-doped TiO 2 (X = F, Cl, Br, and I) is energetically preferred to form under the O-rich rather than under the Ti-rich growth condition, and the formation energy increases in the order I < Br < Cl < F. This indicates that it is more difficult to replace a Ti atom using a smaller and more electronegative X atom.
(iii) Under O-rich growth conditions, it is energetically more favorable to substitute Ti than O using Br and I, while it is energetically more favorable to substitute O than Ti using F and Cl.
(iv) Under Ti-rich growth conditions, it is energetically more favorable to substitute O than Ti using all the X atoms (X = F, Cl, Br, and I).
To qualitatively show the relationship between the formation energy and the growth condition of doped TiO 2 as well as the electronegativities of dopants, we plot the function of the formation energies of nonmetal-doped TiO 2 as the oxygen chemical potential (corresponding to the growth condition of TiO 2 ), see Figure 3. The following conclusions can be obtained.
(i) For high electronegative F (Cl and N), it is energetically more favorable to substitute O than Ti under both O-rich and Ti-rich conditions.
(ii) For low electronegative Si (Ge), it is energetically more favorable to substitute Ti than O under both O-rich and Ti-rich conditions.
(iii) For other nonmetal main group elements X-(X = B, C, S, Se, Te, P, As, Sb, Br, and I) doped TiO 2 , substitutional X-anion-doped TiO 2 is energetically preferred to form under Ti-rich condition while substitutional X-cation-doped TiO 2 is energetically preferred to form under O-rich conditions. This discrepancy can be partially understood from the electronegativity difference between nonmetal dopants and O (Ti) as well as the bond strength of X-O and X-Ti bonds. For high electronegative F (Cl and N), it is preferred to form a F-Ti (Cl-Ti and N-Ti) bond by picking up electrons from Ti rather than form a F-O (Cl-O and N-O) bond by losing electrons. In addition to the bulk doping, a recent computational study shows that F surface adsorption could tune the relative stability of (001) and (101) surfaces of anatase TiO 2 , thus providing one effective approach to synthesize TiO 2 surfaces with desired properties [56][57][58][59]. For low electronegative Si (Ge), it is preferred to form a Si-O (Ge-O) bond by losing electrons to O rather than form a Si-Ti (Ge-Ti) bond by picking up electrons. For other nonmetal main group elements X (X = B, C, S, Se, Te, P, As, Sb, Br, and I), their electronegativities are between Ge and N, and thus both X-O and X-Ti bonds are possible to form, depending on the growth conditions of doped TiO 2 . These results are expected to provide some useful guidance to prepare nonmetal-doped TiO 2 and other semiconductor oxides. Next, let us discuss the possibility of interstitial nonmetal-doped TiO 2 . Generally speaking, only the dopant with a small atomic size can easily form interstitially doped structure. Therefore, B, C, and N are most possible to be located at the interstitial site. First-principles theoretical calculations show that substitutional N-anion and interstitial N-doped structures have nearly the same formation energy under O-rich growth condition [40]. Di Valentin et al. found that interstitial C-doped TiO 2 even has lower formation energy than substitutional C-anion-doped TiO 2 under O-rich growth condition [60]. Geng et al. reported that interstitial B-doped TiO 2 has slightly lower formation energy than substitutional B-anion-doped TiO 2 [61]. Finazzi et al. also suggested that the substitutional-anion doped B dopants can be converted into interstitial B after annealing at high temperature on the basis Catalysts 2020, 10, 972 7 of 34 of the first-principles analysis [62]. In summary, for the nonmetal dopants with small atomic size, both the substitutional (at O site) and interstitial structures are possible to form.

N Doping
Since the discovery that N doping can promote the visible light photocatalytic activity of TiO 2 [9,36], numerous experimental efforts have been made to study the N doping influence on the optical absorption and photocatalytic properties of TiO 2 [39,[70][71][72][73][74][75][76][77][78][79][80][81][82][83]. With the progress of the theoretical and experimental research, the controversy on the origin of the redshift of the optical absorption edge in N-doped TiO 2 appears [7,42,84]. Three different opinions have been proposed to explain the redshift of the optical absorption edge in N-doped TiO 2 : (i) The mixing of N 2p states and valence band (VB) leads to the bandgap narrowing. A representative example is Asahi et al.'s theoretical calculation and experiment [9,36], and it is further confirmed by the later experiments [76,78,85].
In addition, several independent research groups reported that there exists an optimal nitrogen concentration to achieve the highest photocatalytic activity [78,81] or maximize visible light absorption [85]. It is also found that a bandgap narrowing can be realized in high-concentration N-doped TiO 2 [78,85]. As a consequence, it is speculated that the N-doped TiO 2 may show different electronic structure characteristics under different doping concentration, which may lead to the controversy on the origin of the visible light absorption. This speculation was confirmed through the first-principles electronic structure calculations [40]. Yang et al. analyzed electronic structures of N-doped anatase and rutile TiO 2 at different doping levels and found following conclusions [42].
(i) At lower doping concentration (≤~2.1 at.%), N doping introduces some isolated impurity states above the VB in the bandgap. They act as transition levels, and thus the electronic transition among the VB, conduction band (CB), and impurity states may be responsible for the redshift of the optical absorption edge [63,[86][87][88].
(ii) At higher doping levels (≥~4.2 at.%), more N 2p states are introduced, and they mix with the O 2p states, leading to the bandgap narrowing [9].
Zhao et al. also found similar conclusion from first-principles calculations [84]. Therefore, these results can clarify the controversy on the origin of the redshift of the optical absorption edge in N-doped TiO 2 , i.e., bandgap narrowing and N 2p states in the bandgap can be both responsible for the visible light absorption of N-doped TiO 2 , which depends on the N doping levels. It is noted that the same conclusion can be drawn using spin-polarized calculations despite some differences of the positions of N 2p states, which is caused by spin split. It also shows that when nitrogen concentration exceeds 2.1 at.%, the optical energy gap has little further narrowing compared with that at lower doping levels. In contrast, increasing the nitrogen concentration leads to larger formation energy [42]. This is well consistent with the experiment, in which the higher nitrogen concentration does not lead to further optical energy-gap narrowing but makes the growth of N-doped TiO 2 more difficult [85]. In addition, nonmetal doping (C, S, and P) can lower the formation energy of oxygen vacancy [38,[93][94][95], and thus the combined effects of the oxygen vacancy and nonmetal doping cannot be excluded to be responsible for the visible light absorption in N-(C-, S-, and P-) doped TiO 2 .
TiO 2 has three kinds of crystal phases: anatase, rutile, and brookite. Anatase and rutile are two common phases, and thus they are also generally considered to share similar electronic properties despite the different bandgap (3.2 eV for anatase phase [96] and 3.0 eV for rutile phase [97]). It is known that N doping in TiO 2 can induce a redshift of the optical absorption edge [9,36,39,[70][71][72][73][74][75][76][77][78][79][80][81][82][83]. Surprisingly, Diwald et al. observed a blueshift of the optical absorption edge~0.2 eV in N-implanted rutile TiO 2 [98]. Soon after, Di Valentin et al. proposed that the bandgap increasing of approximately 0.08 eV in N-doped rutile TiO 2 can be responsible for the experimentally observed blueshift on the basis of the first-principles calculations [86]. However, this explanation is not in harmony with a basic experimental fact that the redshift is also observed in N-doped rutile TiO 2 [36,72,74,77,82]. In principle, there are three possible N-doping sites, including substitutional N-anion (N at O site, i.e., N@O), N-cation (N at Ti site, i.e., N@Ti), and interstitial N-doped structures. A prior first-principles computational study for all these three possible models shows that N doping either introduces some impurity states in the bandgap for substitutional N-anion-doped and interstitial N-doped TiO 2 or narrows the bandgap for substitutional N-cation doped TiO 2 , which both can lead to a redshift of the optical absorption edge [40] Therefore, the possibility that N doping leads to the blueshift of rutile TiO 2 can be excluded. A phase transition from rutile to anatase by nitrogen doping can easily explain the blueshift of the optical absorption edge~0.2 eV, though Diwald et al. did not observe the anatase phase in N-doped rutile TiO 2 [98] In contrast, Henderson proposed that the blueshift may be explained according to hole trapping effects [8]

