Electronic Band Structures of ATaO3(A = Li, Na ... - ACS Publications

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Electronic Band Structures of ATaO3(A = Li, Na, and K) from First-Principles Many-Body Perturbation Theory Huihui Wang, Feng Wu, and Hong Jiang* Beijing National Laboratory for Molecular Sciences, State Key Laboratory of Rare Earth Materials Chemistry and Applications, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China ABSTRACT: Alkaline tantalates, ATaO3 (A = Li, Na, and K), have attracted a lot of interest in recent years due to their interesting photocatalytic properties and their photocatalytic activity is influenced by a lot of factors, making them ideal model systems for in-depth theoretical investigation. In this work, electronic band structures of alkaline tantalates are investigated based on first-principles many-body perturbation theory in the GW approximation. The band gaps of NaTaO3 and KTaO3 from the GW approach agree very well with experiment; on the other hand, the band gap of LiTaO3 from GW is significantly larger than the experimental values. A strong dependence on crystal structures is observed in LiTaO3, whose band gap in the cubic and rhombohedral structure differs by more than 1.5 eV. Combined with the phenomenological ionic model, it is found that both the Madelung potential and the bandwidth can have strong influences on the band gap. By comparing the structure dependence of LiTaO3 and NaTaO3, it is concluded that the intra-TaO6-octahedron distortion has stronger effects on electronic band structures than the inter-TaO6-octahedron distortion. Possible causes underlying the discrepancy between GW and experiment for LiTaO3 are also analyzed.

1. INTRODUCTION Alkaline tantalates, ATaO3 (A = Li, Na, and K), have attracted intensive interest in recent years due to their photocatalytic activity for solar-driven splitting of water into hydrogen fuels.1
13 Lanthanum-doped sodium tantalate (NaTaO3), combined with NiO as a cocatalyst, currently holds the record of highest quantum yield for direct photolysis of water under ultraviolet light.3 Photocatalytic activities of alkaline tantalates are strongly affected by a lot of factors, such as the presence of excess alkali,2 the loading of NiO as a cocatalyst,2 the doping of lanthanum,3 and the morphology of catalyst particles.7 One of the great challenges is to understand the underlying mechanisms for the influences of various factors. In experiment, all these factors are intertwined so that a definite assignment of determinant factors underlying the photoactivity of a material is anything but trivial. The unprecedented complexity of photocatalytic systems makes theoretical modeling based on first-principles methods extremely important. First-principles modeling can provide atomic and electronic information of the system under study without relying on any empirical input.14 It can also be used to analyze the effects of each individual factor on target properties by investigating “artificial systems” in a well controlled manner. For a photocatalytic semiconductor, the first and most important intrinsic factor is its band gap, which sets the upper bound for the amount of the energy in the solar spectrum that can be absorbed and therefore converted into electric or chemical energy.15,16 In combination with the absolute band positions, i.e., the energy r 2011 American Chemical Society

of the valence band (VB) maximum and conduction band (CB) minimum with respect to the vacuum level, it also determines what kind of chemical reactions can be photocatalyzed. From a theoretical point of view, a first-principles prediction of the band gap is not trivial. Kohn
Sham (KS) density functional theory (DFT) in the local density or generalized gradient approximation (LDA/GGA),17 currently the “standard model” for first-principles modeling of materials,14 suffers from the so-called band gap problem, i.e., the band gaps of typical semiconductors are systematically underestimated by 30
100%.18 The current method of choice for first-principles description of band gaps is provided by many-body perturbation theory, as formulated in terms of oneparticle Green’s function, in the GW approximation.18
23 The band gaps of many semiconductors and insulators have been found to be well described by the GW method with an accuracy that is comparable to the experimental uncertainty.18,21
23 In the past decade, there has been increasing interest to apply the GW method to study electronic properties of solar energy conversion materials (e.g., refs 24
27). Alkaline tantalates have been intensively investigated by firstprinciples approaches in recent years.28
37 Most of previous studies are based on KS-DFT in LDA or GGA, such that electronic band structures of these materials can not be correctly described Received: May 21, 2011 Revised: July 14, 2011 Published: July 14, 2011 16180

