#
Structural, Electronic, and Optical Properties of CsPb(Br_{1−x}Cl_{x})_{3} Perovskite: First-Principles Study with PBE–GGA and mBJ–GGA Methods

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{1−x}Cl

_{x})

_{3}perovskite was investigated in this study. When the chloride (Cl) content of x was increased, the unit cell volume decreased with a linear function. Theoretical X-ray diffraction analyses showed that the peak (at 2θ = 30.4°) shifts to a larger angle (at 2θ = 31.9°) when the average fraction of the incorporated Cl increased. The energy bandgap (E

_{g}) was observed to increase with the increase in Cl concentration. For x = 0.00, 0.25, 0.33, 0.50, 0.66, 0.75, and 1.00, the E

_{g}values calculated using the Perdew–Burke–Ernzerhof potential were between 1.53 and 1.93 eV, while those calculated using the modified Becke−Johnson generalized gradient approximation (mBJ–GGA) potential were between 2.23 and 2.90 eV. The E

_{g}calculated using the mBJ–GGA method best matched the experimental values reported. The effective masses decreased with a concentration increase of Cl to 0.33 and then increased with a further increase in the concentration of Cl. Calculated photoabsorption coefficients show a blue shift of absorption at higher Cl content. The calculations indicate that CsPb(Br

_{1−x}Cl

_{x})

_{3}perovskite could be used in optical and optoelectronic devices by partly replacing bromide with chloride.

## 1. Introduction

_{1−x}Cl

_{x})

_{3}compositions were used for creating various nanophotonic components because they exhibit electroluminescence in the green [12,19] to blue [20] optical ranges. CsPbBr

_{3}exhibits orthorhombic symmetry at temperatures below 88 °C. When the temperature increases, structural distortion occurs and the structure of CsPbBr

_{3}is converted to tetragonal (88 °C < T < 130 °C), and subsequently to cubic at higher temperatures (T > 130 °C) [17,18,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. In comparison, at temperatures below 42 °C, CsPbCl

_{3}exhibits orthorhombic symmetry. When temperature increases, structural distortion occurs and the CsPbCl

_{3}structure is converted to tetragonal (42 °C < T < 47 °C), and subsequently to cubic at higher temperatures (T > 47 °C) [18,39]. The energy band gap (E

_{g}) can be adjusted by adding appropriate materials to the perovskite, which can be designed using theoretical simulations based on density functional theory (DFT) [40]. Recent studies on CsPb(Br

_{1−x}Cl

_{x})

_{3}perovskite thin films, fabricated by sequential deposition technique, revealed an orthorhombic lattice in the case of x = 0.1 and 0.2, whereas for x = 0.4 and 0.6, a cubic phase was observed [41]. The electronic structure of CsPb(Br

_{1−x}Cl

_{x})

_{3}perovskites was studied theoretically and experimentally by Tatiana G. Liashenko et al. [18]. Cl ions, which are the substitute for Br ions in the perovskite crystal lattice at room temperature, do not change its orthorhombic symmetry [18]. Generally, theoretical investigations of electronic and optical properties of organic-inorganic perovskites are often performed by first-principles calculations with the local density approximation (LDA) [42] and Perdew–Burke–Ernzerhof generalized gradient approximation (PBE–GGA) [43,44] using DFT because of their relatively cheap computational cost and reasonable accuracy [45]. The LDA and PBE–GGA potentials failed to calculate the accurate E

_{g}and optical properties because the obtained E

_{g}values were much smaller than the experiment values [43,44,46,47,48] and other possible errors [45]. In addition, the theoretical lattice parameters calculated using PBE–GGA overestimated the experimental lattice constants [45]. LDA potential usually underestimated the lattice constants, which resulted in the underestimation of E

_{g}[45]. To overcome these significant problems of LDA and PBE–GGA potentials, the most accurate potential modified Becke−Johnson GGA (mBJ–GGA) potential was used, which is much more accurate than all other semi-local potentials for strongly correlated systems [49,50]. mBJ–GGA potential can be used for the calculation of E

_{g}with excellent agreement with experimental values thanks to its additional dependence on kinetic energy density [49,50].

_{1−x}Cl

_{x})

_{3}(x = 0.00, 0.25, 0.33, 0.50, 0.66, 0.75, and 1.00) are investigated using PBE–GGA and mBJ–GGA potentials. The calculated values were compared to the previous experimental [51,52,53,54,55,56] and theoretical [27,33,57,58,59,60,61,62,63,64,65,66,67,68] results to verify the validity of the DFT calculation. The effect of spin-orbital coupling (SOC) [57,58,59,60] was included in the calculation because of the heavy lead (Pb) element. By increasing the Cl content x from 0.00 to 1.00, the lattice constants and E

_{g}were calculated. In addition, for these mixed-halide perovskites, the effective masses of charge carriers, the binding energy of the exciton, the absorption coefficients, the optical conductivity, the dielectric constants, and the reflectivity were calculated in detail.

## 2. Computational Method

_{1–x}Cl

_{x})

_{3}(x = 0.00, 0.25, 0.33, 0.50, 0.66, 0.75, and 1.00) were performed using Wu and Cohen (GGA–WC) potential [65]. For the electronic and optical properties, mBJ–GGA [66] and PBE–GGA potentials were used [67]. The mBJ–GGA potential with the SOC effect was included in our DFT calculation because of the heavy Pb element.

_{MT}* k

_{max}value was set at 9.0 (R

_{MT}is the smallest muffin-tin radius in the unit cell and k

_{max}is the maximum value of the reciprocal lattice vectors). The R

_{MT}values were set at 2.5 a.u for (Cs, Pb, and Br) and 2.41 a.u for Cl in such a way that the muffin-tin spheres do not overlap. To ensure the accuracy of our calculations, we considered G

_{max}= 12 and l

_{max}= 10. The irreducible Brillouin zone (IBZ) was produced using 500 k-points (12 × 12 × 3 mesh grids) and the self-consistent convergence of total energy was set at 10

^{−4}Ry.

## 3. Results

#### 3.1. Structural Properties

_{3}and CsPbCl

_{3}have cubic structures with space group Pm$\overline{3}$m (no. 221); the unit cell contains one formula unit. To simulate CsPb(Br

_{1−x}Cl

_{x})

_{3}, a tetragonal 1 × 1 × 4 supercell with 20 atoms was used. For x = 0.00, 0.25, 0.33, 0.50, 0.66, 0.75, and 1.00, a supercell with 0, 3, 4, 6, 8, 9, and 12 atoms of bromide was substituted with chloride atoms, respectively. See the Supplementary Materials, Tables S1–S7, for more details.

_{1−x}Cl

_{x})

_{3}formed by cubic CsPbBr

_{3}and CsPbCl

_{3}.

_{3}and CsPbCl

_{3}structures are in good agreement with recent theoretical and experimental results, thereby proving that our computational parameters are valid.

_{3}(5.8859 Å) and its experimental value of 5.85 (Å) obtained in [69]. Moreover, the value of the lattice parameter for CsPbCl

_{3}was 5.6379 Å, which was in excellent agreement with the experimental value of 5.605 Å obtained in [70]. Theoretical X-ray diffraction (XRD) patterns were obtained using the visualization for electronic and structural analysis (VESTA 3, Ibaraki, Japan) [71] (see Figure 3). The diffraction peaks of CsPbBr

_{3}moved toward CsPbCl

_{3}when x changed from 0.00 to 1.00. As shown in Table 1, when the Cl content x increases from 0.00 to 1.00, the volume of the unit-cell decreases in proportion x with the function of V(x) = 815.29916 – 112.58513x (Å)

^{3}, as shown in Figure 4.

#### 3.2. Electronic Properties

#### 3.2.1. Electronic Band Structure

_{1−x}Cl

_{x})

_{3}were calculated by PBE–GGA and mBJ–GGA potentials without/with SOC. Figure 5 shows the calculated band structures of CsPb(Br

_{1−x}Cl

_{x})

_{3}using the mBJ–GGA potentials without/with SOC. In contrast, Figure 6 shows those using the potential of PBE–GGA without SOC. The band structures have a direct transition character at M, which can improve the photoabsorption coefficient and accelerate the rate of radiative recombination [84]. The calculated E

_{g}for CsPbBr

_{3}, CsPbBr

_{2.75}Cl

_{0.25}, CsPbBr

_{2}Cl, CsPbBr

_{1.5}Cl

_{1.5}, CsPbBrCl

_{2}, CsPbBr

_{0.25}Cl

_{2.75}, and CsPbCl

_{3}based on the mBJ–GGA potential are 2.23, 2.46, 2.40, 2.51, 2.59, 2.64, and 2.90 eV, respectively, whereas the E

_{g}values obtained using the PBE–GGA potential are 1.53, 1.68, 1.56, 1.69, 1.71, 1.77, and 1.93 eV, respectively, as shown in Table 2. The E

_{g}calculated using mBJ–GGA were the closest to the experimental values [51,52,53,54,55].

