XiangLiu-Final

Xiang Liu

=**First-principles calculations on doped perovksites**=

Abstract
First-principle calculations on doped perovskites are introduced in this paper. Some widely used calculation methods are briey discussed. The doping on perovskites can be achieved by bringing new foreign element at A site, B site, oxygen site as well as removing native atom to generate native vacancy.These dopants can introduce signi cant changes to the electronic structures of the perovskites through several dierent ways:(a)they can introduce mid-gap states as well as states in valance band and conduct band, (b)they can distort the original band structures, (c)they can hybridize with the native atomic states, (d)they can shift the Fermi level towards valance band or conductor band, and (e)always they can eectively narrow the band-gap energy. A brief discussion of those eect are discussed in this paper.

1 Introduction
Perovskites, a family of oxides having the general formula ABO 3 where cation A has larger size than B, have received increasing attraction because their diverse applications in superconductors[1], electrocatalysts[2], pollution abatement[3], and chemical sensors[4], etc. Especially,some perovskites have shown high ecient in catalyzing the photodissociation of water molecules[5, 6, 7]. This is a very interesting property because this reaction can provide a green and high ecient fuel, H 2 O, to substitute the fossil fuel of which the reserves are decreasing rapidly, just with consuming the energy from sunlight which is almost unlimited and everlasting.

\ Figure 1.1: The crystal structures of perovskites with two dierent space group:(a)Orthorhombic Pbnm, shown in 1* 1*1 unit cell;(b)Cubic Pm3m, shown in 1* 1*2 unit cells, where Oxygen anions are shown as red balls, A cations are shown as yellow balls, BO 6 octahedrons are shown in blue color. In the perovskite structure, the smaller B cation is 6-fold coordinated and thebigger A cation is 12-fold coordinated with the oxygen anions. The crystal can be considered as that the octahedral BO 6 share their vertexes with each other to form the backbone and the A cations ll into the interstice between those octahedrons. Typical crystal structures of perovskites of two common space groups are shown in the Fig.1.1. The Pbnm in Fig.1(a) can be considered as a distorted structure from the Pm3m in Fig.1(b). Precedent experimental works have proved that various perovskites can be the photocatalysts for water dissociation. Unfortunately, pure perovskites always have low photocatalytic eciency[7, 8].Hence,in order to make these materials valuable for application, some modi cations and improvements are needed to be applied towards pure perovskite materials. For example, Kim et al[6] showed Ni-doped (110)layered perovskite materials can have quantum yields up to 23% and pointed out the catalyzing ability comes from not only the speci c composition but also the layered structure of perovskite. Furthermore, Kato et al[7] found NiO-loaded NaTaO 3 doped with lanthanum can show a even higher activity under UV irradiation, with the highest quantum yield at 56% at 270 nm. The doping of La can eectively decrease the size of the NaTaO 3 particles and create a characteristic nanostep structure at the surface of the particles. It is believed that the high catalysis ecient is attributed from these structural changes as well as the loaded NiO which can act as an active H 2 formation catalyst. Although the perovskites can have high efficient in catalyzing water dissociation, there are still some problems needing further studies. The band gap energies of semiconductive perovskites are always in the UV region of 3.0 -4.0 eV[9]. This is an annoying property for practical application because the UV region only occupies about 4% of the whole solar spectrum arriving on earth. Some eorts have been done aiming at shifting the absorbance band of perovskites into visible region which occupies the majority of the solar spectrum and still keeping their catalysis ability. Miyauchi et al[8] use codoping of La and N to eectively extend the absorption of SrTiO 3 into visible region (400 nm <  < 700 nm ). The doping of nitrogen will narrow the band gap and extend the absorption into visible region but will result in oxygen vacancies, in order to maintain electronic neutral, that will decrease the photocatalytic eciency and the codoping of lanthanum can suppress the generation of such oxygen vacancies. Zhou et al[10] utilized small metal cations(Mn, Fe, and Co) to substitute the B site of SrTiO 3 and NaTaO 3. The band gaps of doped perovskites had been reduced into the region of 2. 0 -3.0 ev and the absorption bands had been extended to 650 nm for Fe-doped NaTaO 3.

Therefore, the doping can significantly change both the electronic structures andthe morphologies of perovskites. Numerous theoretical studies have been carried out in an eort to get some insights into the interaction between the subtract perovskite and the foreign dopant. This articles aims at reviewing previous works done in this field.

