#
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

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**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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## Share and Cite

**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