_{g}values are smaller than the experimental by approximately 1.23 and 1.28 eV for pure CsPbBr

_{3}and CsPbCl

_{3}, respectively, and result in more reasonable band dispersions [85,86]. The SOC causes the conduction band (CB) to decrease by splitting it into a twofold degenerated state (p

_{1/2}) corresponding to light electrons and a fourfold degenerate state (p

_{3/2}) corresponding to heavy electrons at this point [57,87,88]. In contrast, the valance band (VB) showed no significant change in this area [57,87,88]. The correction was thus applied to the E

_{g}with the following equation [78,84,89]:

_{g}corrections for the CsPb(Br

_{1−x}Cl

_{x})

_{3}, CsPbBr

_{3}, and CsPbCl

_{3}compounds, respectively. Figure 7 shows the calculated E

_{g}using PBE–GGA, mBJ–GGA, mBJ–GGA + SOC, and corrected mBJ–GGA + SOC(C). The calculated E

_{g}by mBJ–GGA and mBJ–GGA + SOC(C) are in good agreement with the experimental values [53,55]. The small differences between the theoretical and experimental values are mainly attributed to the changed size for different mixed-halide [84], as depicted in the XRD patterns and the small 1 × 1 × 4 supercell models.

_{g}and the Cl composition x [78,90,91] using the following equation:

_{g}(A) and E

_{g}(B) are the band gaps of pure A and B, respectively; and E

_{g}(x) is the bandgap of A, B mixed-halide perovskites with the composition x. The dependence of the obtained E

_{g}on the concentration of Cl (x) was given by fitting the nonlinear variation with the quadratic function as follows:

_{g}obtained using PBE–GGA, mBJ–GGA, mBJ–GGA + SOC, and mBJ–GGA + SOC(C), respectively.

^{∗}) was evaluated numerically using the following equations:

^{th}band, $\stackrel{\rightharpoonup}{\mathrm{k}}$ represents the wave vector, and $\hslash $ represents the reduced Planck constant.

_{g}value; however, the previous studies stated that the introduction of SOC increases band dispersion and results in more accurate effective masses with respect to DFT calculation without SOC [23,78,79,86,92,96,100,101,102]. Therefore, we employ mBJ–GGA + SOC to evaluate the effective charge masses. The values of m

_{e}* and m

_{h}* decreased significantly with the increase in Cl concentration up to 0.33 owing to the decrease of parabolic nature of the band structure [103]. The increased parabolic nature caused a drastic increase of the effective mass of carriers for high concentration of Cl [103]. The calculated effective charge masses around the M point of the Brillouin zone obtained by evaluating the second derivatives are shown in Table S8 (Supplementary Materials). The reduced masses ${\mu}_{r}$ were calculated using the following equation:

_{0}) can be defined [99] using the following equation:

_{b}) is given by the following:

_{b}, we need to know the dielectric constant of the material ε(∞) and the reduced masses (μ

_{r}), which can be obtained by DFT calculation. The estimated a

_{0}and E

_{b}values were between 5.6 and 8.9 nm and between 41 and 72 meV, respectively, which were in good agreement with other theoretical [16,75,100,104,105] and experimental [106,107] values. A weaker E

_{b}indicates that the charge carriers behave more like free charge carriers [99].

_{0}and E

_{b}values on the concentration of Cl (x) was determined by fitting the nonlinear variation as Cl concentration x with the linear and quadratic functions as follows:

_{0}decreased with the increase in Cl concentration, as shown in Figure 9a. Furthermore, the bowing parameters $\mathrm{b}=17.41855\text{}\mathrm{and}\text{}49.22733$ meV of ${E}_{b}$ using PBE–GGA and mBJ–GGA indicated the decrease in ${E}_{b}$ with the increase in Cl concentration (x), as shown in Figure 9b.

#### 3.2.2. Density of States (DOS)

^{+}does not contribute to VB maximum (VBM) and CB minimum (CBM), and only maintains overall load neutrality and structural stability [23,26,37,72,75,78,79,82,85,92,93,100,101,108,109]. Therefore, we observed only the states of Pb and halogen elements (Cl and Br), as shown in Figure 11. The VBM originates mainly from the p orbitals of Br and Cl, and a small number of contributions from s orbitals of Pb can also be observed. The CBM originated from the p states of Pb and halogen elements (Cl and Br). The CB structure is relatively similar for all of the compounds, and the CBM for each compound comprises mainly p orbitals of Pb and halogen elements (Cl and Br). The uppermost VB is steep, while the lowermost CB in PDOS is relatively flat.

_{1.5}Cl

_{1.5}, PDOS was plotted on the band structure using the mBJ–GGA potential (Figure 12a). The PDOS (Figure 12b) indicated that the effects of the Cs atoms did not follow any specific rules, whereas it shows that the E

_{g}trends are the result of the effects of Pb and Br [93]. Similar band structures of CsPbBr

_{3}and CsPbCl

_{3}with PDOS are shown in Figure S1 (Supplementary Materials).

_{A}-X

_{B}) is crucial [110], where X

_{A}and X

_{B}are the electro-negativities of the A and B atoms, respectively. The percentage of the ionic character (IC) of the bonding can be obtained from the following equation [111]:

#### 3.3. Optical Properties

_{1−x}Cl

_{x})

_{3}perovskite is essential because of its potential for use in photonic and optoelectronic applications. Calculations of dielectric functions with both real ε

_{1}(ω) and imaginary ε

_{2}(ω) parts, refractive index n(ω), extinction coefficient $\mathrm{k}\left(\mathsf{\omega}\right)$, absorption coefficient α(ω), optical conductivity 𝜎(ω), and reflectivity R(ω) were explored by mBJ–GGA potential. These optical parameters can be attracted by the knowledge of the complex dielectric function ε(ω) = ε

_{1}(ω) + iε

_{2}(ω). The imaginary part of the dielectric function ε

_{2}(ω), according to the perturbation theory, is given by the following equation [113,114]:

_{k}and j

_{k}are the crystal wave functions corresponding to the conduction and valence bands with the crystal wave vector k, respectively. The real part ${\mathsf{\epsilon}}_{1}\left(\mathsf{\omega}\right)$ of the dielectric function can be expressed as follows [114]:

_{g}. The results obey the following equation:

_{3}, ${\mathsf{\epsilon}}_{1}\left(0\right)$ was 3.82, which agrees well with the result obtained in the previous studies [23,72,105]. Figure 14b shows the behavior of ε

_{2}(ω) for all Cl concentrations. For x = 0.00, 0.25, 0.33, 0.50, 0.66, 0.75, and 1.00, the critical points in ${\mathsf{\epsilon}}_{2}\left(\mathsf{\omega}\right)$ occurred at approximately 2.14, 2.25, 2.28, 2.34, 2.45, 2.55, and 2.84 eV, respectively, which were closely related to the direct E

_{g}values of 2.26, 2.47, 2.39, 2.51, 2.58, 2.64, and 2.90 eV, respectively.

_{3}, the calculated n (0) value was 1.96, which agrees well with the previous theoretical and experimental values [56,72]. For CsPbCl

_{3}, n(0) was 1.798, which agrees well with the previous value [72,81]. The calculated n (0) versus the Cl concentration (x) is expressed as follows:

_{2}(ω). The peak value of $\mathrm{k}\text{}\left(\mathsf{\omega}\right)$ shifted to lower energies as Cl concentration increased from 0.00 to 1.00.