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The Journal of Physical Chemistry C due to the intrinsic limitation of LDA/GGA. In particular, the band gaps of these materials are significantly underestimated and wrongly predicted to be in the visible light regime. On the other hand, experimental studies of these materials2,4,6,11,38
42 are also not conclusive such that a comprehensive understanding of their electronic properties is still lacking. Band gaps of these materials, mostly obtained from optical absorption, are highly scattered. For example, the band gap of LiTaO3 falls in a wide range from 3.9 to 5.0 eV. In addition, both LiTaO3 and NaTaO3 exhibit strong structural distortion from the ideal cubic perovskite structure at room temperature, but their experimental band gaps differ significantly, the cause of which is still unknown. In this work, we aim to study electronic band structures of alkaline tantalates by employing the GW method. In addition, by combining quantitative first-principles calculations with qualitative analysis based on the phenomenological ionic model, we hope to clarify main factors that determine the band gaps of these materials. The paper is organized as follows. In the next section, we briefly discuss the method used in this work as well as some computational details of our calculations. In section 3, we present electronic band structures of alkaline tantalates predicted by the GW method. We further investigate the influences of various structural factors on the band gap by combining first-principles calculations with phenomenological analysis based on the ionic model, from which main factors that have significant influences on electronic band structures of these materials are discerned. We also discuss the possible causes for the remaining discrepancies between theory and experiment. In section 4 we summarize the main findings of this work.

2. THEORETICAL METHOD AND COMPUTATIONAL DETAILS Electronic band structures of extended systems correspond to quasi-particle (QP) excitations (removal or addition of a single electron), which can be measured straightforwardly by direct and inverse photoemission spectroscopy (PES/IPS).43 Theoretically, they can be described by many-body perturbation theory based on one-body Green’s function (G), whose poles in the complex frequency domain give QP energies. The key ingredient that determines G is the exchange-correlation self-energy Σxc, which links a noninteracting reference system with the interacting system under study. In the GW approximation,18
20,22 Σxc is approximated as the product of G and the screened Coulomb interaction W in the random-phase approximation (RPA). In practice, the GW method is often applied as a first-order correction to KS single-particle energies, with both G and W calculated from KS orbital energies and wave functions. Further improvement can often be obtained by including partial selfconsistency in the calculation of G using QP energies with a fixed screened Coulomb interaction,44,45 which is the approach employed in this work. DFT calculations in this work are performed employing the WIEN2k package46 in which KS equations are solved in the fullKS-DFT for extended systems.47 The parameters used for the FP-LAPW basis are as follows: muffin-tin (MT) radii RMT for O, Ta, Li, Na, and K are 1.60, 1.80, 1.80, 2.50, and 2.50 bohr, respectively; KS wave functions are expanded by atomic-like functions with the angular quantum number l up to lmax = 10 in the MT spheres, and by plane waves with the energy cutoff determined by RKmax  min RMT  Kmax = 7.0 in the interstitial region. We use a mesh of about 1000/Z-points for the Brillouin

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zone integration, where Z is the number of units of chemical formula (ATaO3) in the crystal unit cell. Unless stated otherwise, all DFT calculations are performed with the Perdew
Burke
Ernzerhof (PBE) GGA functional.48 GW calculations are performed using the FHI-gap (Greenfunction with augmented planewaves) package, a recently developed all-electron GW code interfaced to WIEN2k.49,50 The Brillouin zone is sampled with 4  4  4 k-points for cubic ATaO3, and 3  3  3 for rhomboshedral LiTaO3; unoccupied states with energy up to ∼260 eV were taken into account. The GW DOS is calculated from GW QP energies, first calculated on the sparse k-mesh and then interpolated to the fine k-mesh (∼1000) using the Fourier interpolation technique.51