S Doping
Experiments show that there are two possible substitutional doping sites for S dopants in TiO 2 , i.e., S anion at O 2− site (S@O) and S cation at Ti 4+ site (S@Ti) [50][51][52][53][54][99][100][101][102][103], and both the S-anionand S-cation-doped TiO 2 show the high photocatalytic activity under visible light. For example, Umebayashi et al. prepared S-anion-doped anatase and rutile TiO 2 by oxidation annealing of titanium disulfide (TiS 2 ) and ion implantation in the rutile single crystal, respectively, and observed the visible absorption spectrum in these samples [53,54,99]. Similar visible light absorption property of S-anion-doped TiO 2 was also reported by several other groups [101,102], and the S doping concentration was found to have a great influence on the visible light photocatalytic activity of S-anion TiO 2 [100]. For substitutional S-cation-doped TiO 2 , interestingly, Yu et al. observed a bactericidal effect under visible light irradiation [50]. Ohno et al. even found that it exhibits stronger visible light absorption than N-, C-, and the S-anion-doped TiO 2 [51,52,103]. As in the case of N-doped TiO 2 , the controversy on the mechanism of the visible light absorption in S-anion-doped TiO 2 also exists. Earlier theoretical calculations indicated that S-anion doped TiO 2 has an obvious bandgap narrowing [53,99,104,105], while the later first-principles calculation shows that S dopants introduce S 3p impurity states above the VB, which might be responsible for the redshift of the absorption edge of S-anion doped TiO 2 [106].
First-principles studies revealed that S-anion-and S-cation-doped TiO 2 show different mechanisms for the visible light absorption [43]. For S-anion-doped TiO 2 , different doping levels lead to the different electronic structure characteristics. At lower S doping concentration (≤2.08 at.%), the bandgap narrows slightly but some S 3p localized states are introduced in the bandgap. Therefore, electron excitations from these occupied S 3p states to CB might lead to a more significant redshift of the optical absorption edge than the slight bandgap narrowing [100]. At higher S doping concentration (≥4.17 at.%), the mixing of the S 3p states with the VB causes an obvious bandgap narrowing (~0.7 eV), thus leading to a substantial redshift of absorption spectra.
For S-cation-doped TiO 2 , S dopants introduce some occupied impurity states consisting of S 3s and O 2p states in the bandgap despite the unchanged bandgap. This indicates that the S dopant has an electron configuration resembling a S 4+ (s 2 p 0 ) ion in TiO 2 . Therefore, these impurity states can act as transition levels, and the electron excitations from these transition levels to CB may be responsible for the experimental redshift of the optical absorption edge. Similar S doping influence also appear in rutile TiO 2 [43].

P Doping
Visible light photocatalytic activity was also reported in P-anion-and P-cation-doped TiO 2 , respectively. Shi et al. [107] found that P-cation-doped (P@Ti) TiO 2 nanoparticles exhibit a stronger visible light absorption than undoped sample, which is thought to be induced by the impurity states in the bandgap. Furthermore, their X-ray photoelectron spectroscopy (XPS) measurements also indicated that the doped P ions are in the pentavalent-oxidation states (P 5+ ) [107]. Yu et al. also found that P-doped TiO 2 shows a better photocatalytic ability than that the pure TiO 2 [108]. Li et al. observed a visible light absorption in substitutional P-anion-doped (P@O) anatase TiO 2 , and attributed it to a narrowed bandgap [109]. On the contrary, Yu et al. found a larger bandgap in phosphor-modified TiO 2 than that of pure TiO 2 [110].
To understand the mechanism underlying these inconsistent experimental observations, first-principles calculations for P-anion (P@O) and P-cation (P@Ti) doped TiO 2 were performed [43]. It is found that P-anion doping does not cause a large bandgap narrowing but introduce P 3p states in the bandgap. In contrast, P-cation doping neither narrows the bandgap nor introduces impurity states in the bandgap. The first-principles results indicated that the high photocatalytic activity in P-cation-doped TiO 2 may be caused by the large surface area and the crystallinity of TiO 2 instead of the formation of an impurity energy level in the bandgap [108,110].

