#
Calculation of Mechanical Properties, Electronic Structure and Optical Properties of CsPbX_{3} (X = F, Cl, Br, I)

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}(X = F, Cl, Br, I) to explore its physical and chemical properties, including its mechanical behavior, electronic structure and optical properties. Calculations show that all four materials have good stability, modulus of elasticity, hardness and wear resistance. Additionally, CsPbX

_{3}demonstrates a vertical electron leap and serves as a semiconductor material with direct band gaps of 3.600 eV, 3.111 eV, 2.538 eV and 2.085 eV. In examining its optical properties, we observed that the real and imaginary components of the dielectric function exhibit peaks within the low-energy range. Furthermore, the dielectric function gradually decreases as the photon energy increases. The absorption spectrum reveals that the CsPbX

_{3}material exhibits the highest UV light absorption, and as X changes (with the increase in atomic radius within the halogen group of elements), the light absorption undergoes a red shift, becoming stronger and enhancing light utilization. These properties underscore the material’s potential for application in microelectronic and optoelectronic device production. Moreover, they provide a theoretical reference for future investigations into CsPbX

_{3}materials.

## 1. Introduction

_{3}and named it perovskite [1]. Perovskite is a class of mineral compounds characterized by the general formula ABO

_{3}[2]. Halide perovskites possess a similar crystal structure characterized by the general formula ABX

_{3}. Organic–inorganic halide chalcogenides combine the solution-processable characteristics of organic materials and the excellent optoelectronic properties of inorganic materials with excellent photophysical properties, such as high absorption coefficients, long exciton diffusion distances, high carrier mobility, low exciton binding energies, etc., and the photovoltaic devices constructed with them have a simple preparation process, inexpensive production costs, excellent flexibility and outstanding optoelectronic performance [3]. Their structure typically consists of octahedral coordination, where A and B represent two different-sized cations (usually A = Rb

^{+}or Cs

^{+}, B = Sn

^{+}or Pb

^{+}) and X represents an anion (halides: X = F

^{−}, Cl

^{−}, Br

^{−}and I

^{−}) [4]. CsPbX

_{3}(X = Cl, Br, I) is a chalcogenide optoelectronic material consisting of Cs

^{+}and Pb

^{+}ions and X

^{−}ions (Cl

^{−}, Br

^{−}, I

^{−}), focusing on its electronic structure and luminescence properties [5].

_{2}reduction [9] and aquatic hydrogen [10]. For instance, in 2016, Nam et al. first reported the application of metal halide perovskite materials as catalysts in photocatalysis research [11]. The practical application of organic–inorganic hybrid perovskite materials (such as MAPbX

_{3}or FAPbX

_{3}) is limited due to their structural instability, which arises from the volatile nature of the organic cations. These materials are prone to irreversible decomposition in polar solvents, forming PbX

_{2}(X = I, Br, Cl) precipitates, organic cations (MA/FA) and halide anions. It has been shown that the photothermal stability of the material can be significantly improved if an all-inorganic chemical structure is adopted, such as when an organic cation at the A

^{−}site is replaced by inorganic Cs

^{2+}[12]. A higher photoluminescence quantum yield, excellent stability and narrow-band emission properties are achieved compared to the organic–inorganic hybrid analogues. Inorganic CsPbX

_{3}(X = Cl, Br, I) perovskite materials, being characterized by high photoluminescence efficiency (>90%), a broad absorption range, long electron–hole diffusion length and stable crystal structure, have garnered significant attention in the field of photocatalysis [13,14,15,16]. Sebastian et al. successfully synthesized CsPbBr

_{3}and CsPbCl

_{3}perovskite crystals and analyzed their band gaps through photoluminescence spectroscopy [17]. Heidrich et al. [18,19] carried out extensive studies on the electronic structure and spectra of CsPbCl

_{3}and CsPbBr

_{3}and determined that CsPbI

_{3}, CsPbBr

_{3}, CsPbCl

_{3}and CsPbF

_{3}have lattice constant values of 6.2894 Å, 5.874 Å, 5.605 Å and 4.7748 Å, respectively [20,21,22]. Murtaza and Ahmed used the FP-LAPW method to analyze the structural [23], electronic and optical properties of CsPbM

_{3}(M = Cl, Br, I) and found that the lattice constant increases as the halide ion transitions from Cl to I. The calculated band gap values were 2.55 eV, 2.19 eV and 1.75 eV for CsPbF

_{3}, CsPbCl

_{3}and CsPbBr

_{3}, respectively [24]. However, the vast majority of these studies focused on one aspect only and neglected comprehensive analysis, thus lacking a comprehensive understanding of the properties of CsPbX

_{3}(X = F, Cl, Br, I) and in-depth investigation and comprehensive comparison.