’ RESULTS AND DISCUSSIONS Crystal Structures. Alkali tantalates belong to the family of perovskite oxides, which have a general chemical formula of ABO3 with A and B being two cations of different sizes.52 In the ideal cubic (c) structure, ABO3 has the symmetry of Pm3m, in which each B cation is octahedrally coordinated by six oxygen anions, and the BO6 octahedra are connected in a corner-sharing way. Most of perovskite compounds are distorted from the ideal cubic symmetry either due to the cation displacements within the octahedra or the tilting of octahedra with respect to each other. For ATaO3, only KTaO3 is stable in the cubic structure at room temperature. LiTaO3 takes a rhombohedral (r) structure with R3c symmetry, which is ferroelectric up to ∼1400 K. NaTaO3 shows even more versatile polymorphic properties.4,53 Using the neutron power diffraction technique, Kennedy et al.53 found that the crystal structure of NaTaO3 evolves continuously from orthorhombic (Pbnm) at room temperature, to monoclinic (Cmcm) at 720 K, tetragonal (P4/mbm) at 835 K, and finally cubic (Pm3m)for T > 893 K. In addition, the morphology of NaTaO3 also depends on the synthetic method. NaTaO3 prepared by the sol
gel and solid-state method was found to have different crystal structures as well as different surface areas.4 In this work, we mainly focus on electronic band structures properties of alkali tantalates. Considering the rich structural properties of these materials, we are also interested in probing the relation between structural and electronic properties. The electronic band structures of a material are closely related to the nature of chemical bonding. The latter is strongly dependent on the local structure involving atoms that contribute dominantly to highest VBs and lowest CBs, which are O and Ta, respectively, for alkali tantalates. The local structure of different polymorphic phases of ATaO3 can be mainly characterized by two factors: (1) how the internal structure of each TaO6 octahedron is distorted from that of the perfect octahedron, and (2) how the connection between neighboring TaO6 octahedra is distorted from the ideal linear configuration. As shown below, these two factors influence electronic band structures of ATaO3 differently. For LiTaO3, we consider both cubic and rhombohedral phases (denoted as c-LiTaO3 and r-LiTaO3, respectively). For NaTaO3, we consider several observed polymorphic phases with the symmetry of Pbnm, Cmcm, P4/mbm and Pm3m, respectively.53 For KTaO3, we consider only the cubic phase. Experimental structural parameters from refs 54 (for r-LiTaO3), 53 (for NaTaO3), and 55 (for KTaO3) are used in all these calculations except for c-LiTaO3, which is not observed experimentally. For the latter, we use the cubic lattice constant optimized by the recently developed PBE-for-solids (PBEsol) GGA functional,56 which 16181

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Table 1. Band Gaps of ATaO3 (A = Li, Na, K) in Different Crystal Structures Obtained from PBE-GGA, GW and Experimenta materials crystal symmetry PBE LiTaO3 NaTaO3

KTaO3

GW

expt.

Pm3m

2.43

3.80

R3c

3.93

5.58

Pm3m

2.41

Pbnm

2.89

(4.28) 4.0b,c,j, 3.94d, 3.96f, 4.35g,4.1I

Cmcm P4/mbm

2.61 2.41

(4.00) (3.80)

Pm3m

2.22

3.51

4.7b, 3.93e, 5.06g, 4.6h, 4.9I

3.80

3.6b, 3.42f, 3.76g, 3.7I, 3.5j

a

The GW band gap of NaTaO3 in a crystal structure other than the cubic one, presented in the parentheses, is obtained by adding the GW correction obtained in the cubic structure to the corresponding PBE gap. b Reference 2. c Reference 4. d Reference 38. e Reference 39. f Reference 6. g Reference 40. h Reference 41. I Reference 11. j Reference 42.