_{3}(x = 0.00) and CsPbCl

_{3}(x = 1.00), respectively, as shown in Figure 16. The maximum reflectivity peaks of 48%, 46.7%, 47.8%, 48.5%, 48.7%, 48.6%, and 51% occurred at energy values of 15.88, 15.97, 16.00, 16.10, 16.16, 16.18, and 16.29 eV, respectively, and then began to fluctuate and decrease at higher energies. The value of R (0) decreased with the increase in Cl concentration (x), as shown in Figure 17 and presented in Table 3. The calculated R (0) versus Cl concentration (x) was fitted as follows:

## 4. Conclusions

_{1−x}Cl

_{x})

_{3}using DFT. When the Cl content x was increased from 0.00 to 1.00, a decrease in unit-cell volume was observed. Theoretical XRD analyses revealed that the peak shifts to larger angles when the concentration of Cl increases. An increase in E

_{g}was observed with an increase in the concentration of Cl. The E

_{g}values calculated using the PBE–GGA potential were between 1.53 and 1.93 eV, while those calculated using the mBJ–GGA potential were between 2.23 and 2.90 eV. The increase in E

_{g}with the increase in Cl content was due to the fact that the hybridization of Cl 3p states with Pb-s states was stronger than that with Br 4p states, which leads to a downshift of VBM and a decrease in the lattice constant. The calculated E

_{g}and exciton binding energy E

_{b}using mBJ–GGA potential best matched the previously reported experimental and theoretical values. The effective masses of electron and hole (m

_{e}* and m

_{h}*) are correlated with the energies of E

_{g}. The calculated photoabsorption coefficients display a blue shift of the absorption at a higher Cl concentration.

## Supplementary Materials

_{1−x}Cl

_{x})

_{3}Perovskite, Table S8: Effective mass of electron (m

_{e}*) and hole (m

_{h}*), reduced mass (µ

_{r}), bohr diameter (a

_{0}), dielectric constant (ε), and exciton binding energy (E

_{b}) values calculated by PBE–GGA, mBJ–GGA, and mBJ–GGA + SOC potentials, Figure S1: Band structures and PDOS of (a) CsPbBr