_{3}(X = Cl, Br, I) based on first-principle calculations, providing a theoretical reference for further studies on CsPbX

_{3}materials in the future.

## 2. Results and Discussion

#### 2.1. Mechanical Properties

_{ij}is calculated using the stress–strain method, and the bulk modulus of elasticity (B), Young’s modulus (E), shear modulus (G), Poisson’s ratio (ν) and hardness (H) are calculated with the Voigt–Reuss–Hill (VRH) approximation [25,26]. The calculated equations are as follows (Equations (1)–(11)). Hardness is estimated using Tian’s model [27], and elastic constants are solved via the stress–strain method. The elastic constant is an important parameter in the study of the mechanical and dynamic characteristics of materials and can be used to estimate the hardness and stability of materials.

_{ij}> 0

_{ii}+ C

_{jj}− 2C

_{ij}> 0

_{11}+ C

_{22}+ C

_{33}+ 2(C

_{12}+ C

_{13}+ C

_{23}) > 0

_{ij}. The stability criterion confirms that all three of these systems, CsPbF

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}, satisfy the mechanical stability conditions for orthorhombic crystal structures. And the small difference between the calculated and actual values proves the results’ reliability.

_{12}, C

_{13}and C

_{23}of the elastic constants of CsPbF

_{3}are very small, so its bulk modulus is the smallest among all three materials.

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}are hard materials.

_{3}exhibit relative ductility, while CsPbF

_{3}is brittle.

^{1.137}G

^{0.708}

_{3}has the highest Vickers hardness and wear resistance, while the other three materials are similar.

#### 2.2. Electronic Properties

_{3}structures. PBE underestimates the band gap compared to the experimental data, so we used SCAN to perform a new calculation of the band gap of the material. Although the band gap obtained from PBE calculations underestimates the band gap, the other results obtained from PBE calculations are reliable [29].

_{3}> CsPbCl

_{3}> CsPbBr

_{3}> CsPbI

_{3}, even though their predictions are quite different from each other.

_{3}. The band gaps of CsPbF

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}obtained using the PBE method are 3.222 eV, 2.722 eV, 2.174 eV and 1.816 eV, respectively (Figure 2a–d), while the band gaps calculated using the SCAN method are 3.600 eV, 3.111 eV, 2.538 eV and 2.085 eV, respectively (Figure 2e–h). Furthermore, as can be seen in Figure 2, the minimum value of the conduction band (CB) and the maximum value of the valence band (VB) in all four structures are obtained near the G point, which characterizes the band gap structure as a direct band gap.

_{3}semiconductors, the conduction band is mainly composed of Pb and Cs, while the valence band is mainly occupied by halogen elements (F, Cl, Br, I), but exhibits different forbidden bandwidths due to the halogens, which vary in relation to the dispersion of the halogens at the Fermi energy level near the conduction band; the lower the dispersion, the smaller the forbidden bandwidth, leading to different excitation capabilities of the photogenerated electron–hole pairs.

#### 2.3. Optical Properties

_{1}(ω) and the imaginary part ε

_{2}(ω). These quantities can be determined by analyzing the jump matrix and the dielectric function relationship [34].

