Structural, Electronic, and Optical Properties of BiOX1−xYx (X, Y = F, Cl, Br, and I) Solid Solutions from DFT Calculations

Six BiOX1−xYx (X, Y = F, Cl, Br, and I) solid solutions have been systematically investigated by density functional theory calculations. BiOCl1−xBrx, BiOBr1−xIx, and BiOCl1−xIx solid solutions have very small bowing parameters; as such, some of their properties increase almost linearly with increasing x. For BiOF1−xYx solid solutions, the bowing parameters are very large and it is extremely difficult to fit the related calculated data by a single equation. Consequently, BiOX1−xYx (X, Y = Cl, Br, and I) solid solutions are highly miscible, while BiOF1−xYx (Y = Cl, Br, and I) solid solutions are partially miscible. In other words, BiOF1−xYx solid solutions have miscibility gaps or high miscibility temperature, resulting in phase separation and F/Y inhomogeneity. Comparison and analysis of the calculated results and the related physical–chemical properties with different halogen compositions indicates that the parameters of BiOX1−xYx solid solutions are determined by the differences of the physical–chemical properties of the two halogen compositions. In this way, the large deviation of some BiOX1−xYx solid solutions from Vegard’s law observed in experiments can be explained. Moreover, the composition ratio of BiOX1−xYx solid solutions can be measured or monitored using optical measurements.


Results and Discussions
Formation energy. Isolated atoms can combine to form a crystal because the combined system has lower energy. That is, the free atoms combined in a crystal will release energy, or decomposition of a crystal requires energy. This energy is referred to the binding energy (E b ) defined as following for BiOX 1−x Y x solid solutions: is the total energy of the BiOX 1−x Y x solid solution. If x = 0, it is the total energy of pure BiOX. n i and E i (atom) are the number and energy of an isolated i atom, respectively. The definition of E b indicates that the crystal structure of a solid solution is more stable when the value of the binding energy is larger. The calculated binding energies of the BiOX 1−x Y x solid solutions are shown in Fig. 1. In the cases of the pure BiOX compounds, BiOF has the largest binding energy. The binding energy decreases with increasing atomic number of the halogen. This may be related to fluorine having the highest electronegativity, resulting in the van der Waals interactions between the [Bi 2 O 2 ] 2+ slabs of BiOF are the largest. So, the lattice constant of BiOF along the c axis is the smallest. The van der Waals interactions between [Bi 2 O 2 ] 2+ slabs of BiOI are the smallest, and its lattice constant along the c axis is the largest. For BiOX 1−x Y x solid solutions, when Y is incorporated into BiOX, the value of E b increases, indicating that the incorporation of heavy halogen atoms will decrease the stability of the crystal. With increasing Y content, the binding energies of all of the BiOX 1−x Y x solid solutions linearly increases.
In the complete range of the Y content, the variation of the binding energy of the BiOCl 1−x Br x solid solutions is the smallest (~0.19 eV/atom), while that of the BiOF 1−x I x solid solutions is the largest (~0.95 eV/atom). These calculated results indicate that only small energy is needed to incorporate Y into the BiOX (X = Cl and Br) crystal matrix to form solid solution. The above phenomena are quantitatively reflected by the fitting curves in Fig. 1 and the fitting parameters in Table 1: the correlation coefficients for all of the linear fitting curves are greater than 0.99 and the slope of the curve is smallest for the BiOCl 1−x Br x solid solutions and largest for the BiOF 1−x I x solid solutions.
