Theoretical insights of codoping to modulate electronic structure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TiO}_2$$\end{document}TiO2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SrTiO}_3$$\end{document}SrTiO3 for enhanced photocatalytic efficiency

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TiO}_2$$\end{document}TiO2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SrTiO}_3$$\end{document}SrTiO3 are well known materials in the field of photocatalysis due to their exceptional electronic structure, high chemical stability, non-toxicity and low cost. However, owing to the wide band gap, these can be utilized only in the UV region. Thus, it’s necessary to expand their optical response in visible region by reducing their band gap through doping with metals, nonmetals or the combination of different elements, while retaining intact the photocatalytic efficiency. We report here, the codoping of a metal and a nonmetal in anatase \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TiO}_2$$\end{document}TiO2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SrTiO}_3$$\end{document}SrTiO3 for efficient photocatalytic water splitting using hybrid density functional theory and ab initio atomistic thermodynamics. The latter ensures to capture the environmental effect to understand thermodynamic stability of the charged defects at a realistic condition. We have observed that the charged defects are stable in addition to neutral defects in anatase \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TiO}_2$$\end{document}TiO2 and the codopants act as donor as well as acceptor depending on the nature of doping (p-type or n-type). However, the most stable codopants in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SrTiO}_3$$\end{document}SrTiO3 mostly act as donor. Our results reveal that despite the response in visible light region, the codoping in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TiO}_2$$\end{document}TiO2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SrTiO}_3$$\end{document}SrTiO3 cannot always enhance the photocatalytic activity due to either the formation of recombination centers or the large shift in the conduction band minimum or valence band maximum. Amongst various metal-nonmetal combinations, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Mn}_\text {Ti}\hbox {S}_\text {O}$$\end{document}MnTiSO (i.e. Mn is substituted at Ti site and S is substituted at O site), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {S}_\text {O}$$\end{document}SO in anatase \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TiO}_2$$\end{document}TiO2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Mn}_\text {Ti}\hbox {S}_\text {O}$$\end{document}MnTiSO, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Mn}_\text {Sr}\hbox {N}_\text {O}$$\end{document}MnSrNO in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SrTiO}_3$$\end{document}SrTiO3 are the most potent candidates to enhance the photocatalytic efficiency of anatase \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {TiO}_2$$\end{document}TiO2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SrTiO}_3$$\end{document}SrTiO3 under visible light irradiation.


Results
Stability of codoped TiO 2 and SrTiO 3 . The stability has been determined by calculating the defect formation energy using hybrid DFT and ab initio atomistic thermodynamics [68][69][70] . The defect configuration, having minimum formation energy, is the most stable defect. For a defect X with charge state q, the formation energy E f (X q ) is evaluated as follow 64,68,71 : where E tot (X q ) and E tot (pristine 0 ) are the energies of defected supercell (charged) and pristine supercell (neutral), respectively, calculated using hybrid DFT. n i is the number of species i added to or removed from the pristine supercell and µ i 's are the corresponding chemical potentials, which is selected with reference to the total energy ( E tot (i 0 ) ) of species i. Therefore, µ i = �µ i + E tot (i) (i = N, S, O, Mn, Rh, Sr, or Ti), where �µ i 's are chosen according to the environmental growth conditions. µ e is the chemical potential of the electron, which is the energy required to exchange electrons between the system and the electrons' reservoir. It is varied from VBM to CBm of the pristine supercell. V is the core level alignment between pristine neutral and defected supercell.
In our previous findings, we have seen that the substitutional defect is more favorable in comparison to interstitial and also observed that the monodopants in general are not suitable for photocatalytic application 64,65 . Therefore, we have considered here the codoped cases [metal (Rh or Mn) substituted at Ti or Sr site, and nonmetal (N or S) substituted at O site].
To know the stability of different charged states at different environmental growth conditions, first we have shown the 2D formation energy plot for Mn-S related defect in SrTiO 3 (see Fig. 1a-c). For, Mn Sr S q O defect in SrTiO 3 , the formation energy is given by: (1)  , where the last term is the zero point energy of O 2 molecule. And for, Mn Ti S q O defect in SrTiO 3 , the formation energy is given by: The chemical potential of a species incorporates the effect of temperature and pressure 72 . For oxygen, the �µ O as a function of temperature (T) and the partial pressure of oxygen ( p O 2 ) is calculated as follow 69 : where m is the mass, I A is the moment of inertia of O 2 molecule, ν OO is the O-O stretching frequency, M is the spin multiplicity and σ is the symmetry number.