_{2}(ω) is expressed as follows [37]:

_{1}(ω) is expressed as follows [38]:

_{1}(ω) and ε

_{2}(ω). The specific formulae are as follows [39,40,41]:

_{1}and imaginary part ε

_{2}curves of the dielectric function of CsPbX

_{3}with energy are shown in Figure 4a,b. The imaginary part of the dielectric function is related to the electronic transition, so the imaginary part of the dielectric function can reflect the strength of the electron-stimulated transition and thus indicate the strength of the optical absorption capacity. The peaks of ε

_{1}and ε

_{2}are distributed in the low-energy region (≤15 eV). In the high-energy region (>15 eV), ε

_{1}tends to 1 and ε

_{2}tends to 0. Currently, CsPbX

_{3}absorbs very few incident photons and has a high transmittance. The values on the vertical axis at zero energy are the static dielectric constants of the materials, which are 2.22, 3.81, 4.97 and 6.95 for CsPbF

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}, respectively (Figure 4a), showing a gradual increase in the static dielectric constant. With the replacement of different halogen elements, the static permittivity function gradually increases, which is due to the band gap value gradually decreasing. Furthermore, electrons require very little energy to jump from the top of the valence band to the bottom of the conduction band and are easily polarized.

_{3}system. The imaginary part of the dielectric function indicates the intensity of electron transitions, thereby reflecting the absorption capacity of light. There are four dielectric peaks between 0 and 15 eV for CsPbF

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}, respectively. As the band gap decreases, the photon energy required for the electron leap decreases, so the dielectric peaks move towards the lower-energy region and the peaks in the dielectric imaginary part move towards the lower-energy region.

_{3}system. The absorption edge of the system gradually extends into the infrared band; i.e., the absorption spectra are all significantly red-shifted, and the magnitude of the edge values follows the order CsPbI

_{3}< CsPbBr

_{3}< CsPbCl

_{3}< CsPbF

_{3}, increasing the absorption range of the system in the infrared. It can be seen from the graph that the absorption coefficient of the system increases with the transformation of the halogen atoms with larger ionic radii, and CsPbI

_{3}has the largest red shift and the strongest light absorption intensity. This shows that by changing the halogen atoms in the CsPbX

_{3}system, it is possible to enhance its light absorption in the infrared region, which provides a theoretical basis for the application of the system in optics.

_{3}system, reflecting the trend of the reflectance as the photon energy changes. The reflectance of CsPbF

_{3}, CsPbCl

_{3}and CsPbBr

_{3}has a maximum peak at photon energies of 14.391 eV, 17.852 eV and 14.637 eV, and the reflectance of CsPbCl

_{3}and CsPbBr

_{3}is greater than 0.25, but the reflectance of CsPbF

_{3}is smaller and less than 0.15. The reflectance of both CsPbCl

_{3}and CsPbBr

_{3}is less than 0.25 at 13.844 eV. Although the trend of the increasing reflectance of CsPbI

_{3}remains the same, its reflectance is at a maximum energy of 3.638 eV. The reflectance of the CsPbX

_{3}system is less than 0.33, so a small portion of the incident light is reflected back. The reflectivity tends to zero as the photon energy increases above the 21.614 eV range.

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}are 1.49, 1.95, 2.22 and 2.63, respectively, and the static refractive index of CsPbI

_{3}is the largest. The corresponding energies are 3.16 eV, 3.41 eV, 2.90 eV and 2.79 eV. Between 3.5 eV and 19 eV, the refractive index of the whole system tends to decrease with increasing photon energy, and the curves almost coincide after 34 eV, after which the refractive coefficient of the system remains almost constant around 0.93. Figure 5c shows the extinction coefficient curve, which has five peaks, decreasing towards zero after 24.14 eV, and remaining constant at almost zero after reaching 37.43 eV. The overall trend of the curve in Figure 5c shows that the curve tends to increase as the ionic radius of the replacement halogen atoms becomes larger, indicating that the change in the system increases the extinction coefficient, indicating that the energy loss increases as the light waves propagate through the absorbing medium, weakening the utilization of light by the material.

_{3}, with a exhibiting the real part of the photoconductivity and b exhibiting the imaginary part of the photoconductivity. Photoconductivity is an important parameter in the study of optoelectronic materials and describes the phenomenon of light-induced changes in the electrical conductivity of a semiconductor. As shown in Figure 6a, the starting energy of the photoconductivity corresponds to the same trend as the change in the band gap size ratio shown in the previous energy band structure. As the atomic radius of element X becomes larger, the conductivity peak moves towards the bottom energy region, with five conductivity peaks appearing in all systems. The peaks of the other three systems (CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}) increase with an increasing atomic radius of X. These changes are caused by changes in the energy band structure and are consistent with the pattern of changes in the imaginary part of the dielectric function. Furthermore, the peak of the imaginary part of the photoconductivity is located along the falling edge of the real part of the photoconductivity, and the valley of the imaginary part of the photoconductivity occurs along the rising edge of the real part of the photoconductivity, with its peak near the positions of 2.91 eV and 15.04 eV (Figure 6b).