_{3}and (b) CsPbCl

_{3}obtained using the mBJ–GGA potential.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Haddad, J.; Krogmeier, B.; Klingebiel, B.; Krückemeier, L.; Melhem, S.; Liu, Z.; Hüpkes, J.; Mathur, S.; Kirchartz, T. Analyzing Interface Recombination in Lead-Halide Perovskite Solar Cells with Organic and Inorganic Hole-Transport Layers. Adv. Mater. Interfaces
**2020**, 7, 2000366. [Google Scholar] [CrossRef] - Zhao, Y.; Duan, J.; Wang, Y.; Yang, X.; Tang, Q. Precise stress control of inorganic perovskite films for carbon-based solar cells with an ultrahigh voltage of 1.622 V. Nano Energy
**2020**, 67, 104286. [Google Scholar] [CrossRef] - Duan, J.; Wei, J.; Tang, Q.; Li, Q. Unveiling the interfacial charge extraction kinetics in inorganic perovskite solar cells with formamidinium lead halide (FAPbX
_{3}) nanocrystals. Sol. Energy**2020**, 195, 644–650. [Google Scholar] [CrossRef] - Ouedraogo, N.A.N.; Chen, Y.; Xiao, Y.Y.; Meng, Q.; Han, C.B.; Yan, H.; Zhang, Y. Stability of all-inorganic perovskite solar cells. Nano Energy
**2020**, 67, 104249. [Google Scholar] [CrossRef] - Feng, J.; Zhu, X.; Yang, Z.; Zhang, X.; Niu, J.; Wang, Z.; Zuo, S.; Priya, S.; Liu, S. (Frank); Yang, D. Record Efficiency Stable Flexible Perovskite Solar Cell Using Effective Additive Assistant Strategy. Adv. Mater.
**2018**, 30, e1801418. [Google Scholar] [CrossRef] - Wang, K.; Jin, Z.; Liang, L.; Bian, H.; Bai, D.; Wang, H.; Zhang, J.; Wang, Q.; Shengzhong, L. All-inorganic cesium lead iodide perovskite solar cells with stabilized efficiency beyond 15%. Nat. Commun.
**2018**, 9, 1–8. [Google Scholar] [CrossRef] - Kim, H.S.; Lee, C.R.; Im, J.H.; Lee, K.B.; Moehl, T.; Marchioro, A.; Moon, S.J.; Humphry-Baker, R.; Yum, J.H.; Moser, J.E.; et al. Lead iodide perovskite sensitized all-solid-state submicron thin film mesoscopic solar cell with efficiency exceeding 9%. Sci. Rep.
**2012**, 2, 591. [Google Scholar] [CrossRef] [Green Version] - Eperon, G.E.; Paternò, G.M.; Sutton, R.J.; Zampetti, A.; Haghighirad, A.A.; Cacialli, F.; Snaith, H.J. Inorganic caesium lead iodide perovskite solar cells. J. Mater. Chem. A
**2015**, 3, 19688–19695. [Google Scholar] [CrossRef] - Lin, K.; Xing, J.; Quan, L.N.; de Arquer, F.P.G.; Gong, X.; Lu, J.; Xie, L.; Zhao, W.; Zhang, D.; Yan, C.; et al. Perovskite light-emitting diodes with external quantum efficiency exceeding 20 per cent. Nature
**2018**, 562, 245–248. [Google Scholar] [CrossRef] - Matsushima, T.; Bencheikh, F.; Komino, T.; Leyden, M.R.; Sandanayaka, A.S.D.; Qin, C.; Adachi, C. High performance from extraordinarily thick organic light-emitting diodes. Nature
**2019**, 572, 502–506. [Google Scholar] [CrossRef] - Zhang, Q.; Tavakoli, M.M.; Gu, L.; Zhang, D.; Tang, L.; Gao, Y.; Guo, J.; Lin, Y.; Leung, S.F.; Poddar, S.; et al. Efficient metal halide perovskite light-emitting diodes with significantly improved light extraction on nanophotonic substrates. Nat. Commun.
**2019**, 10, 727. [Google Scholar] [CrossRef] - Veldhuis, S.A.; Boix, P.P.; Yantara, N.; Li, M.; Sum, T.C.; Mathews, N.; Mhaisalkar, S.G. Perovskite Materials for Light-Emitting Diodes and Lasers. Adv. Mater.
**2016**, 28, 6804–6834. [Google Scholar] [CrossRef] [PubMed] - Tan, Z.K.; Moghaddam, R.S.; Lai, M.L.; Docampo, P.; Higler, R.; Deschler, F.; Price, M.; Sadhanala, A.; Pazos, L.M.; Credgington, D.; et al. Bright light-emitting diodes based on organometal halide perovskite. Nat. Nanotechnol.
**2014**, 9, 687–692. [Google Scholar] [CrossRef] [PubMed] - Even, J.; Pedesseau, L.; Jancu, J.M.; Katan, C. DFT and k · p modelling of the phase transitions of lead and tin halide perovskites for photovoltaic cells. Phys. Status Solidi Rapid Res. Lett.
**2014**, 8, 31–35. [Google Scholar] [CrossRef] [Green Version] - Droseros, N.; Longo, G.; Brauer, J.C.; Sessolo, M.; Bolink, H.J.; Banerji, N. Origin of the Enhanced Photoluminescence Quantum Yield in MAPbBr
_{3}Perovskite with Reduced Crystal Size. ACS Energy Lett.**2018**, 3, 1458–1466. [Google Scholar] [CrossRef] [Green Version] - Protesescu, L.; Yakunin, S.; Bodnarchuk, M.I.; Krieg, F.; Caputo, R.; Hendon, C.H.; Yang, R.X.; Walsh, A.; Kovalenko, M.V. Nanocrystals of Cesium Lead Halide Perovskites (CsPbX
_{3}, X = Cl, Br, and I): Novel Optoelectronic Materials Showing Bright Emission with Wide Color Gamut. Nano Lett.**2015**, 15, 3692–3696. [Google Scholar] [CrossRef] [Green Version] - Nedelcu, G.; Protesescu, L.; Yakunin, S.; Bodnarchuk, M.I.; Grotevent, M.J.; Kovalenko, M.V. Fast Anion-Exchange in Highly Luminescent Nanocrystals of Cesium Lead Halide Perovskites (CsPbX
_{3}, X = Cl, Br, I). Nano Lett.**2015**, 15, 5635–5640. [Google Scholar] [CrossRef] - Liashenko, T.G.; Cherotchenko, E.D.; Pushkarev, A.P.; Pakštas, V.; Naujokaitis, A.; Khubezhov, S.A.; Polozkov, R.G.; Agapev, K.B.; Zakhidov, A.A.; Shelykh, I.A.; et al. Electronic structure of CsPbBr
_{3-x}Cl_{x}perovskites: Synthesis, experimental characterization, and DFT simulations. Phys. Chem. Chem. Phys.**2019**, 21, 18930–18938. [Google Scholar] [CrossRef] - Zhang, L.; Yang, X.; Jiang, Q.; Wang, P.; Yin, Z.; Zhang, X.; Tan, H.; Yang, Y.M.; Wei, M.; Sutherland, B.R.; et al. Ultra-bright and highly efficient inorganic based perovskite light-emitting diodes. Nat. Commun.
**2017**, 8, 15640. [Google Scholar] [CrossRef] - Song, J.; Li, J.; Li, X.; Xu, L.; Dong, Y.; Zeng, H. Quantum Dot Light-Emitting Diodes Based on Inorganic Perovskite Cesium Lead Halides (CsPbX
_{3}). Adv. Mater.**2015**, 27, 7162–7167. [Google Scholar] [CrossRef] - Zhang, X.; Jin, Z.; Zhang, J.; Bai, D.; Bian, H.; Wang, K.; Sun, J.; Wang, Q.; Liu, S.F. All-Ambient Processed Binary CsPbBr
_{3}-CsPb_{2}Br_{5}Perovskites with Synergistic Enhancement for High-Efficiency Cs-Pb-Br-Based Solar Cells. ACS Appl. Mater. Interfaces**2018**, 10, 7145–7154. [Google Scholar] [CrossRef] - Stoumpos, C.C.; Malliakas, C.D.; Peters, J.A.; Liu, Z.; Sebastian, M.; Im, J.; Chasapis, T.C.; Wibowo, A.C.; Chung, D.Y.; Freeman, A.J.; et al. Crystal growth of the perovskite semiconductor CsPbBr3: A new material for high-energy radiation detection. Cryst. Growth Des.
**2013**, 13, 2722–2727. [Google Scholar] [CrossRef] - Ahmad, M.; Rehman, G.; Ali, L.; Shafiq, M.; Iqbal, R.; Ahmad, R.; Khan, T.; Jalali-Asadabadi, S.; Maqbool, M.; Ahmad, I. Structural, electronic and optical properties of CsPbX
_{3}(X=Cl, Br, I) for energy storage and hybrid solar cell applications. J. Alloys Compd.**2017**, 705, 828–839. [Google Scholar] [CrossRef] - Yaffe, O.; Guo, Y.; Hull, T.; Stoumpos, C.C.; Tan, L.Z.; Egger, D.A.; Zheng, F.; Szpak, G.; Semonin, O.E.; Beecher, A.N.; et al. The nature of dynamic disorder in lead halide perovskite crystals (Conference Presentation). Phys. Chem. Interfaces Nanomater. XV
**2016**. [Google Scholar] [CrossRef] - Rodova, M.; Brozek, J.; Knizek, K.; Nitsch, K. Phase transitions in ternary caesium lead bromide. J. Therm. Anal. Calorim.
**2003**, 71, 667–673. [Google Scholar] [CrossRef] - Bechtel, J.S.; Van Der Ven, A. First-principles thermodynamics study of phase stability in inorganic halide perovskite solid solutions. Phys. Rev. Mater.
**2018**, 2, 045401. [Google Scholar] [CrossRef] - Zhang, M.; Zheng, Z.; Fu, Q.; Chen, Z.; He, J.; Zhang, S.; Yan, L.; Hu, Y.; Luo, W. Growth and characterization of all-inorganic lead halide perovskite semiconductor CsPbBr
_{3}single crystals. CrystEngComm**2017**, 19, 6797–6803. [Google Scholar] [CrossRef] - He, Y.; Matei, L.; Jung, H.J.; McCall, K.M.; Chen, M.; Stoumpos, C.C.; Liu, Z.; Peters, J.A.; Chung, D.Y.; Wessels, B.W.; et al. High spectral resolution of gamma-rays at room temperature by perovskite CsPbBr 3 single crystals. Nat. Commun.
**2018**, 9, 1–8. [Google Scholar] [CrossRef] - Qiao, B.; Song, P.; Cao, J.; Zhao, S.; Shen, Z.; Di, G.; Liang, Z.; Xu, Z.; Song, D.; Xu, X. Water-resistant, monodispersed and stably luminescent CsPbBr3/CsPb2Br5 core-shell-like structure lead halide perovskite nanocrystals. Nanotechnology
**2017**, 28, 445602. [Google Scholar] [CrossRef] - Bertolotti, F.; Protesescu, L.; Kovalenko, M.V.; Yakunin, S.; Cervellino, A.; Billinge, S.J.L.; Terban, M.W.; Pedersen, J.S.; Masciocchi, N.; Guagliardi, A. Coherent Nanotwins and Dynamic Disorder in Cesium Lead Halide Perovskite Nanocrystals. ACS Nano
**2017**, 11, 3819–3831. [Google Scholar] [CrossRef] - Dos Reis, R.; Yang, H.; Ophus, C.; Ercius, P.; Bizarri, G.; Perrodin, D.; Shalapska, T.; Bourret, E.; Ciston, J.; Dahmen, U. Determination of the structural phase and octahedral rotation angle in halide perovskites. Appl. Phys. Lett.
**2018**, 112, 071901. [Google Scholar] [CrossRef] [Green Version] - Cottingham, P.; Brutchey, R.L. On the crystal structure of colloidally prepared CsPbBr3 quantum dots. Chem. Commun.
**2016**, 52, 5246–5249. [Google Scholar] [CrossRef] - Wolf, C.; Lee, T.W. Exciton and lattice dynamics in low-temperature processable CsPbBr3 thin-films. Mater. Today Energy
**2018**, 7, 199–207. [Google Scholar] [CrossRef] - Zhang, M.; Zheng, Z.; Fu, Q.; Chen, Z.; He, J.; Zhang, S.; Chen, C.; Luo, W. Synthesis and single crystal growth of perovskite semiconductor CsPbBr
_{3}. J. Cryst. Growth**2018**, 484, 37–42. [Google Scholar] [CrossRef] - Goesten, M.G.; Hoffmann, R. Mirrors of Bonding in Metal Halide Perovskites. J. Am. Chem. Soc.