Under equilibrium growth condition, the chemical potentials are related to enthalpy of formation of SrTiO 3 (�H f (SrTiO 3 )) by: Similarly, for TiO 2 , the equilibrium growth condition is: In addition, the chemical potentials are bounded by the following relation with enthalpy of formation to hinder the formation of secondary phases: While solving, we have taken the equal sign in Eq. (7), except for the last bound. Therefore, we can see that the chemical potentials could be determined by imposing bounds on the formation of the precursors ( MnO 2 , TiO 2 , TiS 2 , Rh 2 O 3 ) or the secondary phases and are inter-related.
We  Fig. 1c). This implies that formation of Mn Sr S O in SrTiO 3 is easier in O-poor condition. Moreover, from Fig. 1a-c, we can see that Mn Ti S O is stable only in O-rich condition near CBm (n-type) with charge state −1 , whereas, the positive charge states are stable for Mn Sr S O codoped SrTiO 3 . +2 charge state is stable near VBM and thereafter in between (+ 1) charge state is most stable. This indicates that the defect will act as a donor. Near CBm, −2 charge state is most stable. This whole result of 2D formation energy plot at different growth conditions can be summarized by a 3D phase diagram that shows only the most stable phases having minimum formation energy (see Fig. 2g). In 3D phase diagram, �µ O is varied according to environmental growth condition (on x-axis) and µ e is varied throughout the band gap from VBM to CBm (on y-axis). On z-axis, we have shown the stable phases using colored surfaces, which have minimum formation energy. From Fig. 2g, we can easily see that Mn Sr S O is stable in + 2, + 1 and − 2 charge states in all the conditions, which is also confirmed from Fig. 1. Whereas, Mn Ti S O is only stable in − 1 charge state at O-rich condition near CBm, which can also be seen from Fig. 1a. Figure 2 shows the stable phases of codoped TiO 2 and SrTiO 3 having minimum formation energy. Note that in case of codoped TiO 2 , for different charge states, the dependence on chemical potential of oxygen is same for a particular kind of defect (for e.g. Mn substituted at Ti and S substituted at O). Since we have shown only one kind of defect i.e., only substitution at particular sites (a metal at Ti and a nonmetal at O) with different charge

Electronic density of states (DOS).
The defect states could be seen by means of electronic density of states. Figure 3 shows the atom projected density of states for pristine and codoped TiO 2 as well as SrTiO 3 . In the pristine systems, near Fermi level, the valence band is contributed by O 2p orbitals and the conduction band is contributed by Ti 3d orbitals (see Fig. 3a,e). The DOS is symmetric w.r.t. the spin alignments ascribed to the paired electrons in the system. On doping, the DOS becomes asymmetric attributable to unpaired electrons. These unpaired electrons tend to result in finite magnetic moment. The details of magnetic moment for codoped systems are given in Supplementary Tables S1 and S2 (see Section I of SI  . 3b). However, this shift will lower down the CBm and thus, not efficient for reduction of water to produce hydrogen. The VBM is elevated in Rh Ti S O codoped TiO 2 , which is caused by the S and Rh orbitals contribution (see Fig. 3c). Furthermore, the CBm is also shifted down due to the unoccupied states of S and Rh orbitals. Therefore, despite its spectral response in visible region, it cannot be used for producing oxygen via water reduction ascribed to the large shift in VBM. The band gap reduction is induced by the S orbitals in www.nature.com/scientificreports/ S O doped TiO 2 as the S orbitals energies lie higher than the N orbitals (see Supplementary Fig. S1b). Also, in Mn Ti S O codoped TiO 2 , the S orbitals elevate the VBM as they have higher energy than the O orbitals and the Mn orbitals contribute to CBm (see Fig. 3d). This will enhance the photocatalytic efficiency, as the band gap becomes 2.2 eV in both the aforementioned cases and the band edges straddle the redox potentials of water. Note that the energy gap is defined as band gap between the highest occupied and lowest unoccupied band, provided that defect level is not far away from VBM and CBm.  