B Doping
There are three possible doping sites for boron in TiO 2 , i.e., substitutional B for O (B@O), substitutional B for Ti (B@Ti), and the interstitial B site. The atomic radius of B is larger than that of O (0.85 Å vs. 0.6 Å) but smaller than that of Ti (0.85 Å vs. 1.4 Å). Therefore, substitutional B-anion (B@O) doping and interstitial B doping are expected to cause a lattice expansion of TiO 2 , while substitutional B-cation (B@Ti) doping causes a lattice shrinking. Variable-cell structure optimizations for the three B-doped anatase models were carried out to qualitatively study the B doping influence on the lattice structure of TiO 2 [64]. Compared with the undoped anatase model, the volumes of the substitutional B-anion and interstitial B-doped models expand approximately 7.7% and 7.0%, respectively. In contrast, the volume of substitutional B-cation-doped model shrinks by~5%. These results are consistent with Jung et al.'s experiment in which the grain size of anatase phase was enlarged when the incorporation of boron oxides is more than 10% [111]. However, Chen et al. found that the doping of boron ions inhibited the crystal size [112]. Although substitutional B-cation doping can lead to a decrease of the volume of TiO 2 , this kind of structure is not consistent with Chen et al.'s experiment in which the B ions are sited at an interstitial position. Therefore, substitutional B-cation doping cannot account for Chen et al.'s experimental phenomenon.
The calculated electronic properties of three B-doped TiO 2 models show that, for substitutional B-anion-doped TiO 2 , some impurity states mostly consisting of B 2p states are introduced in the bandgap [64] The electron excitation energy from VB to the unoccupied gap states above the Fermi level decreases about 0.3 eV with respect to the optical bandgap of undoped TiO 2 . This is consistent with the experimental redshift of the optical absorption edge in B-doped TiO 2 [113]. It is noted that the substitutional B-anion doping leads to a spin-polarized electron state [62,64,114,115], and thus forms a paramagnetic defect consisting of B and adjacent Ti ions [62]. A detailed discussion can be found in Section 4. For the substitutional B-cation-doped TiO 2 , its optical bandgap decreases by~0.3 eV due to the downward shift of the CB. On the contrary, for the interstitial B-doped TiO 2 , its optical bandgap increases by~0.2 eV for anatase TiO 2 and 0.3 eV for rutile TiO 2 because of the well-known "band-filling mechanism" [116] or "Moss-Burstein shift" [117,118], which is often associated with the optical absorption shift in n-type semiconductors. This is in good agreement with the experimental blueshift of the optical absorption edge in interstitial B-doped TiO 2 , in which the optical absorption energy increases by~0.12 eV. Therefore, the blueshift of optical absorption edge in B-doped TiO 2 can be attributed to the intrinsic property of interstitial B-doped structure instead of the quantum size effects [111,112]. Similar doping effects also occur in B-doped rutile TiO 2 [64,119]. It is also noted that the several different interstitial positions of B dopants should exist [62], and both the standard and hybrid DFT calculations have been carried out to study the trigonal-planar-coordinated [BO 3 ] and pseudo-tetrahedral-coordinated [BO 4 ] species [62,120]. It is found that the [BO 3 ] species leads to an increase in the bandgap, while the [BO 4 ] species leads to a decrease in the bandgap [120]. However, the [BO 3 ] is shown to be more stable than [BO 4 ] [62].

C Doping
C doping can extend the optical absorption of TiO 2 effectively. Interestingly, different degrees of the reduction of the optical bandgap have been observed in C-doped TiO 2 [10,[121][122][123][124][125][126][127][128][129][130][131][132][133][134]. Khan et al. firstly found that C-anion-doped rutile TiO 2 exhibits two optical absorption thresholds at 535 and 440 nm, which corresponds to the optical bandgap decrease of approximately 0.18 and 0.68 eV, respectively [10]. Soon after, more experimental studies confirmed that C-anion doping could induce different degrees of the redshift of the optical absorption edge of TiO 2 . For example, the optical bandgap reduction of approximately 0.3 eV and 0.45 eV was observed [121][122][123][124][125], and more pronounced optical bandgap reduction of approximately 0.72 eV [126], 0.86 eV [127,128], 0.95 eV [129,130], and 1.0 eV [131][132][133] was also observed. Xu et al. found two regions of photo-response from ultraviolet (UV) to 450 nm and 575 nm in C-doped anatase TiO 2 , which equals to the optical bandgap decrease of 0.45 eV and 1.05 eV, respectively [134]. These experiments both indicate that substitutional C-anion doping can lead to several different optical absorption thresholds in TiO 2 . In addition, carbon can also be incorporated into TiO 2 as a cation at a Ti site because of its low electronegativity, and the C-cation doping influence on the optical absorption spectra of TiO 2 is open. Kamisaka et al.'s first-principle calculations indicate that C-cation doping neither introduces gap states nor induces visible light absorption [135]. On the contrary, Ren et al. observed a visible absorption in the 400-450 nm range in their C-cation-doped TiO 2 sample using UV-Vis diffuse reflectance spectroscopy [136].
Spin-polarized first-principles generalized gradient approximation (GGA) plus U calculations were carried to study the electronic and optical absorption properties of C-anion and C-cation-doped TiO 2 [65]. With respect to the undoped TiO 2 , the calculated bandgap of C-anion-doped TiO 2 changes slightly but some spin-polarized impurity states are introduced in the bandgap. Correspondingly, associated electron excitations among the VB, the CB, and the impurity states can be responsible for the various visible light absorption thresholds in C anion-doped TiO 2 . The electron excitation energy from the occupied gap states just above the valence band maximum (VBM) to the conduction band minimum (CBM) decreases approximately 0.33-0.53 eV, which is consistent with the reduction of the optical absorption energy approximately 0.30-0.45 eV [121][122][123][124][125]134]. The electron excitation energies from the occupied gap states to the CB and from the VB to the empty gap states reduce by approximately 0.63, 0.83, and 1.18 eV. This may be responsible for the large redshift (approximately 0.70-1.05 eV) of the absorption edge in C-doped anatase TiO 2 and TiO 2 nanotubes [126][127][128][129][130][131][132][133][134]. C-doped rutile TiO 2 also shows same electronic characteristics [65]. Figure 4 shows density of states (DOS) plots of undoped and C-cation-doped anatase TiO 2 . For C-cation-doped TiO 2 , the calculated bandgap in the frame of GGA+U is approximately 2.85 eV, less than that of the undoped anatase TiO 2 by approximately 0.18 eV [65]. This can explain the experimental redshift of the optical absorption and enhanced visible light absorption in the range of 400 to 450 nm [136]. Similar electronic structure modifications induced by C-cation doping also occur in C-doped rutile TiO 2 . It is worth mentioning that the local structure around the C dopant in anatase TiO 2 significantly influences its electronic properties. Kamisaka et al. performed first-principles calculations for C-cation-doped TiO 2 , in which the cell size and shape were fixed, and found that the C dopant forms a planar CO 3 species. Neither in-gap impurity states nor visible light absorbance is found in their calculations [135].  Interstitial C-doped TiO 2 has also been studied using first-principles calculations [38,137]. It is found that the interstitial C dopants introduce impurity states in the bandgap, which can lead to the visible light absorption in C-doped TiO 2 .