First established in 1960s[11, 12], the density functional theory (DFT) has been successfully applied in diverse dierent theoretical quantum mechanical computational studies. Unlike in traditional Hartree-Fock method where the ground-state properties are determined by the wavefunctions, in DFT, the ground-state properties of a quantum many-body system are only determined by using functionals of the spatially dependent electron density n ( ** r ** ). The Hohenberg-Kohn theorems (H-K) proved in 1964[11] gives a rm theoretical footing to DFT. H-K theorem 1 : For N interacting electrons in an external potential v ( ** r ** ), there is a universal functional of the density F [ n ( ** r ** )]. H-K theorem 2 : The density functional is minimized by the ground state electron density n 0 ( ** r ** ), and then the E is the ground state energy.

Within this framework, an intractable many-body problem with interacting electrons in an external potential can be reduced to a tractable problem of non-interacting electrons in an eective potentials. In DFT, the total energy E can be decomposed in a formally exact way into three terms: (2.1) where the T s [ n ( ** r ** )] is the kinetic energy of a system of non-interacting electrons with density n ( ** r ** ), the rst two terms in the left-hand side of Eq. 2.1 is the electrostatic or Coulomb energy terms and the E XC [ n ( ** r ** )] is called exchange-correlation term. In order to achieve the electron density n ( ** r ** ), it is needed to solve the Kohn-Sham equation which looks similar to the one-particle Schrodinger equation[12]: (2.2) Where, (2.3) (2.4) And setting, (2.5) To solve the Kohn-Sham equation, the wavefunctions needs to be expanded in a basis set as: (2.6) Therefore, in order to utilize DFT to study molecules, clusters and solids, we need to de ne the electronic structures including the basis set and potential shape, etc., as well as the exchange-correlation energy  XC. I will discuss in details about various choices of those terms in following sections. 2.2 Implementations of electron structures

Table 1: Some common choices for potentials and basis sets in DFT Potential Basis Sets Table 1 has listed some common implementations for the potentials and basis sets that can be applied in DFT. Because this article is focusing on the studies on perovskite which is a crystal, all the DFT calculations discussed in this article have periodic boundary conditions. The pseudopotential method just concerns the valence electrons of the atoms and the core electrons are considered to be bounded inside the atom whereas the full potential method concerns all electrons of the atoms. Hence, the treatment of the potentials is closely related to the choice of the basis set. I will focus on three typical methods utilized in most calculation studies to perovskites: (1)fullpotential linearized augmented plane-wave method [17];(2)pseudopotential method[13];(3)projected augmented-wave method[16]. **Full-potential linearized augmented plane-wave method**

In full-potential linearized augmented plane-wave method (FLAPW), all contributions to the potential are taken into account in the Hamiltonian matrix[17]. In thismethod, the unit cell is separated into two parts:the sphere around each atom where the wavefunctions are rapidly varying and atomic-like; and the interstitial region between the spheres, where the wavefuntions are more smooth and non-atomic-like. Then the basis functions are constructed by smoothly connecting the plane-wave functions in the interstitial region and the linear combination of atomic-like functions in the sphere. Because this method explicitly include all the electrons and even perhaps without any further shape approximations[17], the FLAPW can give enough accurate and reliable solutions for any atomic arrangements, such as close-packed, open, highsymmetry or low, surfaces or bulk. Nevertheless, because of its explicitly calculation on all electrons, the calculation involved in FLAPW is always expensive. **Pseudopotential method** The basic notion in pseudopotential method (PS) is that the electrons in atoms have been separated into two dierent categories: the core electrons with approximate spherical symmetry and tight-binding character and the valence electrons experiencing the core as screened by the core electrons[18]. Furthermore, in PS, only the valence electrons are considered to be responsible to the physical phenomena while the core electrons are unresponsive. Therefore, during the calculation process, the PS can just treat the valence electrons and ignore the core electrons explicitly[18].Obviously, the calculation can be much more efficient and faster which is a valuable advantage over the all-electron method like FLAPW where all the electrons need to be calculated and, in most cases, the results can still be accurate and reliable[19]. However, when dealing with the rst-row atom and transition-metal systems, where the distinctions between core electrons and valence electrons are difficult to difine because of the high dispersion of valence electrons into near-core space, the PS would become inaccurate and unreliable results[13]. In order to extend the more efficient PS method to those systems, some efforts has been made,such as the ultrasoft pseudopotential method established in 1990s[13]. In simple PS method, a norm-conserving criteria is satis ed, that is, outside the core, the real and pseudo- potentials should have the same charge density[20] (2.7) where the AE is the real all-electron wavefunction and the ps is the pseudo wavefunction. However, in ultrasoft PS,to generate a smoother wavefunction, this normconserving criteria is released by splitting the wavefunction into two parts: ultrasoft valence wavefunction that do not fullfil the norm-conserving criteria as and a core augmented charge as [13]. The ultrasoft PS method needs some extra operations compared to the simple PS method but is still more computationally ecient than the all-electron methods and it can successfully applied to the rst-row elements and transition metal systems such as perovskites[21].
 * Projector augmented-wave method **