To describe the miscibility of the BiOX 1−x Y x solid solutions, the formation enthalpy was also calculated: Scientific RepoRts | 6:31449 | DOI: 10.1038/srep31449 where E BiOX and E BiOY are the total energies of pure BiOX and BiOY with the same size of supercell, and E BiOX1−xYx is the total energy of the BiOX 1−x Y x solid. The calculated results are shown in Fig. 2. The formation enthalpy Δ H f (x) describes the energy cost of mixing X and Y halogens in a certain lattice. It is clear that all of the BiOX 1−x Y x solid solutions have an upward bowing in their Δ H f dependence on x, which indicates they prefer decoherent phase separation into BiOX and BiOY at zero temperature. Comparing these calculated curves, the formation enthalpies of the BiOX 1−x Y x solid solutions are in the order BiOCl 1−x Br x < BiOBr 1−x I x < BiOCl 1−x I x < BiOF 1−x Cl x < BiOF 1−x Br x < BiOF 1−x I x for the same x. This indicates that halogen mixing is easier with halogens with similar size. It is worth noticing that there is only one peak for BiOCl 1−x Br x , BiOBr 1−x I x , and BiOCl 1−x I x , while there is more than one peak for the other three types of BiOF 1−x Y x solid solution. These peaks are located on the light-halogen-rich side of the Δ H f against the x curve (x < 0.5), indicating that limited solubility might occur    on the heavy-halogen-rich side (x > 0.5). According to conventional solid-solution theory, the solid-solution formation enthalpy is almost a quadratic function of x: 36 where Ω is the interaction parameter, which is an indicator of the solid-solution solubility. A larger Ω indicates a smaller solubility. The fitting curves are shown in Fig. 2 20,22,25,31 in Figures S1 to S6 (in the Supporting Information). The calculated results are consistent with the experimental results, especially the variation trend with x, indicating that the supercell models chose in the present work are basically reasonable. For the DFT calculations, the size of supercell and the occupying sites of Y in the supercell also impact the final total energy of models and the lattice parameters. So, it should carefully choose the configuration of solid solution structure. In the present work, we found that if the model has higher symmetry, the total energy per cell is smaller. In this configuration, the Y atoms gather together at the same plane as much as possible. Thus, it is could be assumed the interaction between same halogen atoms (X-X or Y-Y) or different halogen atoms (X-Y) lead the variation of total energy per cell, as well as the variation of lattice constants. However, the interaction between halogen atoms are very small as mention in ref. 20., so the differences between the possible supercells for the same composition are relatively small (< 0.5 eV/cell, one cell contains 6 atoms). Of course, there are some differences between experimental measurements and DFT calculation, which can be ascribed to the following two aspects: (1) the uncertainty of the method of experimental measurement. The XRD characterization is the most conventional method in experimental measurement, which is a statistical method of analysis, and the accuracy of the results depends on the degree of proficiency of researcher. Fox example, in the Figure S6, the lattice parameter of BiOCl 1−x I x solid solution has two different values at the x = 0.4. (2) The choice of supercell model (including size and occupation pattern) and ordered solid solution have limitation, impacting the accuracy of DFT calculations. For example, the choice of models is not equal interval based on the value of x, so the continuity of variation trend cannot be determined in the present work. However, from the general situation, the variation trends of experimental measurements and DFT calculations are still basically consistent with each other. Thus, the calculated results in the present work can partly explain some experimental phenomena observed previously. Figure 3 shows how the lattice constants vary with x. In the present work, the relationship between the lattice constants and the composition do not follow a linear relationship (i.e., a first-order function 20 ). Therefore, the second-order function of Vegard's law was used to fit these data: 38 where θ is the bowing parameter. The fitting curves are plotted in Fig. 3, and the detailed fitting parameters are listed in Table 2 Lattice constant c:   In the previous experimental report ref. 20, the authors found the BiOCl 1−x I x solid solutions also show abrupt changes for the cell parameters, and they assumed that the strong bowing or deviation of c(x) is mainly due to the weakness of anion-anion interaction across the interface between two vicinal sandwiches. In our present work, we have strengthened this view, and found that the interactions between the halogen atoms, as well as the interaction between the halogen atom and [Bi 2 O 2 ] 2+ layers, are closely related to the nature of the halogen atom itself. So we list these parameters and their differences in Table 3. Furthermore, at the different solubility or content of Y/(X + Y), these differences have different influence on the behavior of solid solutions as shown in Fig. 4. Comparison of these parameters, one can be found that the greater the difference, the more obvious bowing or abruption. Because this explanation is valid for the experimental findings of BiOCl 1−x I x solid solutions, so it could speculate that it also valid for BiOF 1−x X x solid solutions.