Si Doping
Visible light photocatalytic activity of Si-doped TiO 2 has also been reported [138][139][140][141]. For example, Oh et al. found that Si doping at low level could improve the photocatalytic activity of TiO 2 , while a high doping over 2% could decrease its photocatalytic activity [138]. Yan et al. observed that substitutional Si-cation doping (Si at Ti site) could cause a redshift of the absorption spectra of TiO 2 and favor its photocatalytic activity [139]. Ozaki et al. prepared nitrogen-doped silica-modified TiO 2 and found a high visible light photocatalytic activity [140,141]. First-principles calculations were carried out to understand the mechanism of extended visible light optical absorption [44]. In this work, substitutional Si-anion-(Si@O) and Si-cation-(Si@Ti) doped TiO 2 are modeled, respectively. Compared with the undoped anatase TiO 2 , it is found that substitutional Si-cation doping leads to a bandgap narrowing of approximately 0.25 eV, which is consistent with the experimental visible light optical absorption [138][139][140][141]. The calculations also show that high silicon concentration doping cannot lead to further bandgap narrowing, but requires larger formation energy [47]. For the substitutional Si-anion-doped anatase TiO 2 , visible light absorption is expected because it shows the lower photon absorption energy than that of undoped anatase TiO 2 [142].
To clarify the halogen doping influences on the photocatalytic properties of TiO 2 under the UV/Vis light, it is essential to understand whether the doping introduces impurity states in the bandgap and how the doping affects the CB and VB edges, i.e., the CBM and VBM of TiO 2 . First-principle calculations of halogen-doped TiO 2 have been done to understand its electronic properties and origin of the associated visible light photocatalytic activity [45]. The calculated bandgap of the undoped anatase TiO 2 is approximately 2.10 eV. For F-, Cl-, Br-, and I-doped TiO 2 , the calculated bandgaps are approximately 2.06, 1.90, 1.80, and 1.40 eV, respectively. The calculated bandgap of the F-doped TiO 2 is slightly less than that of the undoped TiO 2 (by approximately 0.04 eV). As a result, it is understandable that either a very slight decrease (~0.05 eV [145]) or no optical bandgap change was found in experiments [149,[151][152][153][154][155][156]. It is also noted each F atom (s 2 p 5 ) requires one less electron from TiO 2 than does each O atom (s 2 p 4 ), and thus the substitution of one F − ion for one O 2− introduces one additional electron in the TiO 2 lattice. As a result, the treatment of standard DFT calculations for the additional electron leads to the n-type conductive property of F-doped TiO 2 [160]. In contrast, in the hybrid DFT or GGA+U calculations, the additional electron is shown to reduce one Ti 4+ (d 0 ) to Ti 3+ (d 1 ) ion and introduces localized gap states in the bandgap [161][162][163]. However, the lack of the visible light absorption in F-doped TiO 2 indicates that the electron transition from these occupied gap states to the CB is not effective to lead to visible light absorption. Therefore, F is not a good dopant to extend the optical absorption edge of TiO 2 into the visible light region. In contrast, an obvious bandgap narrowing (approximately 0.2 and 0.3 eV) is observed in Cl-and Br-doped TiO 2 , respectively. This is consistent with the experimentally observed reduction of the bandgap by approximately 0.2-0.3 eV in Cl/Br co-doped TiO 2 [157]. For I-doped TiO 2 , the bandgap narrows by approximately 0.7 eV, and thus a significant visible light optical absorption is expected.
The photocatalytic ability of a semiconductor, i.e., the ability to transfer photon-excited electron-hole pairs to the absorbed species on the surface of the semiconductor, can be partially determined by the relative positions of its CBM and VBM with respect to the redox potentials of the adsorbate [2]. Therefore, the photocatalytic ability of TiO 2 can be qualitatively evaluated by the positions of its CBM and VBM. Thermodynamically, the VBM of a photocatalytic semiconductor should lie below the redox level of the adsorbed species, while the CBM should lie above the redox level, so that the photoinduced hole in VB can capture an electron from the adsorbed species and the photoexcited electron in CB can be transferred to the adsorbed species. Figure 5 shows the comparison of the calculated VBM and CBM positions of X-anion doped anatase TiO 2 (X = F, Cl, Br, and I) with the experimental values of undoped TiO 2 . A scissor operation of 1.10 eV was applied to make the calculated bandgap comparable to the experimental value. The corrected bandgaps for F-, Cl-, Br-, and I-doped TiO 2 are 3.16, 3.00, 2.90, and 2.50 eV, respectively. For F-doped TiO 2 , its CBM and VBM shift downwards by 0.20 and 0.16 eV with respect to the undoped TiO 2 . This indicates that F-doped TiO 2 should have a stronger oxidation ability, which can explain the experimentally observed higher photocatalytic activity in F-doped TiO 2 than that of the undoped TiO 2 [143][144][145]149,155,156]. For Cl-doped TiO 2 , its CBM shifts downwards by approximately 0.15 eV while the VBM shifts upwards by 0.05 eV relative to the corresponding values of undoped TiO 2 . Therefore, Cl doping may reduce the ability of oxidation and reduction of TiO 2 , which is in good agreement with the experimental fact that Cl-doped TiO 2 shows lower water-splitting power than that of undoped TiO 2 under UV irradiation [157]. For Br-doped TiO 2 , its VBM is nearly same with that of undoped TiO 2 , and thus Br-doped TiO 2 should have the same oxidation ability with that of undoped TiO 2 . In contrast, the CMB of Br-doped TiO 2 shifts downwards by approximately 0.3eV compared with that of undoped TiO 2 , so that Br-doped TiO 2 might show lower ability to reduce H + to H 2 than does undoped TiO 2 . For I-doped TiO 2 , its CBM shifts downward slightly but VBM is raised strongly with respect to that of undoped TiO 2 , and thus visible light photocatalytic activity might appear in I-doped TiO 2 . In summary, the changes of the VBM and CBM can qualitatively explain some experimental facts that the photocatalytic ability was improved upon F, Cl, and Br doping [143][144][145]149,[155][156][157]. In addition, the local internal field induced by the dipole moment is considered to be effective to separate the electron-hole pairs and inhibit their recombination [164,165], and thus the photocatalytic performance of doped TiO 2 can be evaluated from the variation of the dipole moment of TiO 6 octahedron adjacent to the dopants [165].
For I-doped anatase TiO 2 , its VBM and CBM are nearly the same as those of undoped TiO 2 . However, a double-occupied bandgap state above the VBM approximately 0.6 eV is introduced in the bandgap. Its PDOS plot shows that this gap state mostly consists of I 5s states and the 2p states of the neighboring O atoms around the I dopant, and that I 5p states contribute to its CB. This indicates that I dopant exists as an I 5+ (s 2 ) cation, as observed experimentally [158,159]. The Fermi level of I-doped TiO 2 is pinned above the CBM approximately 0.2 eV, indicating that iodine is a good n-type dopant, as in the case of I-doped ZnTe [166]. Therefore, the optical bandgap of I-doped anatase TiO 2 decreases approximately 0.4 eV with respect to that of undoped TiO 2 . This is mainly responsible for the optical bandgap decrease of I-doped TiO 2 and the improved photocatalytic efficiency in the UV/Vis region [158,159]. For substitutional X-cation (X = F, Cl, and Br) doped TiO 2 , unlike the case of I-doped TiO 2 , Cl 3s and Br 4s states do not appear above the VBM. In contrast, singly-filled Cl 3p and Br 4p states appear in the bandgap of Cl-and Br-doped TiO 2 , respectively. This indicates that Cl and Br at Ti sites exist as Cl 4+ (s 2 p 1 ) and Br 4+ (s 2 p 1 ) ions, respectively. For F-doped TiO 2 , the F 2p states lie in the VB, and the empty F 2p states lie just above the Fermi level. Therefore, with respect to the Cl-and Br-doped TiO 2 , the doped F atom at a Ti site should exist as a F 3+ (s 2 p 2 ) ion to a first approximation [45].