Supplementary  Fig. S1f,g). However, the reduction in band gap for S O is small in comparison to S O doped TiO 2 . S O doped SrTiO 3 has the band gap of 2.59 eV and hence, responses to visible light irradiation. For Mn Ti monodoped SrTiO 3 , the Mn orbital contributes to the CBm and lowers down it (see Supplementary Fig. S1h). Therefore, its spectral response expands to visible light irradiation (band gap is 2.57 eV). In Rh Ti monodoped SrTiO 3 , the unoccupied states of Rh orbitals appear at VBM, and the difference between highest occupied and lowest unoccupied state is 0.23 eV (see Supplementary Fig. S1i). Thus, it is not a promising candidate for enhanced photocatalytic activity. The lowering of CBm in Rh Sr is occured due to the Rh localized states contribution to CBm and therefore, it doesn't have enough reduction power to produce hydrogen via water splitting (see Supplementary Fig. S1j). There is no reduction in band gap for Mn Sr monodoped SrTiO 3 , as the Mn orbitals contribute deep inside the valence and conduction band (see Supplementary Fig. S1k). Likewise Mn Ti N O codoped TiO 2 , the reduction in band gap of Mn Ti N O codoped SrTiO 3 is occurred by lowering of the CBm as well as elevation of the VBM (see Fig. 3f). However, the shift in CBm is large enough such that its reduction power is deteriorated. In Rh Ti S O and Rh Sr S O codoped SrTiO 3 , the deep trap states arise in the forbidden region, that increase the recombination Optical properties. The optical spectra have been determined by calculating the frequency dependent complex dielectric function ε(ω) = Re(ε) + Im(ε) using HSE06 functional. The real part Re(ε) and the imaginary part Im(ε) are associated with the electronic polarizability and optical absorption of the material, respectively. The sum of all possible transitions from the occupied to the unoccupied states gives the direct interband transition, which is reflected in the imaginary part of the dielectric function. The imaginary and real part for codoped anatase TiO 2 and SrTiO 3 are shown in Fig. 4 (the results for monodoped TiO 2 and SrTiO 3 are shown in Supplementary Fig. S2). Note that anatase TiO 2 has tetragonal structure. Therefore, the optical anisotropy is also associated with it. The detailed discussion of optical anisotropy is already done in our previous work 74 . Therefore, here we have shown only the averaged (x, y, z polarizations) for imaginary and real part of the dielectric function. The imaginary part of dielectric function shows the first peak at 3.56 eV for pristine TiO 2 as shown in Fig. 4a (matching with the previous works, which is 3.8 eV 75 ). The peaks are shifted to lower energy for codoped cases. This enhances the visible light absorption of anatase TiO 2 . The static real part of the dielectric function (at ω = 0 ) for TiO 2 is found to be 5.28 (see Fig. 4b), which is very close to the experimental value i.e., 5.62 76 . On codoping its value is increased. For the case of cubic SrTiO 3 , the spatially average imaginary and real part of dielectric function are shown in Fig. 4c,d, respectively. The static real part of the dielectric function for pristine SrTiO 3 is estimated as 4.7 (experimental value is 5.27 77 ) and its value is increased with codopants (see Fig. 4d). The first absorption peak is observed at 4.08 eV for pristine SrTiO 3 as shown in Fig. 4c (experimental value is 4.7 eV 77 ). Likewise in anatase TiO 2 , the peaks are shifted to visible region for the codoped cases. Note that the optical properties in the high energy range are controlled by the electronic transitions between O 2p states and Ti 3d states. Therefore, the spectra of all the configurations are nearly identical in high energy range. However, the optical properties in low energy range (less than 3 eV) are different, these are affected by the transitions involving the impurity states. The observed visible light absorption could be ascribed to the presence of the dopant states (as shown in DOS near Fermi-level), which reduce the electron transition gap for optical absorption. This leads to a new absorption edge in the visible light region.