**2018**, 140, 12996–13010. [Google Scholar] [CrossRef] - Møller, C.K. The Structure of Perovskite-Like Cæsium Plumbo Trihalides. Mater. Fys. Medd. Dan. Vid. Selsk
**1959**, 32, 1–27. [Google Scholar] - Yang, R.X.; Skelton, J.M.; Da Silva, E.L.; Frost, J.M.; Walsh, A. Spontaneous octahedral tilting in the cubic inorganic cesium halide perovskites CsSnX
_{3}and CsPbX_{3}(X = F, Cl, Br, I). J. Phys. Chem. Lett.**2017**, 8, 4720–4726. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kirschner, M.S.; Diroll, B.T.; Guo, P.; Harvey, S.M.; Helweh, W.; Flanders, N.C.; Brumberg, A.; Watkins, N.E.; Leonard, A.A.; Evans, A.M.; et al. Photoinduced, reversible phase transitions in all-inorganic perovskite nanocrystals. Nat. Commun.
**2019**, 10, 1–8. [Google Scholar] [CrossRef] - Plesko, S.; Kind, R.; Roos, J. Structural Phase Transitions in CsPbCl
_{3}and RbCdCl_{3}. J. Phys. Soc. Jpn.**1978**, 45, 553–557. [Google Scholar] [CrossRef] - Yu, H.S.; Li, S.L.; Truhlar, D.G. Perspective: Kohn-Sham density functional theory descending a staircase. J. Chem. Phys.
**2016**, 145, 130901. [Google Scholar] [CrossRef] - Li, B.; Zhang, Y.; Zhang, L.; Yin, L. PbCl2-tuned inorganic cubic CsPbBr3(Cl) perovskite solar cells with enhanced electron lifetime, diffusion length and photovoltaic performance. J. Power Sources
**2017**, 360, 11–20. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev.
**1965**, 140, A1133. [Google Scholar] [CrossRef] [Green Version] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 66, 3865–3868. [Google Scholar] [CrossRef] [Green Version] - Ziesche, P.; Kurth, S.; Perdew, J.P. Density functionals from LDA to GGA. Comput. Mater. Sci.
**1998**, 11, 122–127. [Google Scholar] [CrossRef] - Raju, N.P.; Thangavel, R. Theoretical investigation of spin–orbit coupling on structural, electronic and optical properties for CuAB2 (A = Sb, Bi; B = S, Se) compounds using Tran–Blaha-modified Becke–Johnson method: A first-principles approach. J. Alloys Compd.
**2020**, 2, 154621. [Google Scholar] [CrossRef] - Tran, F.; Blaha, P. Implementation of screened hybrid functionals based on the Yukawa potential within the LAPW basis set. Phys. Rev. B
**2011**, 83, 235118. [Google Scholar] [CrossRef] [Green Version] - Staroverov, V.N.; Scuseria, G.E.; Tao, J.; Perdew, J.P. Tests of a ladder of density functionals for bulk solids and surfaces. Phys. Rev. B
**2004**, 69, 075102. [Google Scholar] [CrossRef] - Kurth, S.; Perdew, J.P.; Blaha, P. Molecular and solid-state tests of density functional approximations: LSD, GGAs, and Meta-GGAs. Int. J. Quantum Chem.
**1999**, 75, 889–909. [Google Scholar] [CrossRef] - Camargo-Martínez, J.A.; Baquero, R. The band gap problem: The accuracy of the wien2k code confronted. Rev. Mex. Fis.
**2013**, 59, 453–459. [Google Scholar] - Tran, F.; Blaha, P. Importance of the Kinetic Energy Density for Band Gap Calculations in Solids with Density Functional Theory. J. Phys. Chem. A
**2017**, 121, 3318–3325. [Google Scholar] [CrossRef] [Green Version] - Akkerman, Q.A.; Motti, S.G.; Srimath Kandada, A.R.; Mosconi, E.; D’Innocenzo, V.; Bertoni, G.; Marras, S.; Kamino, B.A.; Miranda, L.; De Angelis, F.; et al. Solution Synthesis Approach to Colloidal Cesium Lead Halide Perovskite Nanoplatelets with Monolayer-Level Thickness Control. J. Am. Chem. Soc.
**2016**, 138, 1010–1016. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Duan, J.; Yuan, H.; Zhao, Y.; He, B.; Tang, Q. Lattice Modulation of Alkali Metal Cations Doped Cs
_{1−x}R_{x}PbBr_{3}Halides for Inorganic Perovskite Solar Cells. Sol. RRL**2018**, 2, 1800164. [Google Scholar] [CrossRef] - Qaid, S.M.H.; Al-Asbahi, B.A.; Ghaithan, H.M.; AlSalhi, M.S.; Al dwayyan, A.S. Optical and structural properties of CsPbBr
_{3}perovskite quantum dots/PFO polymer composite thin films. J. Colloid Interface Sci.**2020**, 563, 426–434. [Google Scholar] [CrossRef] - Heidrich, K.; Schäfer, W.; Schreiber, M.; Söchtig, J.; Trendel, G.; Treusch, J.; Grandke, T.; Stolz, H.J. Electronic structure, photoemission spectra, and vacuum-ultraviolet optical spectra of CsPbCl3 and CsPbBr3. Phys. Rev. B
**1981**, 24, 5642–5649. [Google Scholar] [CrossRef] - Pandey, N.; Kumar, A.; Chakrabarti, S. Investigation of the structural, electronic, and optical properties of Mn-doped CsPbCl3: Theory and experiment. RSC Adv.
**2019**, 9, 29556–29565. [Google Scholar] [CrossRef] [Green Version] - Yakunin, S.; Protesescu, L.; Krieg, F.; Bodnarchuk, M.I.; Nedelcu, G.; Humer, M.; De Luca, G.; Fiebig, M.; Heiss, W.; Kovalenko, M.V. Low-threshold amplified spontaneous emission and lasing from colloidal nanocrystals of caesium lead halide perovskites lead halide perovskites. Nat. Commun.
**2015**, 6, 8056. [Google Scholar] [CrossRef] - Afsari, M.; Boochani, A.; Hantezadeh, M.; Elahi, S.M. Topological nature in cubic phase of perovskite CsPbI3: By DFT. Solid State Commun.
**2017**, 259, 10–15. [Google Scholar] [CrossRef] - Zhang, Q.; Su, R.; Du, W.; Liu, X.; Zhao, L.; Ha, S.T.; Xiong, Q. Advances in Small Perovskite-Based Lasers. Small Methods
**2017**, 1, 1700163. [Google Scholar] [CrossRef] - Even, J.; Pedesseau, L.; Jancu, J.M.; Katan, C. Importance of spin-orbit coupling in hybrid organic/inorganic perovskites for photovoltaic applications. J. Phys. Chem. Lett.
**2013**, 4, 2999–3005. [Google Scholar] [CrossRef] [Green Version] - Wang, K.; Yang, Q.; Duan, J.; Zhang, C.; Zhao, F.; Yu, H.; Hu, B. Spin-Polarized Electronic Transport through Ferromagnet/Organic–Inorganic Hybrid Perovskite Spinterfaces at Room Temperature. Adv. Mater. Interfaces
**2019**, 1900718. [Google Scholar] [CrossRef] - Blaha, P.; Schwarz, K.; Trickey, S.B. Full-potential, linearized augmented plane wave programs for crystalline systems. Comput. Phys. Commun
**1990**, 59, 399–415. [Google Scholar] [CrossRef] - Madsen, G.K.H.; Blaha, P.; Schwarz, K.; Sjöstedt, E.; Nordström, L. Efficient linearization of the augmented plane-wave method. Phys. Rev. B Condens. Matter Mater. Phys.
**2001**, 64, 195134. [Google Scholar] [CrossRef] - Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev.
**1964**, 136, B864. [Google Scholar] [CrossRef] [Green Version] - Blaha, P.; Schwarz, K.; Madsen, G.K.H.; Kvasnicka, D.; Luitz, J. WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties; Technische Universität Wien: Vienna, Austria, 2001; ISBN 3-9501031-1-2. [Google Scholar]
- Wu, Z.; Cohen, R.E. More accurate generalized gradient approximation for solids. Phys. Rev. B Condens. Matter Mater. Phys.
**2006**, 73, 235116. [Google Scholar] [CrossRef] [Green Version] - Tran, F.; Blaha, P. Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential. Phys. Rev. Lett.
**2009**, 102, 226401. [Google Scholar] [CrossRef] [Green Version] - Städele, M.; Majewski, J.A.; Vogl, P.; Görling, A. Exact Kohn-Sham Exchange Potential in Semiconductors. Phys. Rev. Lett.
**1997**, 79, 2089–2092. [Google Scholar] [CrossRef] - G.F.S. The volume changes of five gases under high pressures. J. Franklin Inst.
**1924**, 197, 98. [Google Scholar] [CrossRef] - Beal, R.E.; Slotcavage, D.J.; Leijtens, T.; Bowring, A.R.; Belisle, R.A.; Nguyen, W.H.; Burkhard, G.F.; Hoke, E.T.; McGehee, M.D. Cesium Lead Halide Perovskites with Improved Stability for Tandem Solar Cells. J. Phys. Chem. Lett.
**2016**, 7, 746–751. [Google Scholar] [CrossRef] [PubMed] - Lim, A.R.; Jeong, S.Y. Twin structure by 133Cs NMR in ferroelastic CsPbCl3 crystal. Solid State Commun.
**1999**, 110, 131–136. [Google Scholar] [CrossRef] - Momma, K.; Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr.
**2011**, 44, 1272–1276. [Google Scholar] [CrossRef] - Murtaza, G.; Ahmad, I. First principle study of the structural and optoelectronic properties of cubic perovskites CsPbM
_{3}(M=Cl, Br, I). Phys. B Condens. Matter**2011**, 406, 3222–3229. [Google Scholar] [CrossRef] - Mahmood, Q.; Hassan, M.; Rashid, M.; Haq, B.U.; Laref, A. The systematic study of mechanical, thermoelectric and optical properties of lead based halides by first principle approach. Phys. B Condens. Matter
**2019**, 571, 87–92. [Google Scholar] [CrossRef] - Jaroenjittichai, A.P.; Laosiritaworn, Y. Band alignment of cesium-based halide perovskites. Ceram. Int.
**2018**, 44, S161–S163. [Google Scholar] [CrossRef] - Chang, Y.H.; Park, C.H. First-Principles Study of the Structural and the Electronic Properties of the Lead-Halide-Based Inorganic-Organic Perovskites (CH
_{3}NH_{3})PbX_{3}and CsPbX_{3}(X = Cl, Br, I). J. Korean Phys. Soc.**2004**, 44, 889–893. [Google Scholar] - Ghaithan, H.M.; AlAhmed, Z.A.; Lyras, A.; Qaid, S.M.H.; Aldwayyan, A.S. Computational Investigation of the Folded and Unfolded Band Structure and Structural and Optical Properties of CsPb(I
_{1−x}Br_{x})_{3}Perovskites. Crystals**2020**, 10, 342. [Google Scholar] [CrossRef] - Ghaithan, H.M.; Alahmed, Z.A.; Qaid, S.M.H.; Aldwayyan, A.S. First principle-based calculations of the optoelectronic features of 2 × 2 × 2 CsPb(I
_{1-x}Br_{x})_{3}perovskite. Superlattices Microstruct.**2020**, 140, 106474. [Google Scholar] [CrossRef] - Chen, X.; Han, D.; Su, Y.; Zeng, Q.; Liu, L.; Shen, D. Structural and Electronic Properties of Inorganic Mixed Halide Perovskites. Phys. Status Solidi Rapid Res. Lett.