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}are at 19.85 eV, 19.33 eV, 21.73 eV and 21.68 eV, respectively, and the energy loss function peaks at 1.56 eV, 3.44 eV, 2.96 eV and 3.05 eV thereafter. After that, the energy loss function spectrum drops sharply to zero. The energy loss function of CsPbX

_{3}shows a sharp increase and then a decrease in the range of 10–25 eV for the photoelectron energy but tends to zero in all other ranges, indicating that there is no excessive energy loss in these ranges.

## 3. Computation Method

_{3}belongs to the Pbnm space group, and its detailed structural parameters are shown in Table 3 below. The optimized crystallographic information file of the four structures of CsPbX

_{3}is provided in the supplementary information. A method based on the generalized gradient approximation (GGA) method proposed by Perdew–Burke–Ernzerhof (PBE) based on first-principle density functional theory is used to solve the exchange–correlation energy generalization [42], which is implemented in the CASTEP code [43]. Geometric optimization is performed using the limited memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) method [44], and core electron interactions are calculated using an ultrasoft pseudopotential [45]. A 2 × 2 × 1 k-point and an energy cut-off of 400 eV were used for geometrical optimization and property calculations. In addition, the energy convergence tolerance during atomic relaxation was set to no more than 1 × 10

^{−5}eV/atom, and the atomic force was limited to less than 0.03 eV/Å with a maximum displacement of 0.001 Å.

## 4. Conclusions

_{3}are calculated and analyzed using a first-principle plane wave pseudopotential approach in the framework of density generalized theory. The results show that CsPbF

_{3}, CsPbCl

_{3}, CsPbBr

_{3}and CsPbI

_{3}are all semiconductor materials with direct band gaps of 3.600 eV, 3.111 eV, 2.538 eV and 2.085 eV, which are close to the experimental values. The bottom of the conduction band is dominated by contributions from Pb and Cs, and the top of the valence band is dominated by contributions from halogenated states. Optical analysis shows that the peaks of both the real and imaginary parts of their dielectric functions occur in the low-energy region and that the dielectric function decreases slowly as the photon energy increases. The absorption spectrum of light shows that the CsPbX

_{3}material absorbs the most UV light, and that with an increase in the radius of the X atom, the light absorption shows a red shift and the light absorption becomes stronger, i.e., the utilization of light increases. These properties establish the usefulness of the material for the fabrication of microelectronic and optoelectronic devices and provide a potential application and a theoretical reference for further research on CsPbX

_{3}materials.

## Supplementary Materials

_{3}. Figure S2: Details of the four structures optimized (CIF). Figure S3: The different bond lengths and bond angles in the four structures. Figure S4: CASTEP Geometry Optimization and Optimization Convergence.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Elastic constants, (

**b**) Poisson’s ratio and B/G, (

**c**) modulus of elasticity and (

**d**) hardness and H/E for CsPbX

_{3}(X = Cl, Br, I). Bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (ν) and hardness (H).

**Figure 2.**(

**a**–

**d**) are the electronic energy band structures calculated using the PBE method, and (

**e**–

**h**) are the electronic energy band structures calculated using the SCAN method.

**Figure 3.**Densities of states and fractional-wave densities of states for (

**a**) CsPbF

_{3}, (

**b**) CsPbCl

_{3}, (

**c**) CsPbBr

_{3}, (

**d**) CsPbI

_{3}.

**Figure 4.**(

**a**) Real and (

**b**) imaginary parts of the dielectric function and (

**c**,

**d**) light absorption intensity (local amplification) of CsPbX

_{3}(X = F, Cl, Br, I).