Another worth notice phenomenon is the symmetry breaking in some cases. In Fig. 4, two extreme examples are provided: the BiOCl 1−x Br x represents almost no symmetry breaking (i.e. fully obey the Vegard's law) solid solutions; the BiOF 1−x I x represents dramatic symmetry breaking (i.e. obviously disobey the Vegard's law) solid solutions. The similar phenomenon could be observed in BiOF 1−x Cl x and BiOF 1−x Br x solid solutions. When the content (x) of Y (Y = Cl, Br, and I) is smaller than 1/12, the symmetry breaking always exists. We think the main possible reason low content (x) of Y means impurity doping, which destroys the integrity of the interaction in the Fanion plane and then make the [Bi 2 O 2 ] 2+ layer distortion, because there is no self-interaction between Y impurities. When the content (x) of Y is increasing, the self-interaction between Y anions is gradually increasing, resulting in two ordered anion (F − and Y − ) planes. On the other hand, when the content (x) of Y is larger than 11/12, the two ordered anion (F − and Y − ) planes are still maintain, owing the larger interaction of Fwith [Bi 2 O 2 ] 2+ layer, so the symmetry of the atomic positions is keeping. This phenomenon is more obvious if the differences between F and Y atomic properties are larger.  Figure 5 shows the band gaps of the BiOX 1−x Y x solid solutions as a function of x. The variation trend of the band gaps calculated by the GGA + U method is similar to those calculated by the GGA method (Figures S7-S12 in the Supporting Information). Furthermore, the calculated results in the present work are consistent with available experimental values (Figures S13-S15) 21,22,[24][25][26][27]30,31,42 . Based on the above comparison of the lattice constants and band gaps, we consider that the calculation method in the present work is reasonable and produces reliable results.
The electronic energy-band parameters of semiconductor solid solutions and their dependence on x are very important. However, investigation of BiOX 1−x Y x -based photocatalysts has been hampered by a lack of definite knowledge about various material parameters. Therefore, it is necessary to investigate and explain the variation trends of the band gaps of BiOX 1−x Y x solid solutions. The expressions of Vegard's law for the band gap or dielectric function constant are the same as Eq. (4) except that the symbol for the bowing parameter is b rather than θ.    parts and one quadratic variation part. Another interesting result is that the bowing parameters are negative for the BiOCl 1−x Br x , BiOBr 1−x I x , and BiOCl 1−x I x solid solutions, while the bowing parameters are positive for the BiOF 1−x Y x solid solutions. In addition, the bowing parameters decrease with increasing atomic number of Y.
In addition to the variation of the band gap with x, the optical properties of the BiOX 1−x Y x solid solutions also show a similar variation trend. As shown in Fig. 6, the static dielectric constant (ε 0 ) and refractive index (n 0 ) of the BiOX 1−x Y x solid solutions increase with increasing x. Interestingly, expect for the BiOF 1−x I x solid solutions, the optical properties of the BiOX 1−x Y x solid solutions obey the second-order function of Vegard's law in the complete range of x with very small bowing parameters. In other words, the optical properties of the BiOX 1−x Y x solid solutions (expect for the BiOF 1−x I x solid solutions) almost linearly increase with increasing x. For the BiOF 1−x I x solid solutions, the optical properties linearly increase with increasing x in the ranges 0 ≤ x ≤ 1/12 and 1/8 ≤ x ≤ 1/2, while the optical properties quadratically increase with increasing x in the range 3/4 ≤ x ≤ 1. Because optical measurements are relatively easy to perform, we fitted the calculated optical parameters as a function of composition x. The fitting equations are provided in Table 3.

Possible explanation.
In the present work, the properties of BiOX 1−x Y x solid solutions have an inherent connection with the differences between the physical-chemical properties of the two halogen components. The two extreme examples are BiOCl 1−x Br x and BiOF 1−x I x . For the BiOCl 1−x Br x solid solutions, the miscibility temperature is very low and its properties (i.e., lattice constants, band gap, and optical properties) obey the second-order function of Vegard's law in the complete range of x with small bowing parameters. In contrast, for the BiOF 1−x I x solid solutions, the miscibility temperature is very high and its properties obey different rules in different ranges of x: at low I content, the downward/upward bowing is so weak that there is almost linear variation, while at high I composition the bowing is stronger. Therefore, the parameters of the BiOF 1−x I x solid solutions cannot be described using a single bowing parameter, which has also been reported for other semiconductor solid solutions 43,44 . It is worth pointing out that BiOF 1−x I x solid solutions may show phase separation, which may change the photon-absorption mechanism.