Established by Blochl in 1994[16],the projector augmented-wave method (PAW) combines the versatility of FLAPW and the formal simplicity of PS by directly working on the full valence and core wavefunctions with pseudopotential approaches. Similarly as in FLAPW, in PAW, the wavefunctions are separated into two parts by some sphere boundaries: the atomic-like waves inside and smooth planewave outside. In PAW, the all-electron (AE) wavefunction can be obtained from the PS wavefunction (2.8) where the quantities are (i) the AE partial waves as the orthogonal solution of the Schrodinger equation for the isolated atoms. (ii) the PS partial wave coinciding with the corresponding AE partial wave outside the sphere boundaries. (iii) the projector function for each PS partial wave localized within the sphere boundaries and obeying the relation [16]. As said by Blochl, PAW is an extension of augmented wave methods and pseudopotential approach, which combine their traditions into a uni ed electronic structure method[16]. This method can be applied to first-row elements as well as transition metal systems with a modest calculation and yield results with similar accuracy to FLAPW. 2.3 The exchange-correlation energy As shown in the Kohn-Sham equation(Eq. 2.2), the exchange-correlation (EC) potential, the first-order derivative of EC energy E XC ( n ( ** r ** )) with respect to the electron density(2.9)

Although the EC terms somehow represent the interactions between two densities, it is generated from mathematical derivations rather than precisely physical observables[22].Therefore, it is nearly impossible to de ne a exact form for the EC functionals and can only be estimated by some approximations. In general, the approximation approaches can be categorized as local approach where the EC energy only depends on local properties such as the electron density n ( ** r ** ) and its gradient and non-local approach where the energy also depends on global properties. The local approach is relatively less accurate but much more computationally efficient, as well as the local method is more accurate but more complicated and computationconsuming. The most widely used local methods are local density approximation (LDA) and generalized gradient approximation (GGA). In LDA, the EC energy is established to depend on just the local density n ( ** r ** ). (2.10) The LDA is spin-independent, a derivation of LDA including the electron spins is called local spin density (LSD). (2.11) where representing the electron spin densities,  and in LDA and LSD respectively as the exchange-correlation energy per partical of uniform electron gas that is well established[23]. As a further development to LDA, the GGA was constructed as that the EX energy depends not only on the local spin densities but also their gradients. (2.12)

Because of the semi-local properties of GGA, the GGA can yield better results than the results of LDA, lisk total energies, atomization energies, etc[24]. In general, these local approach is suciently reliable and accurate in predicting the crystal structures and lattice parameters of semiconductors like perovskites. However, there is well-known issue that these local approaches would signi cantly underestimate the band-gap of semiconductor as a highly-electron-interacting system. In order to correct this issue, numerical eorts have been made, like the GW method[25], the scissor correction method[26], and LDA+U method[27], etc. Because of the formal simplicity and computational eciency, the GGA method has still been the most widely used DFT exchange-correlation approach so far even in studying the perovskite systems.