**2018**, 12, 1800193. [Google Scholar] [CrossRef] - Lang, L.; Yang, J.H.; Liu, H.R.; Xiang, H.J.; Gong, X.G. First-principles study on the electronic and optical properties of cubic ABX
_{3}halide perovskites. Phys. Lett. A**2014**, 378, 290–293. [Google Scholar] [CrossRef] [Green Version] - Wanwieng, N.; Laosiritaworn, Y. Enthalpy of formation of CsSn(Cl
_{x}(Br, I)_{1-x})_{3}and CsPb(Cl_{x}(Br, I)_{1-x})_{3}. Siam Phys. Congr.**2018**, 1144, 012137. [Google Scholar] [CrossRef] - Ghebouli, M.A.; Ghebouli, B.; Fatmi, M. First-principles calculations on structural, elastic, electronic, optical and thermal properties of CsPbCl
_{3}perovskite. Phys. B Condens. Matter**2011**, 406, 1837–1843. [Google Scholar] [CrossRef] - Ilyas, B.M.; Elias, B.H. A theoretical study of perovskite CsXCl
_{3}(X=Pb, Cd) within first principles calculations. Phys. B Condens. Matter**2017**, 510, 60–73. [Google Scholar] [CrossRef] - Møller, C.K. Crystal structure and photoconductivity of cæsium plumbohalides. Nature
**1958**, 182, 1436. [Google Scholar] [CrossRef] - Su, Y.; Chen, X.; Ji, W.; Zeng, Q.; Ren, Z.; Su, Z.; Liu, L. Highly Controllable and Efficient Synthesis of Mixed-Halide CsPbX
_{3}(X = Cl, Br, I) Perovskite QDs toward the Tunability of Entire Visible Light. ACS Appl. Mater. Interfaces**2017**, 9, 33020–33028. [Google Scholar] [CrossRef] - Yi, Z.; Fang, Z. Theoretical studies on the structural, electronic and optical properties of orthorhombic perovskites CH
_{3}NH_{3}PbX_{3}(X = I, Br, Cl). J. Phys. Chem. Solids**2017**, 110, 145–151. [Google Scholar] [CrossRef] - Umari, P.; Mosconi, E.; De Angelis, F. Relativistic GW calculations on CH
_{3}NH_{3}PbI_{3}and CH_{3}NH_{3}SnI_{3}Perovskites for Solar Cell Applications. Sci. Rep.**2014**, 4, 4467. [Google Scholar] [CrossRef] [Green Version] - Ghaithan, H.M.; Alahmed, Z.A.; Qaid, S.M.H.; Hezam, M.; Aldwayyan, A.S. Density Functional Study of Cubic, Tetragonal, and Orthorhombic CsPbBr
_{3}Perovskite. ACS Omega**2020**, 5, 7468–7480. [Google Scholar] [CrossRef] [Green Version] - Sun, P.P.; Li, Q.S.; Yang, L.N.; Li, Z.S. Theoretical insights into a potential lead-free hybrid perovskite: Substituting Pb2+ with Ge2+. Nanoscale
**2016**, 8, 1503–1512. [Google Scholar] [CrossRef] - Yin, W.J.; Yan, Y.; Wei, S.H. Anomalous alloy properties in mixed halide perovskites. J. Phys. Chem. Lett.
**2014**, 5, 3625–3631. [Google Scholar] [CrossRef] - Bellaiche, L.; Wei, S.H.; Zunger, A. Localization and percolation in semiconductor alloys: GaAsN vs GaAsP. Phys. Rev. B Condens. Matter Mater. Phys.
**1996**, 54, 17568–17576. [Google Scholar] [CrossRef] [Green Version] - Wei, S.H.; Zunger, A. Giant and composition-dependent optical bowing coefficient in GaAsN alloys. Phys. Rev. Lett.
**1996**, 76, 664–667. [Google Scholar] [CrossRef] [Green Version] - Jishi, R.A.; Ta, O.B.; Sharif, A.A. Modeling of lead halide perovskites for photovoltaic applications. J. Phys. Chem. C
**2014**, 118, 28344–28349. [Google Scholar] [CrossRef] [Green Version] - Mao, X.; Sun, L.; Wu, T.; Chu, T.; Deng, W.; Han, K. First-Principles Screening of All-Inorganic Lead-Free ABX
_{3}Perovskites. J. Phys. Chem. C**2018**, 122, 7670–7675. [Google Scholar] [CrossRef] - Paul, T.; Chatterjee, B.K.; Maiti, S.; Sarkar, S.; Besra, N.; Das, B.K.; Panigrahi, K.J.; Thakur, S.; Ghorai, U.K.; Chattopadhyay, K.K. Tunable cathodoluminescence over the entire visible window from all-inorganic perovskite CsPbX
_{3}1D architecture. J. Mater. Chem. C**2018**, 6, 3322–3333. [Google Scholar] [CrossRef] - Diroll, B.T.; Zhou, H.; Schaller, R.D. Low-Temperature Absorption, Photoluminescence, and Lifetime of CsPbX
_{3}(X = Cl, Br, I) Nanocrystals. Adv. Funct. Mater.**2018**, 28, 1800945. [Google Scholar] [CrossRef] - Castelli, I.E.; García-Lastra, J.M.; Thygesen, K.S.; Jacobsen, K.W. Bandgap calculations and trends of organometal halide perovskites. APL Mater.
**2014**, 2, 081514. [Google Scholar] [CrossRef] [Green Version] - Feng, J.; Xiao, B. Effective masses and electronic and optical properties of nontoxic MASnX3 (X = Cl, Br, and I) perovskite structures as solar cell absorber: A theoretical study using HSE06. J. Phys. Chem. C
**2014**, 118, 19655–19660. [Google Scholar] [CrossRef] - Feng, J.; Xiao, B. Crystal Structures, Optical Properties, and E ff ective Mass Tensors of CH
_{3}NH_{3}PbX_{3}(X = I and Br) Phases Predicted from HSE06. J. Phys. Chem. Lett.**2014**, 5, 1278–1282. [Google Scholar] - Jong, U.G.; Yu, C.J.; Ri, J.S.; Kim, N.H.; Ri, G.C. Influence of halide composition on the structural, electronic, and optical properties of mixed CH
_{3}NH_{3}Pb(I_{1-x}Br_{x})_{3}perovskites calculated using the virtual crystal approximation method. Phys. Rev. B**2016**, 94, 125139. [Google Scholar] [CrossRef] [Green Version] - Qian, J.; Xu, B.; Tian, W. A comprehensive theoretical study of halide perovskites ABX
_{3}. Org. Electron. Phys. Mater. Appl.**2016**, 37, 61–73. [Google Scholar] [CrossRef] - Yuan, Y.; Xu, R.; Xu, H.T.; Hong, F.; Xu, F.; Wang, L.J. Nature of the band gap of halide perovskites ABX
_{3}(A = CH_{3}NH_{3}, Cs; B = Sn, Pb; X = Cl, Br, I): First-principles calculations. Chinese Phys. B**2015**, 24, 116302. [Google Scholar] [CrossRef] - Fang, Z.; Yi, Z. First principles study on mixed orthorhombic perovskite CH
_{3}NH_{3}Pb(I_{1-x}Br_{x})_{3}. Chem. Phys. Lett.**2017**, 687, 19–22. [Google Scholar] [CrossRef] - Rahman, N.M.; Adnaan, M.; Adhikary, D.; Islam, M.; Alam, M.K. First-principles calculation of the optoelectronic properties of doped methylammonium lead halide perovskites: A DFT-based study. Comput. Mater. Sci.
**2018**, 150, 439–447. [Google Scholar] [CrossRef] - Kang, Y.; Han, S. Intrinsic Carrier Mobility of Cesium Lead Halide Perovskites. Phys. Rev. Appl.
**2018**, 10, 044013. [Google Scholar] [CrossRef] [Green Version] - Tomanová, K.; Čuba, V.; Brik, M.G.; Mihóková, E.; Martinez Turtos, R.; Lecoq, P.; Auffray, E.; Nikl, M. On the structure, synthesis, and characterization of ultrafast blue-emitting CsPbBr
_{3}nanoplatelets. APL Mater.**2019**, 7, 011104. [Google Scholar] [CrossRef] [Green Version] - Wu, K.; Liang, G.; Shang, Q.; Ren, Y.; Kong, D.; Lian, T. Ultrafast interfacial electron and hole transfer from CsPbBr
_{3}perovskite quantum dots. J. Am. Chem. Soc.**2015**, 137, 12792–12795. [Google Scholar] [CrossRef] - Du, W.; Zhang, S.; Shi, J.; Chen, J.; Wu, Z.; Mi, Y.; Liu, Z.; Li, Y.; Sui, X.; Wang, R.; et al. Strong Exciton-Photon Coupling and Lasing Behavior in All-Inorganic CsPbBr
_{3}Micro/Nanowire Fabry-Pérot Cavity. ACS Photonics**2018**, 5, 2051–2059. [Google Scholar] [CrossRef] - Jin, H.; Im, J.; Freeman, A.J. Topological insulator phase in halide perovskite structures. Phys. Rev. B Condens. Matter Mater. Phys.
**2012**, 86, 121102. [Google Scholar] [CrossRef] [Green Version] - Sutton, R.J.; Filip, M.R.; Haghighirad, A.A.; Sakai, N.; Wenger, B.; Giustino, F.; Snaith, H.J. Cubic or Orthorhombic? Revealing the Crystal Structure of Metastable Black-Phase CsPbI
_{3}by Theory and Experiment. ACS Energy Lett.**2018**, 3, 1787–1794. [Google Scholar] [CrossRef] - Reshak, A.H.; Alahmed, Z.A.; Bila, J. Phase transition in BaThO
_{3}from Pbnm to Ibmm turn the fundamental energy band gap from indirect to direct. J. Alloys Compd.**2019**, 771, 607–613. [Google Scholar] [CrossRef] - Callister, W.D., Jr.; Rethwisch, D.G. Fundamentals of Materials Science and Engineering, 5th ed.; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2018; ISBN 978-1-119-17550-6. [Google Scholar]
- Pitriana, P.; Wungu, T.D.K.; Herman; Hidayat, R. The characteristics of band structures and crystal binding in all-inorganic perovskite APbBr
_{3}studied by the first principle calculations using the Density Functional Theory (DFT) method. Results Phys.**2019**, 15, 102592. [Google Scholar] [CrossRef] - Ambrosch-Draxl, C.; Sofo, J.O. Linear optical properties of solids within the full-potential linearized augmented planewave method. Comput. Phys. Commun.
**2006**, 175, 1–14. [Google Scholar] [CrossRef] [Green Version] - Dar, S.A.; Sharma, R.; Srivastava, V.; Sakalle, U.K. Investigation on the electronic structure, optical, elastic, mechanical, thermodynamic and thermoelectric properties of wide band gap semiconductor double perovskite Ba
_{2}InTaO_{6}. RSC Adv.**2019**, 9, 9522–9532. [Google Scholar] [CrossRef] [Green Version] - Adachi, S. Properties of Semiconductor Alloys: Group-IV, III-V and II-VI Semiconductors; John Wiley & Sons: Hoboken, NJ, USA, 2009; ISBN 9780470744383. [Google Scholar]
- Benchehima, M.; Abid, H.; Sadoun, A.; Chabane Chaouche, A. Optoelectronic properties of aluminum bismuth antimony ternary alloys for optical telecommunication applications: First principles calculation. Comput. Mater. Sci.
**2018**, 155, 224–234. [Google Scholar] [CrossRef] - Amin, B.; Ahmad, I.; Maqbool, M.; Goumri-Said, S.; Ahmad, R. Ab initio study of the bandgap engineering of Al
_{1-x}Ga_{x}N for optoelectronic applications. J. Appl. Phys.**2011**, 109, 023109. [Google Scholar] [CrossRef] [Green Version] - Kushwaha, A.K.; Laref, A.; Laref, S. First-Principles Investigation of Structural, Electronic, Optical, and Magnetic Properties of Ternary Mixed Compound CsTe
_{x}S_{1−x}. J. Electron. Mater.**2019**, 48, 3479–3489. [Google Scholar] [CrossRef]