has been shown to describe equilibrium lattice constants of solids much more accurately than LDA or PBE functionals. As a confirmation, we have also optimized r-LiTaO3 by PBEsol, from which we obtain lattice constants that are in good agreement with experiment. Band Gaps and DOS. Table 1 shows the band gaps of different polymorphic phases of ATaO3 obtained from PBE and GW calculations. Experimental band gaps of ATaO3, mainly obtained from optical absorption spectroscopy measured at room-temperature, are also collected. In general, the QP corrections account for a systematic increase of the band gaps by an amount in the range of 1.29
1.65 eV, indicating that the inclusion of QP corrections is necessary for a quantitative description of electronic properties of these materials. The GW band gap of KTaO3 is in good agreement with the experimental values (3.51 eV vs 3.42
3.76 eV). For NaTaO3, the PBE band gap exhibits a noticeable dependence on the crystal structure. The high-temperature cubic (Pm3m) phase has a minimal indirect band gap of 2.41 eV. As the structure is distorted from the cubic one, the band gap increases, and reaches 2.89 eV in the orthorhombic phase (Pbnm) that is stable at room temperature. Experimental band gaps are usually measured in the room temperature phase, but a direct GW calculation of this structure is still quite a challenge due to its relatively large unit cell (20 atoms/cell). The GW calculation is performed only for the cubic phase, from which the band gap opens up by 1.39 eV. By assuming same GW corrections in different polymorphic phases, we obtain the GW band gap of the orthorhombic phase to be 4.28 eV, falling in the range in which experiment values scatter (3.96
4.35 eV). We can therefore conclude that band gaps of NaTaO3 and KTaO3 are well described by the GW method. By contrast, LiTaO3 exhibits very different features. The band gap of c-LiTaO3 is nearly identical to that of c-NaTaO3, in both PBE and GW, which can be attributed to the fact that the two materials have nearly identical lattice constants. On the other hand, the GW band gap (5.4 eV) for r-LiTaO3 is 1.8 eV larger than that of c-LiTaO3. The experimental values for LiTaO3 scatter considerably, falling in the range between 3.93 and 5.06 eV. Both the strong dependence of the band gap on crystal structure and dramatic difference between GW and experiment for materials such as LiTaO3 are quite remarkable and call for more in-depth analysis.

Figure 1. DOS in c-LiTaO3, r-LiTaO3, c-NaTaO3, and c-KTaO3 obtained from PBE and GW.

Figure 1 shows electronic density of states (DOS) in c-LiTaO3, r-LiTaO3, c-NaTaO3, and c-KTaO3, obtained from both PBE and GW band energies. By comparing PBE and GW DOS, several features that are common in the four materials can be clearly observed. The VB, which is dominated mainly by the O-2p character, has slightly larger width in GW than that in PBE. The semicore states arising from O-2s, Ta-4f, Na-2p, and K-3p are significantly shifted toward lower energy (larger binding energy) by the GW correction. The shift is about 2 eV for O-2s and K-3p, more than 5 eV for Ta-4f, and more than 6 eV for Na-2p. Similar effects of GW corrections on semicore states were also observed previously in HfO2.45 For unoccupied states, mainly of Ta-5d character, the GW correction opens the band gap, but the shape of DOS changes very little. When in the same cubic structure, the three alkaline tantalates have very similar DOS in the VB and low-lying CBs, except that KTaO3 has slightly smaller VB width, which can be attributed to the fact that KTaO3 has slightly larger lattice constant. In addition, the high-lying CBs in KTaO3 show features arising from K-3d. The most remarkable feature in Figure 1 is the difference between cubic and rhombohedral phases of LiTaO3. Besides the difference in the band gap that we have discussed, the width of the VB and the low-lying CB differs considerably in the two phases. Quantitatively, the VB and CB width in the r-LiTaO3 are about 0.8 and 1.9 eV smaller than the cubic phase, respectively. The bandwidth is mainly determined by the extent to which orbitals on neighboring atoms overlap, i.e., the nature of chemical bonding. As we have pointed out before, in r-LiTaO3, the TaO6 octahedron is strongly distorted from its ideal structure. In particular, the internal O
Ta
O bond angle is reduced to 169.2, and the interoctahedron Ta
O
Ta angle is distorted even more strongly to 144°. The deviation from the linear configuration results in reduced interatomic orbital overlap, which explains the 16182

