Four crystal structures of bismuthates are illustrated in Fig. 1. The coordination numbers in these bismuthate compounds are 6 for Bi atoms, 2 for O atoms (in AgBiO₃, the coordination number of O is 3), 0 for Li (Na, K), and 3 for Ag atoms. LiBiO₃ has an orthorhombic structure with space group of Pccn (No. 56) and local symmetry ; NaBiO₃ and AgBiO₃ have a trigonal structure with space group of (No. 148) and local symmetry ; and KBiO₃ has a cubic structure with space group of (No. 201) and local symmetry . It is known that their structure has rather regular or slightly distorted [BiO₆] octahedrons linked together at the corners and forming a network structure. In the cases of LiBiO₃ and KBiO₃, the octahedrons form a tunneled structure, and the alkaline-earth metals insert in the tunnels. In the cases of NaBiO₃ and AgBiO₃, the octahedrons form a layered structure, and Na atoms insert in the space between layers, the whole Ag atoms are bonded with oxygen atoms forming [AgO₃] tetrahedrons.

Fig. 1.

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	Fig. 1. The crystal structure models of four bismuthate compounds: (a) LiBiO3, (b) NaBiO3, (c) KBiO3, and (d) AgBiO3.

All the crystal lattice parameters for these four bismuthates are tabulated in Table 1, which are nicely consistent with experimental data,^[20–22] indicating that the calculation methods in the present work are reasonable. Compared with the crystal structures of NaBiO₃ and AgBiO₃, one could find that their symmetries are identical, and the crystal structures are very similar. The main difference is the stronger interaction between Ag atoms and O atoms in AgBiO₃ than that in NaBiO₃, resulting in the Ag–O bond length (2.2974 Å) being shorter than the Na–O bond length (2.4608 Å). Hence, the lattice constants of AgBiO₃ are also slightly smaller than the lattice constants of NaBiO₃. For these four bismuthate compounds, the binding energy is decreasing with the increase of atomic number of M, which is calculated by the following equation: (where E_total is the total energy of bismuthate model, n_i and E_i are respectively the number and energy of a single atom in 15×15×15 ų box for the i atom in the bismuthate model). Resulting from the tunneled or layered structures, the decrease of binding energy implies that the interaction between [BiO₆] octahedron is weakening along with the increase of atomic number of M. For the application of photocatalysis, the tunneled and layered could improve the transfer behavior of photo-generated carriers. In particular, resulting from the small ionic radii of alkaline-earth metals, the ions of Li⁺, Na⁺, or K⁺ maybe easily transfer in the tunnels or the space between layers. Thus, these ions could also carry the energy excited by the solar light. Although there is not alkaline-earth metal in AgBiO₃, the space between layers is large enough to allow the ions or radicals (such as H⁺, OH⁺, etc.) to transfer easily. Therefore, from the viewpoint of micro-structure, these bismuthates are potentially used as an efficient photocatalyst. Although this point is just a conjecture at the current stage, Wei et al. recently investigated the transport behavior of Li-ion in layered cathode materials (the kinetics of Li-ion diffusion versus selection of the advantageous paths, and their effective diffusion barriers depended on activation energies and Li-slab space) by using DFT calculations, and clarified how these factors tune the kinetics of Li diffusion in a delithiation process.^[23] This published work provides great inspiration to us. In this paper, we research the transport behavior of the ions in MBiO₃ by using similar research ideas.

Table 1.

Table 1.

Table 1. The lattice constants and parameters, and the atomic average net charge from Mulliken population analysis of bismuthate compounds. . LiBiO3 NaBiO3 KBiO3 AgBiO3 Symmetry orthorhombic trigonal cubic trigonal Lattice constant (cal.) a = 8.9007 Å, a = b = 5.6122 Å, a = b = 5.7038 Å, b = 4.9615 Å, c = 16.0733 Å a = b = c = 10.0432 Å c = 15.5061 Å, c = 10.8220 Å α = β = 90°, α = β = γ = 90° α = β = 90°, α = β = γ = 90° γ = 120° γ = 120° Lattice constant (exp.)a a = 8.8278 Å, a = b = 5.6572 Å, a = b = 5.6411 Å, b = 4.9135 Å, c = 15.9890 Å a = b = c = 10.0194 Å c = 16.118 Å, c = 10.6914 Å α = β = 90°, α = β = γ = 90° α = β = 90°, α = β = γ = 90° γ = 120° γ = 120° Crystal volume/molecule·Å3 59.7394 73.0715 84.4192 72.8125 Density/g·cm−3 7.3360 6.3622 5.8239 8.3205 Binding energy/eV·atom−1 5.7936 5.6910 5.5921 5.4276 Average bond length/Å Li–O: 2.0018 Na–O: 2.4608 K–O: 2.6887 Ag–O: 2.2974 Bi–O: 2.1345 Bi–O: 2.1370 Bi–O: 2.1347 Bi–O: 2.1679 Average net charge/e Li: 0.84 Na: 0.43 K: 0.28 Ag: 0.32 Bi: 1.96 Bi: 2.05 Bi: 0.64 Bi: 1.97 O: −0.94 O: −0.82 O: −0.31 O: −0.76 a Refs. [20]–[22].