3 Types of dopants
For semiconductors, doping can always be categorized as two dierent types:(1)n-type dopant, which brings in extra electrons;(2)p-type dopant, which brings in extra holes.For metal oxides such as perovskite, the n-type dopant can be achieved by doping foreign metal cation with lower charges like substituting Sr(II) with La(I)[28] and the p-type dopant can be achieved by substitute the original metal cation with a cation having higher charges like doping Mn(IV) for Ba(II)[29]. Furthermore, for perovskite ABO 3, the doping can occur with two dierent types:( 1) The dopant substitutes the A metal cation that has a larger ion radius[30];(2)The dopant substitutes the B metal cation that had a smaller ion radius[31]. And sometimes these two dierent types can occur at the same time[32]. Besides, the doping system can also be categorized by the dopant elements. Typically, the dopants are transition metals when the doping would occur on A or B site. And the dopant can be main-group elements as nitrogen when the nitrogen would replace the oxygen. Otherwise, the vacancy of native elements can also be considered as a kind of doping. The formation energy of a dopant in a compound depends on the atomic chemical potentials of the system and the fermi energy of the system. In general, the formation energy can be calculated by  (3.1) where the E total [ D; q ] is the total energy of the supercell with one dopant D in charge state q and E total [ perfect ] is the total energy of the perfect-crystal supercell. n i is the number of atoms of type i being added to ( n i > 0) or removed from ( n i < 0) in the perfect-crystal supercell and is the atomic chemical potential. E f is the Fermi level[29, 33]. In this equation, the atomic chemical potential are free variables but still subjected to some constraints. For example, in NaTaO 3 doped with La, in order to prevent the spontaneous formation of O 2, Na, Ta as well as Na 2 O and Ta 2 O 5 , those chemical potentials must satisfy where the zeros mean the formation energy of their reference elements or oxides. In practical computation, those references are de ned by the total energy of the most stable of those elements or oxides in a particular condition. From these we can see that the chemical potentials can only lie in a particular region con ned by the above inequalities.

3.1 Doping by group V-A elements
As nitrogen-doping on other semiconductors has been extensively studied and especially the success of nitrogen-doping on TiO 2 [34], what could happen if dope nitrogen on perovskite has rationally attract interesting in scientists. How the nitrogen dopant affect the perovskite will be discussed here in theoretical approach. Figure 3.1: Calculated DOS for each element:(a)SrTiO 3 ;(b)SrTiO 2.75 N 0.125 from Miyauchi et al[8]. Miyauchi used ultrasoft pseudopotential with GGA to calculate the electronic structure of SrTiO 3-2x N x systems. The density of states of perfect SrTiO 3 and doped SrTiO 3-2x N x ( x = 0.125), the author assume that nitrogen-doping would contain oxygen vacancies to maintain charge balance) are shown in Fig.3.1. In Fig.1(b) we can see that the valence and conduction bands of perfect SrTiO 3 consists of O(2p) and Ti(3d) orbitals respectively and the band gap is about 1.6 eV, which underestimates the experimental value at 3.2 eV due to the shortcoming of GGA. In Fig.1(a), an isolated mid-gap band of Ti(3d) orbitals can be observed in the band gap mainly due to the localized electrons of Ti 3+ . It is believed that this mid-gap band can behave as electron-hole recombination centers decreasing the photocatalytic eciency.[8] Mi et al[35] used the same calculation method to study the electronic structure of SrTiO 3 -xNx which is shown in Fig.3.2. For perfect SrTiO 3, Mi and Miyauchi had similar results that valence and conduction bands of perfect SrTiO 3 consists of O(2p) and Ti(3d) orbitals respectively and the band gap in this work was 1.8 eV where the difference might become their dierent supercell and choosing of parameters. When comparing their results for N-substituted SrTiO 3, some similar structures can also be observed:(1)The band gap is slightly reduced by the doping but the change is not signi cant;(2)The valence band mainly consists of O(p) and N(p) orbitals. However, an obvious dierence can be found about Ti(d) orbitals, in Fig.3.2(b), the character midgao band in Miyauchi's work cannot be observed here. A possible reason is that the charge balance is satis ed by removing one extra O atom generating a O vacancy in Miyauchi's work (N:O ration is 1:22)[8] but it is done by just charge transfer without O vacancy in Mi's(N:O ration is 1:23)[35]. Besides, in interstitial model, the valence band is a hybridization of Ti(d) and N(p) orbitals. Figure 3.2: Local DOS of (a)perfect SrTiO 3 ;(b)nitrogen-substituted SrTiO 3 ;(c)SrTiO 3 with interstitial N atoms from Mi et al[35]. A more reliable calculation was done by Onishi [36] by using hybrid DFT method which is more reliable in predicting strong correlated semiconductors[37]. It showed that the band-gap reduction is caused only by nitrogen-doping and the charge transition can occur from the impurity level to the conduction of Ti(d) orbital and O vacancy is irrelevant to the band-gap reduction as well as the band-gap reduction depends on the concentration of dopants where the reduction would not occur under high dopants concentration[36]. Table 2: Some well studied perovskite substrates with dierent transition-metal dopants