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significantly reduced bandwidth in the r-phase. As we will show below, the significantly larger band gap in r-LiTaO3 is also related to the greatly reduced bandwidth. Determinant Factors for Electronic Band Structures of ATaO3. In the following, we will combine first-principles calculations with a phenomenological ionic model to investigate various factors that can have significant effects on electronic band structures of alkaline tantalates. Qualitatively, the fundamental band gap of an ionic crystal can be regarded as the energy needed to transfer an electron from the highest occupied orbital of the anion to the lowest unoccupied state of the cation. Taking the influences of the crystalline environment into account, the band gap of an ionic crystal can be estimated as follows:57,58 Eg ¼ ðAa
I c Þ
þ e jVMa j þ jVMc j
ðΔEapol 1 þ ΔEcpol Þ
ðWVB þ WCB Þ ð1Þ 2 In eq 1, e denotes the absolute charge of electrons, “a” indicates the anion whose states dominate the high-lying VB, and “c” indicates the cation whose states dominate the low-lying CB. For ATaO3, “a” and “c” refer to oxygen and tantalum, respectively. The first term is the contribution from free ions, where A and I are the electron affinity and ionization potential of the atoms corresponding to the anion and cation, respectively. The second term accounts for the contribution of the Madelung potentials (VM) at the anionic and cationic sites. The third term is the contribution from the electrostatic polarization. The last term is the contribution of the VB and CB width that is determined by the overlap of orbitals on neighboring atoms. The ionic model as exemplified by eq 1 has been used to estimate the band gaps of alkaline halides, which are generally in good agreement with experiment.57 It is, however, not straightforward to employ the model to estimate the band gaps of materials whose chemical bonds fall between ionic and covalent, such as alkaline oxides studied in this work. In a simplistic picture, ATaO3 can be regarded as ionic crystals composed of A+, Ta5+, and O2
. The population analysis based on Bader’s atoms-inmolecules (AIM) method59 gives the charges on Li, Ta, and O to be 0.90 (0.94), 2.91 (2.94), and
1.27 (
1.29) in the rhombohedral (cubic) LiTaO3, which deviate quite significantly from the nominal charges in the ideal ionic picture, especially for Ta and O. Nevertheless, we can still use eq 1 to analyze, in a qualitative way, the main factors that affect the band gaps of materials with strong ionicity. We expect that the polarization (third) term is mainly determined by the polarizability of constituent ions, and therefore should be influenced little by the variation of the crystal structure. The second and fourth terms, on the other hand, are strongly dependent on crystal structures: Madelung potentials are inversely proportional to the distances between ions, and the bandwidth is determined by the interatomic overlap of orbitals, which depends on the bond length exponentially. In the following, we will mainly focus on these two terms. To further reveal the physics that underlies the difference of the band gap in different crystal structures, we perform a series of calculations to investigate how electronic band structures of LiTaO3 evolves as (1) the lattice constant of cubic LiTaO3 changes and (2) the crystal structure changes gradually from the ideal cubic to the experimentally observed rhombohedral structure. Since, compared to the difference between the two phases, the dependence of the GW correction on the crystal structure is

Figure 2. PBE DOS of cubic LiTaO3 obtained with different lattice constants (in units of bohr). The left and right insets show the PBE band gap and the contribution of the Madelung potentials as a function of a, respectively.

relatively weaker (∼0.3 eV), we base our investigations on PBE results. Figure 2 shows the DOS of cubic LiTaO3 calculated at three different lattice constants, a = 7.0, 7.4, and 7.8 bohr (as a reference, the equilibrium lattice constant a0 = 7.404 bohr). The band gap, as shown in the left inset, decreases as a increases. This feature can be well understood in terms of the contributions of the bandwidth and the Madelung potential. As we can see from the DOS plot, increasing lattice constant, as a result of weakened interatomic orbital overlap, leads to smaller bandwidth, which, according to eq 1, contributes to increasing the band gap. On the other hand, the contribution from the Madelung potential decreases as the lattice is expanded. The latter is found to be dominant. Using the ionic charges obtained from Bader’s AIM analysis, we calculate the Madelung potentials on the oxygen and tantalum sites. As the lattice constant increases from 7.0 bohr to 7.8 bohr, the contribution to the band gap from the Madelung potential decreases by as much as ∼6 eV. We have already shown before (see Figure 1) that the DOS of LiTaO3 in the cubic and rhombohedral structures are remarkably different. To see the influences of the structural change more clearly, we consider a series of structures that fall between the ideal cubic structure and the experimental rhombohedral structure, each characterized by the interpolation fraction x, i.e., x = 0.0 denoting the cubic structure, and x = 1.0 denoting the experimental rhombohedral structure. As the crystal structure of LiTaO3 evolves from cubic to rhombohedral, partial charges on Li, Ta, and O atoms decrease slightly, indicating that the ionicity is only slightly weakened, which justifies the use of the ionic model to analyze the influences of different factors. Figure 3 shows evolution of the PBE DOS as a function of x, which reveals the effects of the bandwidth variation. We can see clearly that the width of the VB and low-lying CB, mainly of Ta-5d character, decrease continuously as the crystal structure evolves from the ideal cubic one to the experimental rhombohedral one. This factor alone would, according to eq 1, result in an increase of the band gap by about 1.4 eV. On the other hand, the distances between Li, Ta, and O change quite significantly in the course of structural variation. For example, the distance between nearest neighboring Li and O is 2.78 Å in the cubic structure, but it is significantly reduced in the experimental rhombohedral structure to 2.04 and 2.31 Å. Therefore the change of Madelung potentials may also have some influences on the band gap. As shown in the right inset 16183