Table 1. The lattice constants and parameters, and the atomic average net charge from Mulliken population analysis of bismuthate compounds. .

The band structures of these four bismuthates are illustrated in Fig. 2. In the present work, we found that the spin-up states and the spin-down states are completely coincident, owing to the pairing electrons in these four bismuthate compounds. For LiBiO₃, the valence band maximum (VBM) is located at the k-point line of Γ → Z, while the conduction band minimum (CBM) is located at the Γ k-point, which means that LiBiO₃ is an indirect band gap semiconductor. The calculated band gap, the distance between VBM and CBM, is 1.7 eV, which is highly consistent with the experimental measurement (E_g =∼ 1.8 eV).^[11] The band gap of NaBiO₃ (2.550 eV, which is highly consistent with the experimental measurement, 2.6 eV^[24]) is larger than that of LiBiO₃, and also belongs to an indirect band gap semiconductor (the VBM is located at the k-point line of F → Γ, while the CBM is located at the Γ k-point). In the case of KBiO₃, the VBM is located at the M k-point, while the CBM is located at the Γ k-point. Hence, the band gap of KBiO₃, 2.3 eV (which is strongly consistent with the experimental measurement, 2.1 eV^[11]), belongs to the indirect band gap.

In the case of AgBiO₃, the highest VB and the lowest CB contact together, indicating that the minimum value of band gap is 0 eV. Above the lowest CB, one could see another forbidden band with an indirect pseudo-band gap of 2.563 eV, in which the pseudo-VBM/CBM is located at Γ/Z k-point. The band-gap value of AgBiO₃ is a very controversial issue. The sample color of AgBiO₃ is black,^[25,26] implying its band gap is very small. Based on the UV–vis spectra, Takei et al. estimated its band gap is 0.87 eV,^[26] while Yu et al. estimated its band gap is 2.5 eV.^[27] On the contrary, using different calculation methods (linear muffin-tin orbital method and discrete-variational X α molecular orbital method), Mizoguchi et al.^[25] and Takei et al.^[26] independently obtained the band gap of AgBiO₃ is 0 eV. Carefully comparing the experimental measurements with theoretical calculations, we considered that the band gap of AgBiO₃ should be determined as 0 eV, and the experimental measurements might come from the electron transition deviated from Γ k-point or the electron transition between pseudo-VBM and pseudo-CBM in the second forbidden band.