Hydrogen Impurities in TiO 2
As a ubiquitous impurity, hydrogen (H) widely exists in metal oxides, which forms either deep gap states or shallow donor levels [66][67][68][69]. Owing to its small ionic radius, H at an interstitial site is more stable than that at a substitutional site [167], though in principle, it can form substitutional structure by replacing the O atom. First-principles calculations show that the interstitial H introduces a shallow donor level in TiO 2 [66][67][68][69]. This is consistent with the recent experiments in which the H is identified as a either a shallow donor or metastable donor in rutile TiO 2 single crystal [168,169]. Interstitial H introduces an additional electron in TiO 2 , and thus the n-type conductivity can be understandable. However, recent GGA+U and hybrid functional calculations show that the additional electrons can be trapped by the Ti ions, reducing Ti 4+ to Ti 3+ , and introduce the gap states [161,170], although the position of the gap states strongly depends on the choice of U parameters [162]. First-principles calculations also show that substitution of H for O can lead to the n-type conductivity, though its structure is not stable [167].
It is also interesting to explore the effects of H impurity on the structural stability and electronic property of nonmetal-doped TiO 2 [167,171,172]. With respect to the N-doped TiO 2 , the combinational doping of (N, H) can enhance the structural stability and lead to a significant bandgap narrowing for anatase and brookite phases of TiO 2 [167,171]. For N-anion-doped TiO 2 , the doped N dopant exists as N 2− (s 2 p 5 ) ion, and introduces an unoccupied N 2p impurity state in the bandgap [173]. For (N, H)-doped TiO 2 , the interstitial H atom adjacent to the N dopant introduces an additional electron into the lattice, and forms the N-H bond by transferring the additional electron to one empty N 2p orbital. As a result, all the N 2p impurity states become occupied, leading to either an obvious bandgap narrowing or double-filled N 2p impurity states above the VB. This type of charge compensation mechanism also occurs in (N, Ta)-and (N, F)-doped TiO 2 [60,174], in which the substitutional Ta-cation (Ta 5+ ) and F-anion (F − ) doping introduce one additional electron into the TiO 2 lattice separately, which fills the empty N 2p orbital. For (C, H)-doped TiO 2 , it is thermodynamically more stable than C-doped TiO 2 , and a bandgap narrowing also occurs [172]. However, in this case, one electron donated by the interstitial H cannot compensate the two holes of C 2− (s 2 p 4 ) dopant [175], and thus this incomplete charge compensation results in an impurity level just below the CB [172]. It is expected that two interstitial H and one substitutional C at an O site can satisfy full charge compensation. In addition, recent experimental and theoretical studies show that hydrogenation on the surface of TiO 2 can lead to a structural disorder and introduce mid-gap states, which is responsible for the improved solar-driven photocatalytic activity [176,177]. In summary, the influence of hydrogen impurity on the structural stability and electronic property of TiO 2 is one interesting topic, and further work is worthy to be done.

Co-Doping
Introducing impurity states in the bandgap by nonmetal or metal doping is effective to extend the optical absorption edge of TiO 2 into the visible light region; however, the electron-hole recombination center, which generally refers to the impurity-related states near the middle of the bandgap, inhibits the photocatalytic efficiency of TiO 2 . Therefore, narrowing bandgaps without creating mid-gap states is necessary to maximize the photocatalytic performance of TiO 2 under the visible light irradiation [178][179][180][181]. To this aim, co-doping can be an effective approach because it has three important effects: (1) Co-doping can eliminate impurity states in the bandgap via charge compensation between different dopants, which can reduce the number of recombination centers and promote the separation of electron-hole pairs.
(2) Co-doping can facilitate the mixing between the impurity states and VB (CB) by adjusting the position of impurity states in the bandgap, and narrow the bandgap effectively.
(3) Co-doping can reduce the formation energy of the combination defect with respect to the monodoped TiO 2 , and thus improving the solution of ideal dopants in TiO 2 or other metal oxides becomes possible.
Herein, we divided the co-doping into three classes, i.e., anion-anion co-doping, anion-cation co-doping and cation-cation co-doping, according to the ionic type (anions or cations) of dopants, and summarized the recent theoretical research progress of the co-doped TiO 2 , mainly from the viewpoint of charge-compensation.

Anion-Anion Co-Doping
One typical example of anion-anion co-doping is (N, F) co-doped TiO 2 [60,150,151,[182][183][184], in which N and F both replace O ions. In N-anion monodoped TiO 2 , one N at an O site introduces an accept level in the bandgap and exists as a N 2− ion, while in F-anion monodoped TiO 2 , one F at O site introduces one donor level by donating one additional electron into the TiO 2 lattice. As a result, in (N, F) co-doped TiO 2 , this additional electron can occupy the N 2p accept level, and then all the N impurity states become occupied. These occupied N 2p states are located just above the VB, and do not act as the electron-hole recombination center, thus improving the visible light photocatalytic activity of TiO 2 with respect to that of the N-anion monodoped TiO 2 [60,185]. In addition, (N, F) co-doping reduces the formation energy with respect to that of the N monodoped TiO 2 , and thus a higher nitrogen concentration doping can be realized. Similar anion-anion charge compensation effect is expected to occur in (N, Cl), (N, Br), (N, I), (P, F), (P, Cl), (P, Br), and (P, I) co-doped TiO 2 .