3.2 Doping by transition metal
Since experiments showed that transition-metal-doped perovskites have high photocatalytic efficiency[6, 7], studying the interaction between the transition-metal dopants and the perovskites is quite useful in designing future photodissociation catalysts. Some theoretical approaches in studying the eect of transition-metal dopants will be discussed here. As shown in Table 2, doping of transition metal can occur on both A and B sites of perovskites which will be discussed separately. From Table 2, we can see DFT with local exchange-correlation energy(GGA in most cases) can give a close lattice parameters compared to experimental values, however, the band gaps are drastically underestimated by 1-2 eV. Therefore, some extra correction method will be needed in order to achieve accurate band-gap energies. **• Transition-metal dopants on A site ** Choi et al[33] studied the La-impurities in NaTaO 3 via PAW with GGA exchangecorrelation energies. As shown in Fig.3(a), in neutral charge state, La Na is energetically preferred under most conditions (2-6) whereas La Ta is more stable only under O-rich limit (1) with too high formation energy to form a substantial concentration. The formation energies as a function of Fermi level are shown in Fig.3(b). In O-poor condition (5), La Ta is more stable than La Na only in a highly n-type region and it s positive formation energy throughout the whole range of the Fermi level make it difficult to form a high concentration either. Figure 3.3: Calculated formation energies of La-doped NaTaO 3 where La Ta means La dopes in Ta site and La Na means La dopes in Na site. :(a)the condition 1-6 corresponds from Na-poor and O-rich conditions to O-poor conditions in neutral charge states;(b)the formation energies as a function of fermi level under condition 1 (solid) and 5 (dashed)[33]. Figure 3.4: Site-decomposed electron DOS in the vicinity of La Na and La Ta along with the total DOS of the perfect crystal. The red curves are the DOS of La and the grey curves are DOS of its neighbors. The dashed lines indicate the position of CBM and VBM[33]. The Fig.3.4 give the one-electron structures of the doping systems. We can see La Na introduce no localized mid-band states in the calculated band gaps and it does not change the band-gap energies consistently with other works[39, 40]. Kanhere et al[32] used PAW with GGA-PBE to study the Bi-doped NaTaO 3. The calculated total DOS is shown in Fig.3.5. Comparing Fig.3.5(a) with Fig.3.5(c), we can see that the A-site doping of Bi causes relatively small change to the band gap structures with just 0.15 eV reduction. From the PDOS in Fig.3.6, we can see the reduction of band gap is mainly due to the Bi(6s) state that is slightly above the VBM. Figure 3.5: Total DOS of (a) perfect NaTaO 3, (b)NaTa 1-x Bi x O 3 , (c)Na 1-x Bi x TaO 3 and (d)Na 1-x Bi x Ta 1-x Bi x O 3 (x=0.0625)[32]. Figure 3.6: Electronic partial density of states (PDOS) of Na 1-x Bi x TaO 3 (x=0.0625)[32]. •** Transition-metal dopants on B site ** Besides the dopant on the inter-octahedral site A, the dopant can replace the cation on the intra-octahedral site B either. Typically, such dopant prefers to be smaller cations like Bi[32],Nb[41],Fe[10],Mn[10],Co[10],etc. Similar to the A-site dopants discussed in previous subsection, the B-site dopants can introduce change to the electronic structures of perovskite either and even more greatly. In Fig.3.4, we can see a character dierent from the La Na is the deep defect state of La Ta located at ~11 eV resulted from the coupling of La 5p, O 2s,and O 2p[33]. This defect state is believed to work as a electron-hole recombination center. This can explain the experimental phenomenon that the photocatalytic eciency is decreased above some doping level[7], because with a high concentration of La, the La Ta can be formed and the defect state as recombination center will be formed decreasing the photocatalytic efficiency. In Fig.3.5(b) with the PDOS in Fig.3.7, a noticeable state forms (in the red rectangular) under CBM after the Bi-dopants replace on the Ta sites. As shown in PDOS, this under-CBM state consists of Bi(6s), O(2p) and Ta(5d), therefore a transition from O(2p) to Bi(6s) can occur during photoexcitation[32]. Furthermore, the band-gap energy is signi cantly reduced by this under-CBM state by about 0.85 eV. Figure 3.7: Electronic partial density of states (PDOS) of NaTa 1-x Bi x O 3 (x=0.0625)[32]. Luo et al[42] compared the n-doped (V for Ti) and p-doped (Sc for Ti) SrTiO 3 by LDA with pseudopotential approach. In their work, n-type dopant can lead to significant changes for the conduction bands but the density of states at the Fermi level is not enhanced signi cantly. Contrarily, the p-dopant introduce no significant change to the original VBM and CBM but the Fermi level shifts downwards into the original valence bands and the density of states at Fermi level is largely increased, and the doped SrTiO 3 becomes metallic[42]. Wang et al[43] use LDA+U method to study doping Cr for Ru on SrRuO 3. They found with more and more Ru was replaced by Cr as increasing the dopant concentration, the metallic character of SrRuO 3 gradually decreased and nally became a insulator over SrRu 0.5 Cr 0.5 O 3 [43]. Figure 3.8: DOS of (a)pure SrTiO 3 ,(b) Mn at Ti site, (c) Fe at Ti Site, and (d) Co at Ti site. The corresponding PDOS are shown in a'-d'. The green curve is for O(2p) state,red curve for Ti(3d) state and blue curve for dopant metal(3d) state repectively[10]. In Fig.3.8, Fe, Mn, Co dopants on SrTiO 3 were studied by Zhou et al[10] by PAW with GGA method. All of these three dopants signi cantly changed the electronic structures, but only Fe-dopant shifted the Fermi level towards the valence band as well as the Fe-doped SrTiO 3 had the lowest band-gap energy which is consistent to the experimental results[10]. Furthermore, the author studied the same dopants on NaTaO 3 which yielded the similar results.