The parameters of band structures of these four bismuthate compounds are listed in Table 2. Firstly, the direct band gaps are slightly larger than the minimum band gaps. This situation leads to difficult or certain confusion in the band gap measurement in practice. Importantly, the band structures of these four bismuthate compounds present the feature of multi-band gaps. In other words, their bands near the fundamental band gap are located and separated, which may be caused by the tunneled or layered structures. As shown in Table 2, the alkaline-earth metal’s ns states are located below the valence band (VB) with a large band gap and a small band width. At the same time, the bands related to O-2s states and Bi-6s states are also below VB with relative small band gaps. These bands are not influenced by the extra-situation, and belong to the inner-energy bands. Therefore, they have an insignificant effect on the photocatalytic performance, and are not displayed in Figs. 1 and 2. These four bismuthate compounds have wide VBs with band widths > 4.2 eV, and the VB band width of AgBiO₃ is the largest. The conduction bands (CBs) of these four bismuthate compounds have two distinct parts: lower conduction band (CB_L) and upper conduction band (CB_U), which are separated by > 2.1 eV. In the case of KBiO₃, the CB is divided into three distinct parts: CB_L1, CB_L2, and CB_U. Resulting from the identical crystal structure, the band structure of AgBiO₃ has similar features to that of NaBiO₃. However, because of the bonding between O atoms and Ag atoms (i.e., the strong interaction between O atoms and Ag atoms), the CB_L is noticeably down-shifting to the top of VB, leading to the zero band gap of E_g1. In other words, AgBiO₃ exhibits semi-metallic features. However, the CB_L contains only two energy levels, so this feature is not very prominent in the experimental measurement, and therefore has not attracted much attention in practice (see the discussion of optical properties in subsection 3.3). In the previous experimental works carried out by Takei et al.,^[26] it was observed that only adsorption of methylene blue occurs on the AgBiO₃ and its photocatalytic activity for methylene blue decomposition is very low. However, Yu et al. found that AgBiO₃ has a superior inhibitory effect on the cyanobacteria at relatively low concentrations with natural light, causing irreversible damage to M. aeruginosa, compared with AgNO₃ and NaBiO₃.^[27] Combining their UV–vis diffuse reflectance absorption measurement, we considered that this effect might be caused by the photo-generated electron–hole pairs on the VB, CB_L, and CB_U. Thereby, we suggest that the relationship between the semi-metallic properties and photocatalytic performance of the novel photocatalyst is very worthy of further investigation.

Fig. 2.

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	Fig. 2. The calculated band structures of four bismuthate compounds. (a) LiBiO3, (b) NaBiO3, (c) KBiO3, and (d) AgBiO3.

Table 2.

Table 2.

Table 2. The parameters of band structures of bismuthate compounds, the energy unit is eV. . LiBiO3 NaBiO3 KBiO3 AgBiO3 Top of VB Γ → Z F → Γ M Γ Bottom of CB Γ Γ Γ Γ Eg (minimum)/eVa 1.70, 2.687 2.55, 3.097 2.30, 2.19, 1.49 0, 2.563 Eg (direct at Γ)/eV 1.740 2.613 2.373 0 Band gaps/eV Li-s → O-2s: 22.863 Na-2s → O-2s: 1.401 K-3s → O-2s: 8.211 O-2s → Bi-6s: 3.605 O-2s → Bi-6s: 1.623 O-2s → Bi-6s: 3.260 O-2s → Bi-6s: 3.261 Bi-6s → VB: 1.433 Bi-6s → VB: 0.657 Bi-6s → VB: 1.689 Bi-6s → VB: 1.033 VB → CBL: 0 CBL → CBU: 2.687 CBL → CBU: 3.097 CBL1 → CBL2: 2.190 CBL2 → CBU: 1.490 CBL → CBU: 2.563 Band width/eV M-ns 0.028 0.011 0.727 5.709 O-2s 3.115 3.233 3.295 2.798 Bi-6s 4.425 3.775 2.885 3.305 VB 5.536 4.2160 4.857 6.378 CBL 3.999 2.216 0.496, 3.240, 2.653 CBU 11.838 10.867 6.157 12.090 a The multi-band gaps of MBiO3 are provided as the orders of first, second, and third band gaps as shown in Fig. 2.

Table 2. The parameters of band structures of bismuthate compounds, the energy unit is eV. .

To analyze the chemical bonding information, the total and partial density of states (DOS) of these four bismuthate compounds are plotted and compared in Fig. 3, which are highly consistent with the calculated results by discrete-variational X α molecular orbital method.^[26] In general, the peaks and features of DOS for these compounds exhibit obvious similarity: (i) a deep band consists of Li(Na, K)-ns and O-2s, Bi-6s states; (ii) the lower VB is dominated by the hybridized states of O-2p and Bi-6p states; (iii) the middle and top of VB is dominated by the O-2p nonbonding states; (iv) the CB_L is mainly contributed by the hybridization states of O-2p and Bi-6s states; (v) the CB_U is mainly contributed by the Bi-6p states. In addition to the similarities mentioned above, due to the different crystal structures and compositions, these compounds also show some differences: (i) the K-3s states form the second lower conduction band (CB_L2) in the case of KBiO₃; (ii) due to the strong interaction between O atoms and Ag atoms in the case of AgBiO₃, the Ag-5s states also contribute to the form of VB hybridized with O-2p states, and the Ag-4d states also contribute to the form of CB_U. Because the Fermi energy level (E_F) is fixed at 0 eV in standard DFT calculations, so the band relative-shifting could be directly observed.