Anion-Cation Co-Doping
(N, H int ) co-doped TiO 2 is a typical example of the anion-cation co-doping, in which N replaces O while H locates at an interstitial site. In this system, one electron donated by an interstitial H occupy the N 2p acceptor level, and the bandgap narrows approximately 0.26 eV, much larger than the N-anion monodoped TiO 2 (0.04 eV) [171].
Another two kinds of anion-cation co-doping combinations are (N, X) and (C,Y) [174,179,186,187], in which X can be Ta 5+ or Nb 5+ while Y (Herein Y is not element yttrium.) can be W 6+ or Mo 6+ . It is known that N dopant exists as N 2− (s 2 p 5 ) ion at an O site [173], and introduces one acceptor level in the bandgap, while the pentavalent Ta (Nb) cation at a Ti site donates one electron into TiO 2 lattice, introducing one donor level in the bandgap. As a result, the charge compensation can occur in (N, Ta) [174,186], and (N, Nb) co-doped TiO 2 , and the electron-hole recombination center (i.e., the impurity-related states near the middle of the bandgap) might be removed, thus promoting the rate of electron-hole separation and improving its photocatalytic efficiency. Similarly, the C dopant exists as C 2− (s 2 p 4 ) ion at an O site [175], and introduces two acceptor levels in the bandgap. In contrast, the hexavalent W (Mo) cation at a Ti site donates two electrons into TiO 2 lattice, i.e., introducing two donor levels. Therefore, in (C, W) and (C, Mo) co-doped TiO 2 [179,187], the two acceptor levels introduced by C dopant become occupied by capturing the two electrons donated by W and Mo, and C 2p impurity states are more close to the VB, reducing the number of electron-hole recombination centers. However, it is noted that the full charge compensation only occurs when the doped anion and cation are bonded together, though this configuration generally possesses the lowest total energy [187,188]. This is reasonable because the doped anion and cation can form a strong bond by a direct charge transfer.
Another kind of anion-cation co-doping, such as (N, Mo) co-doping, has been proposed [188], in which two N anions are bonded to one hexavalent Mo cation, forming a near-linear N-Mo-N unit. For convenience, we refer it to as (N-Mo-N) co-doping. In this case, two electrons donated by one hexavalent Mo cation at a Ti site can compensate two holes introduced by two N anions, thus achieving full charge compensation. Liu et al. further studied the difference of the electronic structures between non-passivated and passivated (N, Mo) co-doping [188]. The local geometrical configurations of N and Mo dopants for these two kinds of (N, Mo) co-doped TiO 2 are shown in Figure 6a,b. Their calculated TDOS and PDOS plots are shown in Figure 6c. For non-passivated (N, Mo) co-doping, one N anion is bonded to one Mo cation, and the doping ratio between N and Mo is 1:1. In this case, one hole introduced by one N cannot compensate two electrons donated by one Mo, thus leading to n-type conductivity. In passivated (N, Mo) co-doped TiO 2 , i.e., (N-Mo-N) co-doped system, the two electrons are fully compensated by the two holes introduced by the two N anions, and the N 2p acceptor levels are removed and do not act as an electron-hole recombination centers, thus improving the photocatalytic efficiency. Similar to the case of (N-Mo-N) co-doped TiO 2 , full charge compensation is also expected in (N-W-N), (Ta-C-Ta), and (Nb-C-Nb) co-doped TiO 2 .

Cation-Cation Co-Doping
Cation-cation co-doping has also been proposed to narrow the bandgap of TiO 2 , and two criteria of choosing cations are suggested [180]. First, the cation should have a closed-shell electronic configuration like d 0 or d 10 . Second, to keep the semiconductor characteristic of TiO 2 , co-doping with cations A x+ and B y+ should satisfy a simple rule, i.e., x + y = 8. On the basis of the criteria, the cation-cation co-doping combination of (Mo 6+ , Zn 2+ /Cd 2+ ) and (Ta 5+ , Ga 3+ /In 3+ ) is expected to cause an effective bandgap narrowing without introducing gap states. In these two kinds of cation-cation co-doped TiO 2 , the number of the holes introduced by Zn 2+ /Cd 2+ (Ga 3+ /In 3+ ) equals to that of electrons introduced by Mo 6+ (Ta 5+ ), and thus full charge compensation can be obtained. In addition, the electronic configuration of d 0 or d 10 of the cations can guarantee that the d states of dopants either are fully unoccupied or occupied, and thus no d-orbital-related impurity states can be created in the bandgap. First-principles calculations show that charge compensation does occur in the (Ta 5+ , Ga 3+ /In 3+ ) and (Mo 6+ , Zn 2+ /Cd 2+ ) co-doped TiO 2 , and with respect to cation monodoped TiO 2 , the mid-gap states are passivated.
In summary, co-doping can be an effective approach to narrow the bandgap of semiconductor photocatalyst and get rid of the electron-hole recombination centers through the full charge compensation. To this aim, the number of the holes on acceptor levels should be equal to that of the electrons on donor levels. Therefore, we can define the following equation to be a basic rule for co-doping: N a and N d represent the number of acceptor atoms and donor atoms, respectively. N h represents the number of introduced holes per acceptor atom and N e represents the number of donated electrons per donor atom. It is noted that the Ti (O) ions in TiO 2 and SrTiO 3 have same chemical state, and thus the co-doping combinations mentioned above can also be applied in SrTiO 3 [189][190][191]. For a reference to choose an ideal combination of dopants for co-doping in TiO 2 and SrTiO 3 , we list the common acceptor (donor) atoms as well as the number of the holes (electrons) per acceptor (donor) atom in Table 2.