**Figure 1.**Atomic structures of CsPb(Br

_{1−x}Cl

_{x})

_{3}, with x = 0.00, 0.25, 0.33, 0.50, 0.66, 0.75, and 1.00 for different Cl content (x).

**Figure 2.**Calculated total energy versus volume of (

**a**) CsPbBr

_{3}and (

**b**) CsPbCl

_{3}via (Wu and Cohen generalized gradient approximation (WC–GGA)) potential.

**Figure 3.**(

**a**) Theoretical X-ray diffraction (XRD) patterns of CsPb(Br

_{1−x}Cl

_{x})

_{3}obtained using visualization for electronic and structural analysis (VESTA) software, (

**b**) XRD patterns (2θ = 30°–32.1°), and (

**c**) the peak position versus Cl content (x).

**Figure 5.**Calculated band structures of CsPb(Br

_{1–x}Cl

_{x})

_{3}using the modified Becke−Johnson (mBJ)-GGA potential without/with spin-orbital coupling (SOC).

**Figure 6.**Calculated band structures of CsPb(Br

_{1−x}Cl

_{x})

_{3}using the Perdew–Burke–Ernzerhof (PBE)-GGA potential.

**Figure 7.**Band gaps of CsPb(Br

_{1−x}Cl

_{x})

_{3}using the PBE–GGA and mBJ–GGA potentials with/without SOC. By applying the band gap correction, we get the mixed ratio band gaps of inorganic mixed halide perovskite compared with the experimental results.

**Figure 8.**Effect of Cl concentration on the electron and hole effective masses for CsPb(Br

_{1−x}Cl

_{x})

_{3}perovskites.

**Figure 9.**(

**a**) Bohr diameter a

_{0}(nm) and (

**b**) exciton binding energy E

_{b}(meV) with respect to Cl content (x).

**Figure 10.**Total density of states (TDOS) of CsPb(Br

_{1−x}Cl

_{x})

_{3}calculated using the mBJ–GGA potential.

**Figure 11.**Calculated partial DOS (PDOS) of CsPb(Br

_{1−x}Cl

_{x})

_{3}calculated using the mBJ–GGA potential without SOC.

**Figure 12.**(

**a**) Band structures and (

**b**) PDOS of CsPbBr

_{1.5}Cl

_{1.5}obtained using the mBJ–GGA potential.

**Figure 13.**Calculated electron density in the (001) plane of CsPb(Br

_{1−x}Cl

_{x})

_{3}. (

**a**) x = 0.00, (

**b**) x = 0.25, (

**c**) x = 0.33, (

**d**) x = 0.50, (

**e**) x = 0.66, (

**f**) x = 0.75, and (

**g**) x = 1.00 using the mBJ–GGA potential.

**Figure 14.**Calculated (

**a**) real dielectric function ε

_{1}(ω) and (

**b**) imaginary dielectric function ε

_{2}(ω) of CsPb(Br

_{1−x}Cl

_{x})

_{3}with respect to Cl content (x) using the mBJ–GGA potential.

**Figure 15.**Calculated (

**a**) refraction indices n (ω) and (

**b**) extinction coefficients $\mathrm{k}\text{}\left(\omega \right)\text{}$of CsPb(Br

_{1−x}Cl

_{x})

_{3}with respect to Cl content (x) using the mBJ–GGA potential.