Fig. 3.

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	Fig. 3. The calculated density of states of four bismuthate compounds. (a) LiBiO3, (b) NaBiO3, (c) KBiO3, and (d) AgBiO3.

Fig. 4.

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	Fig. 4. The calculated contour maps of electron densities (right) and electron difference densities (left) of four bismuthate compounds. (a) LiBiO3, (b) NaBiO3, (c) KBiO3, and (d) AgBiO3.

The contour maps of electron densities and electron difference densities of these four bismuthate compounds are plotted and compared in Fig. 4. The characteristics of tunneled or layered structures are well presented in these maps. In the case of LiBiO₃, the electron densities and electron transfer are different around Li atoms and Bi atoms. The covalent bonds between Bi atoms and O atoms are stronger than those bonds between Li atoms and O atoms, and more electrons gather onto Bi–O bonds. By Mulliken population’s analysis, the average net charge of Bi is 1.96e, which is larger than that of Li (0.84e). In the case of NaBiO₃, the bonds between Bi atoms and O atoms exhibit the feature of covalent bond, while the bonds between Na atoms and O atoms exhibit the feature of ionic bond. The interaction between atoms is obviously weaker than that in LiBiO₃, so the average net charge of Bi (2.05e) is the largest in these compounds. In the case of KBiO₃, the situation is similar to that of NaBiO₃. However, the average net charge on atoms is very small: 0.28e for K, 0.64e for Bi, and −0.31e for O, which are the smallest in these compounds. This calculated result indicates that the interactions between atoms in KBiO₃ are very weak. In the case of AgBiO₃, the bonding characteristics between Bi atoms and O atoms are very similar to those in NaBiO₃, and the layered feature is still clearly presented. But the bonding characteristics between Ag atoms and O atoms are different from those between Na atoms and O atoms in NaBiO₃: the Ag–O bonds exhibit an obvious feature of covalent bond. These characteristics of bonding information in the four bismuthate compounds indicate that the crystal structure and composition could determine the electronic structure, and subsequently determine the corresponding photocatalytic performance.

Figure 5 illustrates the calculated absorption spectra of four bismuthate compounds by the polycrystalline model. The optical properties of bismuthate compounds are determined by their compositions, crystal structures, and electronic structures. In turn, the differences of optical properties between these four bismuthate compounds could reflect the subtle differences among their crystal structures and electronic structures. Owing to the feature of multi-band gaps, when electrons on the top of VB absorb the incident photon energy, there are three kinds of electron transition: VB→CB_L, CB_L →CB_U, and VB→CB_U. In these absorption spectra, we could identify the absorption peak centered at 450–600 nm arising from the electron transition between the top of VB to the bottom of CB_L, and the absorption peak centered at 250–300 nm arising from the electron transition between the top of VB to the top of CB_U. This is because the CB_L is an unoccupied band above the Fermi energy level, and in the current first-principles only the electron transition between occupied states and unoccupied states is considered. Therefore, the absorption peak corresponding to the electron transition of CB_L →CB_U and VB→CB_L →CB_U is not calculated. Specifically, in the case of AgBiO₃, because the zero band gap of E_g1 arising from CB_L connects the top of VB, the optical absorption in the visible-light region is very significant, which means that the electrons on the top of VB can be easily excited by lower photon energy.

Fig. 5.

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	Fig. 5. The calculated absorption spectra of four bismuthate compounds by polycrystalline models.

However the absorption spectra calculated by DFT calculations are different from the experimental results, for example in the case of AgBiO₃. The differences may be because only a straightforward transition from occupied energy bands (VB) to unoccupied energy bands (CB_L and CB_U) is considered in DFT calculations, while in practice, owing to the semi-metallic features, the electron transition from CB_L to CB_U could occur. However, their underlying mechanisms are identical to that determined by the electron transfer from the occupied energy bands to the unoccupied energy bands by photon excitation. Taking the experimental literature Ref. [27] as an example, the authors determined the optical band gap, based on the absorption edge in the range of 400–600 nm, which is corresponding to the electron transition from VB → CB_U in Fig. 2. However, in their absorption spectra, one could see that there are other shoulder absorption peaks in the visible-light region, which are contributed by the electron transition of CB_L →CB_U and VB→CB_L →CB_U. That is to say, the absorption spectra calculated here are good reflections of the multi-band-gap characteristics of these four bismuthate compounds.