TiO 2 /Perovskite Interface
The organic-inorganic halide perovskite solar cells have received significant attention in the recent years because of the low cost, high power conversion efficiency, and the flexibility of halide perovskite materials. In the perovskite solar cells, TiO 2 is widely used as electron transport layers and plays a significant role in achieving high efficiency and stability of photovoltaic devices. One major reason is the conduction band of TiO 2 is lower than that of the halide perovskite such as MAPbI 3 and MASnI 3 (MA = CH 3 NH 3 , methylammonium) [16,17]. Therefore, first-principles computational studies of TiO 2 /perovskite interfaces have been an emerging topic recently [16][17][18][19][20][21][22][23][24].
De Angelis's research team has extensively studied the material interfaces between the anatase TiO 2 (denoted as a TiO 2 ) and tetragonal MAPbI 3 and MAPbI 3−x Cl x perovskite from first-principles calculations [18,21,[194][195][196]. To build materials interface models, the first step is generally to choose the most stable surfaces. The authors considered pseudo-cubic (001) and tetragonal (110) perovskite surfaces because they found the (001) surfaces of the cubic and tetragonal phases have an identical topology [21]. That is to say, the (001) surface of the tetragonal phase is essentially same with one of the three equivalent surfaces of the cubic phase. Their calculations show that the isolated (110) surface is energetically more favorable than the (001) surface, with a total energy difference of approximately 0.7 and 0.1 eV for MAPbI 3 and MAPbI 3−x Cl x , respectively. Interestingly, after "depositing" the perovskite on the a TiO 2 (101) surface, the (110) perovskite surface are further stabilized against the (001) surface from the energetic analysis; moreover, the interfacial Cl atoms further increases the interfacial binding energy to TiO 2 in MAPbI 3−x Cl x compared to that in MAPbI 3 . Later, a combined angle-resolved X-ray photoelectron spectroscopy (AR-XPS) and first-principles calculations by the same research team revealed more details at the MAPbI 3−x Cl x /TiO 2 interface. They found that Cl is preferentially located at TiO 2 interface rather than the bulk perovskite [194], and the interfacial Cl is also found to induce the band bending that creates a directional "electron funnel", improving the charge collection efficiency of the photovoltaic devices. This conclusion is also consistent with the previous computational study on Cl-doped bulk TiO 2 in which Cl is thermodynamically favorable to replace O [45].
The strong interfacial interaction between TiO 2 and perovskite is essentially attributed to the charge transfer across the TiO 2 /perovskite interface. To quantify the charge transfer, De Angelis et al. carried out a charge displacement analysis, as shown by calculated charge displacement curves for TiO 2 /MAPbI 3 and TiO 2 /MAPbI 3−x Cl x interfaces in Figure 7 [21]. It clearly shows a strong charge displacement variation across the interfacial region in both cases, indicating a strong interaction between TiO 2 and perovskite and a strong polarization. Furthermore, there is a slight increase in charge donation to TiO 2 in the case of MAPbI 3−x Cl x , suggesting an increased charge accumulation at the TiO 2 /MAPbI 3−x Cl x interface compared to TiO 2 /MAPbI 3 interface. This charge displacement analysis well explains the stronger binding energy for TiO 2 /MAPbI 3−x Cl x than that for TiO 2 /MAPbI 3 . This work indicates that the charge displacement analysis is one effective approach to visualize the charge displacement from the non-interacting two materials fragments to the interacting heterointerface. Figure 7. Illustration of (a) calculated charge displacement and (b) geometrical structure for TiO 2 /MAPbI 3 (red) and TiO 2 /MAPbI 3−x Cl x (blue). Reprinted with permission from the authors of [21]. Copyright (2014) American Chemical Society.
In 2019, Sultana et al. reported first-principles computational studies for the MAPbI 3 perovskite interfaces with TiO 2 , ZnO, and SnO 2 [22]. The interface models were built by placing tetragonal MAPbI 3 along its [001] direction on top of rutile (001) TiO 2 , wurtzite (1010) ZnO, and rutile (110) SnO 2 slabs. To understand the interfacial charge transport properties of these systems, the authors calculated their charge difference density using the following equation.
The authors analyzed the charge transfer from charge density difference plot and found that the charge accumulation and depletion mainly occur around the interfacial Pb and O atoms, indicating that the charge transfer mostly occurs at the interface. Among the three materials interfaces, the TiO 2 -based interface has the highest charge transfer based on the Bader charge analysis, thus suggesting that TiO 2 -based interface has the highest efficiency for the perovskite solar cells. However, the authors found that TiO 2 and ZnO have a stronger binding energy with MAPbI 3 than SnO 2 , but the strong binding may lead to decomposition of CH 3 NH 3 molecules. The authors thus conclude that SnO 2 can be a feasible replacement for TiO 2 and ZnO to improve the stability of perovskite solar cells.
The defect properties at the TiO 2 /perovskite interface were also studied from first-principles calculations. In 2017, Haruyama et al. reported the defect properties at the a TiO 2 (101)/tMAPbI 3 (110) interface using first-principles computational approach, in which anatase TiO 2 (001) was placed on the MAPbI 3 (110) plane [20]. They found that the vacancy defects in the TiO 2 layer create undesired defect levels within the bandgap, which serves as hole traps and recombination centers; while most of the vacancy defects in the MAPbI 3 layer produces no additional states, thus having no influence on the electron-hole separation rates. This computational work indicates a possible route to further improve the performance of perovskite solar cells via interface modification.
Yang et al. studied relative stability and charge transport properties of different interfaces between the lead-free MASnI 3 and TiO 2 , including anatase and rutile phases from first-principles calculations [24], in which (001) surface of MASnI 3 and (001) surfaces of anatase and rutile were used to build models. The authors considered two types of terminations of MASnI 3 and built four interface models-MAI/A, MA/R, SnI 2 /A, and SnI 2 /R, in which A and R represents anatase and rutile phases, respectively, as shown in Figure 8. The authors found that SnI 2 /A interface is more stable than the other three systems and this interface also has a better separation of photoinduced electron-hole pairs from the analysis of the plane-averaged electrostatic potential and density of states. Similarly, by examining four types of interfaces between MAPbI 3 /TiO 2 from first-principles calculations, Gent et al. also concluded that PbI 2 /A interface has a stronger interaction but PbI 2 /R interface model is most efficient for charge separation from the electrostatic potential analysis, which was attributed to the better lattice and atoms arrangement match between MAPbI 3 and rutile TiO 2 [19].
In addition to the extensively studied materials interfaces between the hybrid halide perovskites and TiO 2 , it is noted that the materials heterointerfaces between the classical inorganic perovskite oxides such as SrTiO 3 and TiO 2 has also attracted increasing attention because of the emerging interfacial properties and/or enhanced functionalities [23,197]. One major consideration is that the formed heterointerface has a better stability compared to the isolated bulk materials [23,198]. In 2017, Kitchin's research team explored the possibility to stabilize the epitaxial thin films of anatase and rutile TiO 2 on perovskite (Sr/Ba)TiO 3 by studying relevant bulk and interface energy terms using first-principles calculations [23]. The authors first identified four potential coherent epitaxial interfaces-(001)A/(001)(S/B)TO, (102)A/(011)(S/B)TO, (112)A/(111)(S/B)TO, and (100)R/(111)(S/B)TO-based on the experimental findings, volumetric formation and strain, and areal substrate, and then calculated their interface energies. Their computational models predict a decrease order of anatase stability from (001) to (011) to (111) perovskite substrates, which is in a general agreement with the experimental epitaxial stability investigations. In contrast, rutile is energetically more favorable than anatase on the (111) perovskite substrate. This work indicates that first-principles calculations of interface energy is a promising approach to predict epitaxial polymorph stability and to search for coherent epitaxial interfaces toward the stability of target materials even beyond TiO 2 polymorphs.
The TiO 2 polymorphic structures exhibit different photocatalytic properties and the synthesis of the specific TiO 2 polymorph with target materials properties are desired. One possible way to stabilize the metastable polymorphs is to use the epitaxial synthesis approach, i.e., materials heterointerfaces. the complexity and importance of the interface in a charge process. For example, the surface defect is considered to be passivated by the existence of PbI 2 , which can improve the charge separation efficiency at the interface between TiO 2 and perovskite [36]. Wang et al. proposed that the remaining PbI 2 may slow the interfacial charge extraction [37]. Hysteresis of PSCs has drawn much attention and is also related to the interface and charge processes. The origin of hysteresis behavior of PSCs is thought to be caused by non-radiative recombination induced by a high density of traps in the window/perovskite interface [38,39]. However, the trap mechanism cannot explain the reversible poling behavior observed in PSCs. Researchers have proposed other mechanisms, such as ionic motion, for the origin of hysteresis [40].
Thus, it is necessary to shed light on the structural and electronic properties of perovskite/TiO 2 interfaces for suppressing that for tetragonal CH 3 NH 3 PbI 3 , the (001) surface is the most stable surface among various surfaces [43]. Since CH 3 NH 3 SnI 3 has a similar structure to CH 3 NH 3 PbI 3 , we also select the (001) surface to build CH 3 NH 3 SnI 3 /TiO 2 interface. We consider two types of termination: (1) the CH 3 NH 3 I termination (MAI-T) and (2) the SnI 2 termination (SnI 2 -T). It is reasonable because CH 3 NH 3 SnI 3