System | C_{11} | C_{12} | C_{13} | C_{23} | C_{22} | C_{33} | C_{44} | C_{55} | C_{66} |
---|---|---|---|---|---|---|---|---|---|

CsPbF_{3} | 26.944 | 10.646 | 3.018 | 1.472 | 31.857 | 30.189 | 10.787 | 10.463 | 19.287 |

CsPbCl_{3} | 27.868 | 16.536 | 9.232 | 7.208 | 28.541 | 24.298 | 6.298 | 4.122 | 11.598 |

CsPbBr_{3} | 23.542 | 14.357 | 7.679 | 6.120 | 24.981 | 18.857 | 5.313 | 3.298 | 10.711 |

CsPbI_{3} | 16.193 | 10.061 | 4.949 | 5.355 | 18.191 | 15.175 | 4.266 | 2.126 | 7.425 |

**Table 2.**Mechanical properties of the material: bulk modulus (GPa), shear modulus (GPa), Young’s modulus (GPa), Poisson’s ratio(GPa) and hardness(GPa).

System | Bulk Modulus (B) | Shear Modulus (G) | Young’s Modulus (E) | Poisson’s Ratio (ν) | Hardness (H) | B/G | H/E |
---|---|---|---|---|---|---|---|

CsPbF_{3} | 13.170 | 12.542 | 28.560 | 0.138 | 5.215 | 1.050 | 0.183 |

CsPbCl_{3} | 16.008 | 7.064 | 18.474 | 0.307 | 1.448 | 2.266 | 0.078 |

CsPbBr_{3} | 13.361 | 5.954 | 15.553 | 0.306 | 1.298 | 2.244 | 0.083 |

CsPbI_{3} | 9.870 | 4.295 | 11.253 | 0.309 | 1.002 | 2.298 | 0.089 |

**Table 3.**Crystal structure information and band gap of CsPbX

_{3}(X = F, Cl, Br, I). I and II in the Pb-X and Pb-X-Pb columns of the table represent the different lengths, and different angles, of the bonds formed by the Pb atoms with the X atoms (specific types can be viewed in the Supplementary Material for picture information).

System | Lattice Constant/Å | Unit Cell Volume V/Å ^{3} | Band Gap/eV | Pb-X Bond Length/Å | Pb-X-Pb Bond Angle/° | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

a | b | c | PBE | SCAN | Exp. | I | II | I | II | ||

CsPbF_{3} | 6.950 | 7.030 | 10.095 | 493.149 | 3.222 | 3.600 | 3.68 [30] | 2.494/2.493 | 2.550 | 164.758 | 163.341 |

CsPbCl_{3} | 8.030 | 8.112 | 11.389 | 741.918 | 2.722 | 3.111 | 2.90 [31] | 2.918/2.915 | 2.951 | 156.167 | 149.494 |

CsPbBr_{3} | 8.388 | 8.504 | 11.869 | 846.726 | 2.174 | 2.538 | 2.30 [32] | 3.060/3.056 | 3.097 | 155.108 | 146.726 |

CsPbI_{3} | 8.894 | 9.087 | 12.587 | 1017.39 | 1.816 | 2.085 | ~1.70 [33] | 3.273/3.271 | 3.307 | 152.672 | 144.151 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Fang, C.; Lin, S.; Liu, G.; Zhang, B.; Shi, H.; Dong, N.; Yang, N.; Zhang, F.; Guo, X.;
et al. Calculation of Mechanical Properties, Electronic Structure and Optical Properties of CsPbX_{3} (X = F, Cl, Br, I). *Molecules* **2023**, *28*, 7643.
https://doi.org/10.3390/molecules28227643

**AMA Style**

Liu Y, Fang C, Lin S, Liu G, Zhang B, Shi H, Dong N, Yang N, Zhang F, Guo X,
et al. Calculation of Mechanical Properties, Electronic Structure and Optical Properties of CsPbX_{3} (X = F, Cl, Br, I). *Molecules*. 2023; 28(22):7643.
https://doi.org/10.3390/molecules28227643

**Chicago/Turabian Style**

Liu, Yang, Canxiang Fang, Shihe Lin, Gaihui Liu, Bohang Zhang, Huihui Shi, Nan Dong, Nengxun Yang, Fuchun Zhang, Xiang Guo,
and et al. 2023. "Calculation of Mechanical Properties, Electronic Structure and Optical Properties of CsPbX_{3} (X = F, Cl, Br, I)" *Molecules* 28, no. 22: 7643.
https://doi.org/10.3390/molecules28227643