3.3 Site vacancy as dopants
Generally, a native atom vacancy can be considered as a kind of doping either. In perovskite as a kind of oxide, oxygen vacancy has been well studied. The possible distinct configuration for two and three oxygen vacancies in a 222 supercell of SrTiO 3 as shown in Fig.3.9, which should be general to all Pm3m perovskites[28].They further used LMTO calculation to show with increasing O vacancies, the vacancy clustering seriously distorted the bottom of the conduction band, failure for describing in rigid-band model, and some con gurations of the vacancy clustering lead to the formation of localized mid-gap states (F,G,I,J and K) consistent with the Hall measurements on these compounds[44, 45].

Figure 3.9: Dierent con guration arising from the distribution of two O vacancies (A-G) in the 222 supercell of Pm3m SrTiO 3 and the con gurations arsing from the distribution of three O vacancies (H-K)[28]. Another study on O vancancy of SrTiO 3 by Luo et al[42] showed the similar results that O vacancy as a n-type dopant would lead to a distortion of the CBM and shift the Fermi level into the conduct band. They further pointed out the defect states are of Ti(3d)- e g type. Besides O vacancy, other site vacancies can be introduced to the perovskites. Yao et al[46] studied the vacancies of V Pb, V Ti ,V O 1 and V O 2 in PbTiO 3 where O1 represents the oxygen at the apical location of the TiO 6 octahedra and O2 represents the oxygen on the TiO 2 base plane. The formation energies of these vacancies with dierent charge states are shown in Fig.3.10. By comparing their formation energies shwn in Fig.3.10, the author pointed out the optimal Fermi energies for reducing all types of vacancies as shown in Fig.3.11 to produce a crystal as perfect as possible.

Figure 3.10: Vacancy formation energies as a function of Fermi energy for the vacancies:(a)V O 1,(b)V O 2,(c) V Pb, and (d) V Ti. Dierent lines are for dierent charge states[46]. Figure 3.11: Optimal electron Fermi energies(shaded area) that may simultaneously reduce the concentrationd of all types of vacancies in PbTiO 3 under: (a)O-rich condition,(b)intermediate case, and (c)O-poor condition[46].

4 Summary
(i) By using rst-principle calculation we can get insights into the interaction between the dopants and the substrate. The density of states curves can tell us the information of the contributions from the atomic states to the band structures. (ii)In general, the local exchange-correlation functionals such as LDA and GGA would drastically underestimate the band-gap energy of semiconductors. Even though relative comparison is still rational and reliable, combining them with some correction method would be preferred.

(iii)The dopants can dope in perovskite in several dierent way:(a)dopant at A site,(b) dopant at B site, (c) dopant replacing oxygen, and (d) native vacancy. (iv) The dopants can aect the electronic structures of perovskite in several different way:(a)they can introduce mid-gap states as well as states in valance band and conduct band, (b)they can distort the original band structures, (c)they can hybridize with the native atomic states, (d)they can shift the Fermi level towards valance band or conductor band, and (e)always they can eectively narrow the band-gap energy which is a quite useful character in applying perovskite as photocatalyst. The formation energy of dopant can be calculated with the free independent atomic chemical potentials constrained to some boundary conditions.

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