TiO 2 /BiOI Interface
Bismuth oxyhalide BiOX (X = Cl, Br, and I) has a layer structure and exhibits promising catalytic properties for degrading pollutants under visible light irradiation because of their low bandgaps [22][23][24]194,195]. However, high electron-hole recombination rates in this class of materials severely inhibit their photocatalytic applications. A heterostructure that combines two different semiconductors could transfer charge carriers between a high energy band and a low energy band, which could effectively separate photoinduced electron-hole pairs, and thus greatly enhance the photocatalytic efficiency of semiconductor heterostructure. In a very recent study [199], Qu et al. studied BiOI/TiO 2 heterostructure using first-principles electronic structure calculations, in which the heterostructure model was built by "depositing" (001) BiOI slab on the (001) rutile TiO 2 slab. In this work, two types of materials interfaces including 1I/TiO 2 and BiO/TiO 2 were modeled, as BiOI has two types of surface terminations along the [001] direction. Note that the authors chose the average value of the two separate lattice parameters of the BiOI and TiO 2 as the lattice parameters of the new heterostructure in order to achieve a high lattice match between BiOI (001) and TiO 2 (001) surfaces, leading to a lattice mismatch of 6.63%. The calculated formation energy is −2.21 eV/A and −2.25 eV for 1I/TiO 2 and BiO/TiO 2 models, respectively, suggesting that both two models are likely to be formed and the BiO/TiO 2 is slightly more stable than 1I/TiO 2 . More interestingly, the bandgap of 1I/TiO 2 heterostructure reduces to 0.28 eV while BiO/TiO 2 heterostructure changes into an n-type semiconductor, see Figure 9a. Moreover, as shown in Figure 9c, the built-in potential of the heterostructure can effectively separate the photoinduced electron-hole pair across the interface, thus potentially improving the photocatalytic performance  [199]. Copyright (2018) Elsevier.

TiO 2 /RuO 2 Interface
Ruthenium oxide (RuO 2 ) has various technological applications such as being used as electrodes, supercapacitors, and catalysts because of its promising properties [11]. It has a rutile-like structure and exhibits an excellent electrical conductivity. In recent years, TiO 2 /RuO 2 heterostructure has been proposed as one efficient photocatalyst for water splitting and decomposition of organics [11][12][13][14][15], which is attributed to the high separation rate of photogenerated electron-hole pairs. In 2015, Wei et al. studied the energetic and electronic properties of the TiO 2 /RuO 2 (110) heterostructure in which RuO 2 was "deposited" on the rutile TiO 2 (110) surface using first-principles DFT calculations [200]. The calculated interfacial energy shows that the TiO 2 /RuO 2 interface is exothermic, indicating a strong bonding interaction between TiO 2 and RuO 2 . The calculated projected DOS plots for the TiO 2 /RuO 2 heterostructure show that the valence and conduction band of TiO 2 bend downward toward RuO 2 . Interestingly, upon introducing oxygen vacancies at the interface, there exists an upward band bending toward RuO 2 , in agreement with the experimental finding [11].
For convenience, a brief summary of these materials interfaces and their critical materials properties are listed in Table 3. However, it is worth mentioning that, in addition to the material interfaces discussed above, there exist many other types of TiO 2 -based materials interfaces such as TiO 2 /SnO 2 [201], TiO 2 /WS 2 [202], TiO 2 /WSe 2 [203], TiO 2 /Mo 2 [204], TiO 2 /ZnS [205], TiO 2 /g-C 3 N 4 [206], TiO 2 /MnO x [207], TiO 2 /Mo 3 [208], TiO 2 /C60 [209], and TiO 2 -based plasmonic composite [210]. The central idea of these heterostructures is to enhance the charge separation rate of the photoinduced electron-hole pairs via the interfacial charge transfer mechanism, which generally requires the appropriate band alignment. Among these interfaces, it is particularly worth mentioning that plasmonic composites that consist of plasmonic nanoparticles such as noble metals [210,211] and TiO 2 are emerging as one promising photocatalytic system because of their efficient light absorption. Details on this new class of photocatalytic composites can be found in recent review articles [212][213][214][215][216][217][218][219][220]. Table 3. Summary of TiO 2 -based materials interfaces along with their critical interfacial properties. a TiO 2 and r TiO 2 denote anatase and rutile TiO 2 , respectively.

Interface
Properties Ref.

Conclusions and Outlook
In summary, this review covered first-principles computational and theoretical understanding on the structural, energetic, electronic, and optical properties of TiO 2 -based nanocluster, bulk, and material interface. A fundamental understanding and computational design of novel TiO 2 -based materials structures and/or composites is of great importance for developing highly efficient photocatalytic and photovoltaic applications, in which the first-principles computational approach is expected to play an increasing role. Herein, we discuss several possible future research directions from the viewpoint of first-principles calculations: (i) Nonmetal or metal doping is one effective approach to tune electronic properties of TiO 2 , which, however, will also significantly influence the anatase-to-rutile phase transition. A clear computational and theoretical understanding of the phase-transition mechanism will be useful to further optimize the performance of TiO 2 in photocatalytic and other applications. In this regard, future efforts can be devoted to the development of novel computational and analytical approaches based on first-principles calculations.
(ii) Increasing the number of photoinduced electron-hole pairs is one effective way to achieve high photocatalytic performance of TiO 2 . Researchers have made great efforts to tune the bandgap of TiO 2 via doping engineering or defect engineering, which essentially utilizes light absorption property of TiO 2 for producing electron-hole pairs. Instead of utilizing generated electron-hole pairs in TiO 2 , the emerging TiO 2 -based plasmonic composites that utilize the light absorption of plasmonic nanoparticles instead of TiO 2 are one promising approach to achieve high photocatalytic performance. However, the current plasmonic nanoparticles are typically noble metal particles, and thus future direction could be to search for low-cost alternative plasmonic nanoparticles that can work with TiO 2 and exhibit comparable and even superior photocatalytic properties.
(iii) Increase the separation rate of photoinduced electron-hole pairs is another approach to improve the phtocatalytic activity of TiO 2 -based catalysts. Although several TiO 2 -based materials interfaces have been studied from first-principles, future efforts can be on the design of novel TiO 2 -based material interfaces such as heterostructures or core-shell structures that can significantly promote the separation rate of electron-hole pairs.
(iv) The emerging high-throughput computational materials design approach is becoming one powerful tool to significantly accelerate the materials discovery and development process, which has been successfully applied in the design of bulk materials [221][222][223][224], material interfaces [225,226], and organic-inorganic hybrid materials with target properties [227,228]. It is thus expected that the application of such an approach in the design of TiO 2 -based photocatalysts is likely to yield fruitful results and worth further exploring.

Conflicts of Interest:
The authors declare no conflicts of interest.

Abbreviations
The following abbreviations are used in this manuscript.