In the above examples, Cl and Br are adjacent elements in the periodic table, while F and I are the end elements of the halogen group. In other words, the differences between Cl and Br are very slight, while the differences between F and I are very obvious. Keller et al. considered that Bi-X and Bi-Y bonds of different lengths coexist in a mixed crystal and the weak anion-anion interactions across the interface between two vicinal sandwiches induce the large deviation from Vegard's law 20 , which was confirmed in the present work. We suggest that  Table 4, we extracted the chemical and physical parameters of halogens from ref. 45 and compared their corresponding differences. Except for the differences of the electron affinity, the order of the differences of the other parameters is consistent with the order of the parameters of the BiOX 1−x Y x solid solutions, such as the slope (a) and the interaction parameter (Ω, or miscibility temperature) in Table 1, and the slope or the bowing parameter in Tables 2  and 3. Combining the calculated results in the present work and the data in Table 4, we conclude the following: (1) the atom radii (including van der Waals radii, covalent radii, and ionic radii) directly affect the lattice constants. Because the ratio of lattice constants a/b is mainly determined by the intralayer interaction between Bi and O atoms, it only slightly varies with x in all of the BiOX 1−x Y x solid solutions. However, the c lattice constant is mainly determined by the interlayer interactions, so its variation is related to the differences in the radii of the different halogens, and it significantly varies with x. (2) Both intralayer and interlayer interactions are determined by electron redistribution (i.e., electron gain or loss), which is reflected by the electronegativity or electron affinity. In the present work, we found that the order of the electron affinity difference is not consistent with the order of the parameters of the BiOX 1−x Y x solid solutions. The electron affinity indicates the ability of neutral atoms to accept electrons. In our previous work, we found that the ionic bond is stronger with increasing atomic number of the halogen in BiOX compounds 17 , indicating that BiOF exhibits an obvious mixed-bond feature. Therefore, in BiOCl 1−x Br x , BiOBr 1−x I x , and BiOCl 1−x I x solid solutions, the order of the electron affinity difference is consistent with the order of the solid-solution parameters. However, in BiOF 1−x Y x solid solutions, the electron affinity difference cannot be completely reflected by the variation of the solid-solution parameters. In fact, the electronegativity indicates the binding ability of a neutral atom to a valence electron, accurately reflecting the variation of the solid-solution parameters.

Conclusions
The lattice constants, band gaps, and optical properties of BiOX 1−x Y x solid solutions have been calculated by the GGA + U method. The calculated lattice constants and band gaps of the BiOX 1−x Y x solid solutions agree well with the available experimental values. The calculated results show that: (1) BiOCl 1−x Br x , BiOBr 1−x I x , and BiOCl 1−x I x solid solutions have very small bowing parameters; thus, some of their properties almost linearly vary with x. (2) BiOF 1−x Y x solid solutions have very large bowing parameters. Furthermore, the properties of BiOF 1−x Y x solid solutions cannot be fitted to a single equation. In other words, its properties obey different rules in different ranges of x. For low Y content, the downward/upward bowing is so weak that the variation is almost linear, while at high Y content the bowing is stronger. Consequently, BiOX 1−x Y x solid solutions that do not contain fluorine are highly miscible, while those that contain fluorine are partially miscible. Therefore, BiOF 1−x Y x solid solutions have a miscibility gap or high miscibility temperature, resulting in phase separation and F/Y inhomogeneity. To provide a possible explanation, we compared and analysed the calculated results and the physical-chemical properties for varying halogen compositions, and found that the parameters of BiOX 1−x Y x solid solutions are determined by the differences of the physical-chemical properties between two halogen compositions. In this way, the large deviation from Vegard's law in some BiOX 1−x Y x solid solutions observed in experiments can be explained. Finally, the composition ratio of BiOX 1−x Y x solid solutions can be measured or monitored using optical measurements, because their optical properties approximately linearly vary as a function of x, and the corresponding equations are provided. The band gap of BiOX 1−x Y x solid solutions can be tuned from 1.7 to 4.0 eV by adjusting the halogen composition, which can meet some specific requirements of BiOX-based photocatalysts. The findings in this article provide useful information for designing efficient BiOX-based photocatalysts. Summary, this article achieves the following two purposes: (1) find the underlying mechanism that BiOX 1−x Y x solid solutions obey/ disobey Vegard's law, which partially observed by different experimental researches; (2) provide some available data or formula for future experiments that want to determine or measurement the composition/band gap/optical properties of BiOX 1−x Y x solid solutions.  Table 4. Chemical and physical parameter of halogen elements and the corresponding differences. a Taken from ref. 45. b The Mulliken electronegativity of a neutral atom is the arithmetic mean of the atomic electron affinity and the first ionization energy 50 .