Band edge alignment. The band edge alignment has been performed to obtain the potential candidates for photocatalytic water splitting. The CBm should lie above water reduction potential and VBM should lie below water oxidation potential for overall water splitting. Note that we have adopted the standard methodol- www.nature.com/scientificreports/ ogy to align the band edges as in Refs. 78,79 . First we align the band edges of pristine TiO 2 and SrTiO 3 w.r.t. water redox potential. For pristine SrTiO 3 , the CBm lies 0.8 eV above the water reduction potential and VBM lies 1.25 eV below water oxidation potential, which is reported in the experimental study 49 . Further, we align the band edges of defected configurations by observing the shift in CBm and VBM w.r.t. the pristine system (see Fig. 5). Similarly, we have aligned the band edges of (un)doped TiO 2 . For pristine TiO 2 , the CBm lies 0.4 eV above the water reduction potential 48 . The band edge alignment for monodoped anatase TiO 2 and SrTiO 3 are shown in Supplementary Fig. S3. The monodoped N O is not suitable in both the cases (anatase TiO 2 and SrTiO 3 ), as it results in deep trap states. This increases the recombination and decreases the mobility of photogenerated charge carriers (see Supplementary Fig. S3). Likewise, for Rh dopant (in monodoping as well as in codoping), there is occurrence of trap states. These states degrade the photocatalytic efficiency. Therefore, mono-and codopoing of Rh with a nonmetal could reduce the band gap, but it cannot be an efficient photocatalyst in TiO 2 as well as in SrTiO 3 . The monodoped S O in both anatase TiO 2 as well as SrTiO 3 could enhance the photocatalytic efficiency and split water as their band edges straddle the redox potential of water (see Supplementary Fig. S3). However, in S O monodoped SrTiO 3 , the band gap (2.59) is slightly higher than the desirable band gap ( ∼ 2 eV 51,52 ), and thus, its efficiency will be smaller. Similarly, for Mn Ti monodoped SrTiO 3 , the band gap is 2.57 eV, and due to shift of its CBm towards Fermi level, its reduction power will be degraded (see Fig. S3). On the other hand, for Mn Sr monodoped SrTiO 3 , and Mn Ti monodoped anatase TiO 2 , the slight change in band gap is observed and thus, these can not enhance the photocatalytic activity.
In Rh Ti S O codoped TiO 2 , since there is manifestation of deep unoccupied states as well as the VBM lies above the oxidation potential of water, it could not be utilized for photocatalytic overall water splitting (see Fig. 5). For Rh Ti N O and Mn Ti N O codoped TiO 2 , the CBm lies below the reduction potential of water, and thus, cannot produce hydrogen via water splitting. In TiO 2 , only the Mn Ti S O codoping is the potential candidate for overall photocatalytic water splitting as it has a desirable band gap of 2.22 eV and it does not contain the trap states in forbidden region while retaining the sufficient reduction and oxidation power for hydrogen evolution reaction (HER) as well as oxygen evolution reaction (OER). Similarly, in SrTiO 3 , except Rh Sr N O , the Rh doping does not aid in enhancing the photocatalytic activity ascribed to the formation of recombination centers. Rh Sr N O defect configuration enhances the photocatalytic efficiency, however its band gap (2.69 eV) is little bit larger in comparison to the maximum efficient photocatalyst ( ∼ 2 eV). In Mn Sr S O , since the occupied deep states lie below the CBm and also these are not the shallow impurity levels, this configuration is not a desirable photocatalyst. The reduction in band gap for Mn Ti N O is concomitant with the lowering of CBm, that deteriorates its reduction power. The Mn Sr N O , and Mn Ti S O codoped SrTiO 3 configurations are the potential candidates for overall photocatalytic water splitting attributable to their desirable band gap ( ∼ 2 eV) with congenial band edge positions.
Band structure and effective mass. To see the effect on mobility due to the defects, we have calculated the effective mass of charge carriers (using HSE06) of those systems, which could be promising candidates for overall photocatalytic water splitting (see Table 1).
These are obtained from the relation of effective mass ( m * ) with second derivative of energy with respect to k (wave vector) at the band edges:  [79][80][81] . Except for the heavy hole, all are matching well. For pristine SrTiO 3 , the effective masses are calculated along Ŵ -X high symmetry path. Pristine has degenerate bands at the Ŵ k-point (see Fig. 6a). In contrast to pristine SrTiO 3 , rest of the cases have non-degenerate bands (highest occupied and lowest unoccupied) (see Fig. 6). Note that here, we have shown the total bands containing both the spin up and spin down contribution.  Fig. 6c).