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Figure 3. Evolution of PBE DOS of LiTaO3 as the crystal structure changes from the cubic to experimental rhombohedral structure. The left and right insets show the PBE band gap and the contribution of the Madelung potentials as a function of the interpolation fraction x, respectively.

of Figure 3, the contribution of the Madelung potentials (the second term in eq 1) varies quite mildly as a function of x, in contrast to the situations in Figure 2. We can therefore conclude from the analysis above that the main factor underlying the significantly larger band gap of LiTaO3 in the rhombohedral structure is the greatly reduced bandwidth. The bandwidth is mainly determined by the overlap of orbitals on neighboring atoms, and is therefore directly related to the nature of chemical bonding. The main cause for significant reduction of the bandwidth is obviously related to the fact that the local structure in rhombohedral LiTaO3 is strongly distorted away from the ideal cubic structure. On the other hand, different polymorphic phases of NaTaO3 also arise from distortion from the cubic structure, but the change in the band gap is much smaller, as shown by the data in Table 1. Figure 4 shows the DOS of NaTaO3 in the cubic (Pm3m) and orthorhombic (Pbnm) phases, denoted as c- and o-NaTaO3, respectively. We can see that indeed the difference in the DOS of the two phases, although noticeable, is much smaller compared to that between the cubic and rhombohedral LiTaO3. These features can be understood by the fact that the distortion in r-LiTaO3 and o-NaTaO3 is different. In r-LiTaO3, both the internal structure of each TaO6 octahedron and the connection of the neighboring TaO6 octahedra, characterized by the O
Ta
O and Ta
O
Ta bond angles, respectively, are strongly distorted away from the linear configuration. On the other hand, the distortion in o-NaTaO3 is mainly due to that of Ta
O
Ta bond angles, and the internal structure of TaO6 octahedra changes very little. Comparing the different way the band gap changes in r-LiTaO3 and o-NaTaO3, we can therefore conclude that the intraoctahedron distortion has much stronger effects on electronic band structures than that of the interoctahedral distortion. We note in passing that the influences of the interoctahedral distortion (tilting) on electronic band structures of perovskite oxides have been intensively investigated previously (see, e.g., ref 58 and references therein). It was found that the effects of the octahedral tilting distortion are much stronger in d10-group perovskite oxides (e.g., CaSnO3 and CdSnO360) than d0-group perovskite oxides (e.g., NaTaO3).58 Possible Causes for the Discrepancy between GW and Experiment in LiTaO3. We close this section by discussing the

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Figure 4. DOS of NaTaO3 in the cubic (Pm3m) and orthorhombic (Pbnm) structure calculated by PBE.