**Figure 16.**Calculated reflectivity spectra R (ω) of CsPb (Br

_{1−x}Cl

_{x})

_{3}using the mBJ–GGA potential.

**Figure 17.**Static refractive index, real dielectric function, and reflectivity at zero frequency versus Cl content (x).

**Figure 18.**(

**a**) Calculated absorption spectra α(ω) and (

**b**) conductivity σ(ω) of CsPb(Br

_{1−x}Cl

_{x})

_{3}with respect to Cl content (x) using the mBJ–GGA potential. Inset: absorption spectra in the range from 2.0 to 3.2 eV.

**Table 1.**Calculated structural parameters; lattice constants a, b, and c (Å); unit cell volume V (Å)

^{3}; bulk modulus B (GPa); and its derivative B′ of CsPb(Br

_{1−x}Cl

_{x})

_{3}perovskite by Wu and Cohen generalized gradient approximation (WC–GGA) potential. mBJ, modified Becke−Johnson; LDA, local density approximation; PBE, Perdew–Burke–Ernzerhof.

CsPb(Br_{1−x}Cl_{x})_{3} | Present Work | Other Calculations (Exp.) | |||||
---|---|---|---|---|---|---|---|

a (Å) | V (Å)^{3} | B (GPa) | B′ | a (Å) | B (GPa) | B′ | |

CsPbBr_{3} | a = 5.874 | 810.703 | 20.7379 | 4.881 | 5.84 (WC–GGA) [72] 5.86 (TB–mBJ) * [73] 5.74 (LDA) [74,75] 6.005 (PBE–GGA) [23] 5.87 (PBEsol) [23] 5.87 (PBE–GGA) [76] 5.77 (LDA) [23] 6.0039 (PBE–GGA) [77] (5.874) [36] (5.85) [69] | 23.5 [72] | 5.0 [72] |

CsPbBr_{2.75}Cl_{0.25} | a = 5.801 c = 5.855 | 807.008 | a = 6.005 c = 5.859 (PBE–GGA) [78] | ||||

CsPbBr_{2}Cl | a = 5.784 c = 5.748 | 774.139 | a = 5.708 c = 6.012 (PBE–GGA) [78] | ||||

CsPbBr_{1.5}Cl_{1.5} | a = 5.739 c = 5.7395 | 756.278 | a = 5.718 c = 5.874 (PBE–GGA) [78] | - | - | ||

CsPbBrCl_{2} | a = 5.695 c = 5.6947 | 738.692 | a = 5.725 c = 6.012 (PBE–GGA) [78] | ||||

CsPbBr_{0.25}Cl_{2.75} | a = 5.672 c = 5.6722 | 730.005 | a = 5.728 c = 5.879 (PBE–GGA) [78] | ||||

CsPbCl_{3} | a = 5.605 | 704.347 | 24.2106 | 5.0142 | 5.56 (WC–GGA) [72] 5.61 (TB–mBJ) [73] 5.73 (PBE–GGA) [79] 5.49 (LDA) [75] 5.743 (PBE–GGA) [80] 5.726 (PBE–GGA) [78] 5.728 (PBE–GGA) [81] 5.618 (PBE–GGA)[82] 5.740 (LDA) [74] 5.603 (PBE–GGA) [55] 5.605 [70,83] 5.61 [55] 5.6228 [30] | 25.8 [72] 22.59 [81] 25.447[82] 26.33 [73] | 5.0 [72] 4.33 [81] 4.4 [82] |

**Table 2.**Calculated E

_{g}(eV) values of CsPb(Br

_{1−x}Cl

_{x})

_{3}perovskite using PBE–GGA, mBJ–GGA, and mBJ–GGA + spin-orbital coupling (SOC) potentials, and mBJ–GGA + SOC(C).

CsPb(Br_{1−x}Cl_{x})_{3} | E_{g} (eV) | ||||
---|---|---|---|---|---|

This Work | Other (Exp.) | ||||

PBE–GGA | mBJ–GGA | mBJ–GGA + SOC | mBJ–GGA + SOC (C) | ||

CsPbBr_{3} | 1.53 | 2.23 | 1.05 | 2.28 | 2.34 (GW) [74] 1.61 (PBE–GGA) [23] 2.36 (nTmBj) [23] 2.228 (KTB–mBJ) * [92] 2.08 (GLLB-SC) ** [93] 2.10 (QE) *** [35] 2.50 (mBJ–GGA) [77] (2.36) [51,94] (2.32) [52] (2.282) [53] (2.35) [95] |

CsPbBr_{2.75}Cl_{0.25} | 1.68 | 2.46 | 1.40 | 2.64 | 1.809 (PBE–GGA) [78] |

CsPbBr_{2}Cl | 1.56 | 2.40 | 1.20 | 2.45 | 1.827 (PBE–GGA) [78] (2.59) [94] |

CsPbBr_{1.5}Cl_{1.5} | 1.69 | 2.51 | 1.41 | 2.67 | 1.859 (PBE–GGA) [78] (2.72) [94] |

CsPbBrCl_{2} | 1.71 | 2.59 | 1.52 | 2.78 | 1.881(PBE–GGA) [78] (2.88) [94] |

CsPbBr_{0.25}Cl_{2.75} | 1.77 | 2.64 | 1.53 | 2.80 | 2.05(PBE–GGA) [78] |

CsPbCl_{3} | 1.93 | 2.90 | 1.69 | 2.97 | 2.20 (PBE–GGA) [78,96] 2.829 (KTB–mBJ) [92] 2.92 (HSE) **** [79] 3.406 (PBE–GGA)[82] 2.88 (GW) [74] 2.74 (TB–mBJ) [73] 2.168 (PBE–GGA) [23] 3.10 (nTmBj) [23] (3.00) [54] (2.97) [55] (3.04) [95](2.98) [94] |

**Table 3.**Calculation of static optical parameters ε

_{1}(0), refractive index n(0), and reflectivity R(0) for CsPb(Br

_{1−x}Cl

_{x})

_{3}compounds.

CsPb(Br_{1−x}Cl_{x})_{3} | mBJ–GGA (others) | ||
---|---|---|---|

ε_{1} (0) | n (0) | R (0)% | |

CsPbBr_{3} | 3.82 (4.30) [104] (4.60) [23] (4.63) [72] | 1.96 Exp. (1.85–2.3) [56] (2.152) [72] | 10.50 (13.4) [72] |

CsPbBr_{2.75}Cl_{0.25} | 3.59 | 1.897 | 9.65 |

CsPbBr_{2}Cl | 3.57 | 1.890 | 9.55 |

CsPbBr_{1.5}Cl_{1.5} | 3.56 | 1.882 | 9.52 |

CsPbBrCl_{2} | 3.55 | 1.880 | 9.36 |

CsPbBr_{0.25}Cl_{2.75} | 3.41 | 1.848 | 8.85 |

CsPbCl_{3} | 3.23 (3.69) [104] (3.00) [81] (4.10) [23] (4.43) [72] | 1.798 (1.739) [81] (2.105) [72] | 8.11 (12.7) [72] (10) [82] |

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**MDPI and ACS Style**

Ghaithan, H.M.; Alahmed, Z.A.; Qaid, S.M.H.; Aldwayyan, A.S.
Structural, Electronic, and Optical Properties of CsPb(Br_{1−x}Cl_{x})_{3} Perovskite: First-Principles Study with PBE–GGA and mBJ–GGA Methods. *Materials* **2020**, *13*, 4944.
https://doi.org/10.3390/ma13214944

**AMA Style**

Ghaithan HM, Alahmed ZA, Qaid SMH, Aldwayyan AS.
Structural, Electronic, and Optical Properties of CsPb(Br_{1−x}Cl_{x})_{3} Perovskite: First-Principles Study with PBE–GGA and mBJ–GGA Methods. *Materials*. 2020; 13(21):4944.
https://doi.org/10.3390/ma13214944

**Chicago/Turabian Style**

Ghaithan, Hamid M., Zeyad. A. Alahmed, Saif M. H. Qaid, and Abdullah S. Aldwayyan.
2020. "Structural, Electronic, and Optical Properties of CsPb(Br_{1−x}Cl_{x})_{3} Perovskite: First-Principles Study with PBE–GGA and mBJ–GGA Methods" *Materials* 13, no. 21: 4944.
https://doi.org/10.3390/ma13214944