In pristine TiO 2 the CBm is at Ŵ k-point and there is no degeneracy (see Fig. 6d). The electron's effective mass is 0.39m e along Ŵ -Z high-symmetry path, and the effective mass of hole is −1.57m e along VBM-Z and −1.61m e along VBM-Ŵ direction. For Mn Ti S O codoped TiO 2 and S O monodoped TiO 2 , the electron's effective mass (along Ŵ-Z) is comparable with pristine, whereas the hole's effective mass (along VBM-Z) is increased. These increments are also evident from the smaller curvature of the bands around the band edges (see Fig. 6e,f). For larger effective mass, the mobility will be smaller and the recombination rate will also be greater. Therefore, from Table 1, we can see that in case of Mn Ti S O codoped SrTiO 3 , the mobility of charge carriers will be large, and for rest of the cases, the effective mass values are comparable and the mobility will not be affected much. This is because, the mobility depends on both the effective mass and scattering (relaxation) time. On doping, the scattering rate is expected to get decreased as the degeneracy will be lifted. As a consequence of this, despite of small increment in effective mass, the mobility will not be affected considerably, especially here due to low doping concentration 82 . These effective mass studies should assist future experimental as well as theoretical investigations to tailor the transport properties of the system.

conclusions
In summary, we have evaluated the thermodynamic stability of (un)doped anatase TiO 2 and SrTiO 3 using hybrid DFT and ab initio atomistic thermodynamics. We have found that the codopants in TiO 2 could act as donor (in p-type host) as well as acceptor (in n-type host). However, the most stable codopants (codoping of metal at Sr site and nonmetal at O site) in SrTiO 3 mostly act as donors. The codoping expands the spectral response and induces visible light in both the cases. However, the recombination centers are present in Rh-related defect configurations attributable to Rh localized orbitals in the forbidden region and moreover, there is a large shift in the CBm or VBM. This will lead to degradation in photocatalytic efficiency. The mobility of charge carriers is maximum in Mn Ti S O codoped SrTiO 3 , and in rest of the cases, it is not affected much. Our results reveal that Mn Ti S O codoped, S O monodoped anatase TiO 2 , Mn Ti S O and Mn Sr N O codoped SrTiO 3 are the most favorable candidates for enhancing photocatalytic overall water splitting owing to the passivation of trap states and congenial band edge positions with desirable visible light absorption.

Methods
We have performed the DFT calculations as implemented in Vienna ab initio simulation package (VASP) 83,84 . The projector-augmented wave (PAW) pseudopotentials 85 have been used to describe the interactions between electrons and ions for all the species. For the energy calculations, hybrid exchange-correlation (xc) functional HSE06 86 is used. Note that we have seen in our previous study that GGA+U is not a good functional to predict the correct energetics, albeit it can reproduce the correct band gap with suitable value of U 64 . The exact exchange fractions in HSE06 functional used for TiO 2 and SrTiO 3 are 22% and 28%, respectively (see SI Ref. 64,65 for validation of exact exchange fraction). The band gap of 3.15 eV and 3.28 eV are reproduced for TiO 2 and SrTiO 3 respectively, which are well in agreement with the experimental values 87,88 . To make the defect to be localized, we have used 2 × 2 × 1 (48-atom) and 2 × 2 × 2 (40-atom) supercells by replication of TiO 2 and SrTiO 3 unit Table 1. Effective masses (in terms of free-electron mass m e ) at the band edges. The masses m he , m le , m hh , and m lh correspond to heavy-electron, light-electron, heavy-hole, and light-hole band, respectively. www.nature.com/scientificreports/ cells respectively (for validation of supercell size, see section VI of SI and also, the localized states due to defects can be seen from band structure in Fig. 6). The k-grid for Brillouin zone sampling is generated using Monkhorst-Pack 89 scheme and all results are checked for convergence w.r.t. the mesh size ( 4 × 4 × 4 ). The electronic self-consistency loop for the total energy is converged with a threshold of 0.01 meV. An energy cutoff of 600 eV is used for the plane wave basis set. Note that the spin-polarized calculations have been carried out since the doped systems contain unpaired electrons.