possible causes for the discrepancy between GW and experiment in the band gap of LiTaO3. As we have pointed out in the preceding section, the experimental values for the band gap of LiTaO3, in spite of the fact that all were extracted from optical absorption, are highly scattered, falling in the range between 3.9339 and 5.0640 eV. A possible cause for this large uncertainty is that the band gap of LiTaO3 depends sensitively on the conditions for sample preparation. This is consistent with the strong structure dependence of the LiTaO3 band gap we have found in this work. Considering that four in five optical experiments give band gaps of LiTaO3 that are relatively close to each other, we can take the average of them, 4.8 eV, as the most “accurate” optical experimental gap, compared to which the GW fundamental gap is overestimated by about 0.8 eV. There are two possible factors that might contribute to this discrepancy: (1) the electron
hole interaction (excitonic) effect, and (2) the additional screening of QP excitation due to the electron
phonon coupling (i.e., the polaronic effect). LiTaO3 in the rhombohedral structure has an indirect minimal gap, but the direct gap at Γ is only slightly larger (by ∼0.01 eV). We can therefore expect that the experimental band gap from optical absorption may contain the contribution of the excitonic effect. The latter is not included in our GW band gap, which is the fundamental band gap corresponding to the difference between the ionization potential (the energy needed to remove an electron) and the electron affinity (the energy released when adding an electron). In a recent theoretical study of LiNbO3,61 a material very similar to LiTaO3, it was found that the inclusion of the electron
hole interaction accounted for by the Bethe
Salpter equation (BSE)62 shifts the absorption edge toward lower energy by about 1 eV. This amount of the excitonic effect might be overestimated considering the approximations used in the calculation,61 but it is probably qualitatively true that the excitonic effect is significant in optical absorption of materials such as LiNbO3 and LiTaO3. For highly ionic materials such as LiTaO3, the electron
phonon coupling (the polaronic effect) gives rise to additional screening (polarization) to QP excitation and can therefore also play an important role.25,26,61,63 The role of the polaronic effect in solar energy conversion materials has attracted a lot of interest in recent years,25,26,61,64 but a precise evaluation of its contribution to electronic band structures, including, in particular, the band gap, is not straightforward. For TiO2, estimation based on highly simplified models suggests that the polaronic effect tends to 16184

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The Journal of Physical Chemistry C reduce the band gap considerably, but the magnitude of the reduction varies significantly in the range between 0.1 and 0.7 eV, depending on the model used for analysis (see ref 25 and references therein). Schmidt et al.61 estimated the polaronic effect in LiNbO3 by combining the GW method with a model dielectric function that contains the contribution of the lattice polarizability,63 and they obtained a reduction of the GW band gap by nearly 1.2 eV. A large polaronic effect is consistent with the strong temperature dependence (∼2.7  10
3 eV K
1) of the optical gap in LiNbO3 as reported in ref 39. A weaker temperature dependence (∼5.5  10
4 eV K
1) was observed for LiTaO3, implying a weaker polaronic effect than LiNbO3. Due to the limitation of the theoretical approaches used in this work, it is difficult to make an unambiguous judgment about which factor, the excitonic or polaronic effect, is the main cause for the significant discrepancy between GW and experiment for the band gap of LiTaO3. In a more general context, there are still a lot of open issues regarding the roles that electron
hole and electron
phonon interactions play in electronic properties of solar energy conversion materials. More accurate theoretical approaches that go beyond those used in this work are clearly needed to address these issues.

’ CONCLUSIONS In this work we have investigated electronic band structures of alkaline tantalates, ATaO3 (A = Li, Na, K), based on many-body perturbation theory in the GW approximation. We found that KS-DFT with GGA significantly underestimates the band gaps of all these materials. The GW approach, on the other hand, gives band gaps of NaTaO3 and KTaO3 in good agreement with experiment, but for LiTaO3, a significant overestimation is observed when considering the rhombohedral phase of LiTaO3 that is stable at room temperature. In addition, we found that the band gap of LiTaO3 strongly depends on crystal structures, and that the band gap in the cubic and rhombohedral phases differ by more than 1.5 eV. By combining first-principles calculations with the phenomenological ionic model, we found that the strong structure dependence can be attributed mainly to two factors: the Madelung potential and the bandwidth. By comparing different structure dependence of LiTaO3 and NaTaO3, we found that the distortion of the internal structure of TaO6-octahedron has stronger effects on electronic band structures than that of interoctahedral configuration. We further pointed out that both electron
hole interactions (the excitonic effects) and electron
phonon couplings (the polaronic effects) can be the possible causes for the discrepancy between GW and experiment in the band gap of LiTaO3. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors thank Dr. Guo-Ling Li for helpful discussions. This work is partly supported by the National Natural Science Foundation of China (Project No. 20973009). ’ REFERENCES (1) Kato, H.; Kudo, A. Catal. Lett. 1999, 58, 153. (2) Kato, H.; Kudo, A. J. Phys. Chem. B 2001, 105, 4285.

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