#
Electronic, Optical, Thermoelectric and Elastic Properties of Rb_{x}Cs_{1−x}PbBr_{3} Perovskite

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{3}, where A is an inorganic cation, M is a metal cation, and X is a halide anion, have attracted attention for optoelectronics applications due to their better optical and electronic properties, and stability, under a moist and elevated temperature environment. In this contribution, the electronic, optical, thermoelectric, and elastic properties of cesium lead bromide, CsPbBr

_{3}, and Rb-doped CsPbBr

_{3}, were evaluated using the density functional theory (DFT). The generalized gradient approximation (GGA) in the scheme of Perdew, Burke, and Ernzerhof (PBE) was employed for the exchange–correlation potential. The calculated value of the lattice parameter is in agreement with the available experimental and theoretical results. According to the electronic property results, as the doping content increases, so does the energy bandgap, which decreases after doping 0.75. These compounds undergo a direct band gap and present an energies gap values of about 1.70 eV (x = 0), 3.76 eV (x = 0.75), and 1.71 eV (x = 1). The optical properties, such as the real and imaginary parts of the dielectric function, the absorption coefficient, optical conductivity, refractive index, and extinction coefficient, were studied. The thermoelectric results show that after raising the temperature to 800 K, the thermal and electrical conductivities of the compound RbxCs

_{1−x}PbBr

_{3}increases (x = 0, 0.25, 0.50 and 1). Rb

_{0.75}Cs

_{0.25}PbBr

_{3}(x = 0.75), which has a large band gap, can work well for applications in the ultraviolet region of the spectrum, such as UV detectors, are potential candidates for solar cells; whereas, CsPbBr

_{3}(x = 0) and RbPbBr

_{3}(x = 1), have a narrow and direct band gap and outstanding absorption power in the visible ultraviolet energy range.

## 1. Introduction

_{3}, where A is an organic or inorganic cation (such as ion cesium (Cs

^{+}), ion methyl ammonium (MA), or ion formamidinium (FA)), M is a metal cation (such as Pb

^{2+}or Sn

^{2+}), and X is a halide anion (such as I

^{−}, Br

^{−}, and Cl

^{−}), have attracted much attention in the past few years due to their potential applications as solar-cell absorbers [1,2,3], light-emitting diodes [4,5], photodetectors [6,7,8,9,10,11], and lasers [12]. They possess numerous important technological applications due to their remarkable optical [13,14], electronic [15], and ferroelectric properties [16,17]. Organic–inorganic hybrid perovskite solar cells (PSC) are still adversely affected by poor stability, which reduces their practical applications, as exemplified by [18,19]. These hybrid perovskites degrade because of the considerable impact that moisture, heat, and UV light have on their organic phase.

_{3}have attracted great attention for their usefulness in perovskite solar cell PSC stability improvement. Cesium-based inorganic perovskites have gained popularity due to their significantly increased thermal stability. Formamidinium- or methylammonium-based perovskite films degrade quickly above 200 °C, but the inorganic CsPbI

_{3}and CsPbBr

_{3}perovskite absorbers can maintain their original composition and crystal structure under high temperatures of up to 400 °C, effectively preventing the decomposition of organic groups and further enhancing the performance and stability of the device. In addition, compared to organic–inorganic hybrid perovskite materials, inorganic perovskite materials exhibit superior photoelectric properties, such as high carrier mobility and a long carrier lifetime. When formed from cold-pressed pellets of annealed CsSnI

_{3}polycrystalline material, cesium-based inorganic perovskites have been shown to have a high hole mobility (520 cm

^{2}V/s) and electron mobility (530 cm

^{2}V/s), whereas CsPbBr

_{3}halide single-crystals could achieve an electron lifetime of 2.5 s and an estimated electron mobility of 1000 cm

^{2}V/s [20].

^{+}and Na

^{+}) can enhance perovskite film quality and perovskite solar cells photovoltaic performance. The poor carrier transport properties of inorganic perovskite materials severely restrict the development of the corresponding perovskite solar cell solar performance. Nam et al. partially substituted out Cs

^{+}for K

^{+}, which reduced the PbX

_{6}octahedron volume and increased phase stability [31].

_{3}perovskites that have been created and can be utilized for perovskite solar cells, including CsPbBr

_{3}and CsPbCl

_{3}[32,33,34,35,36]. Additionally, all-inorganic perovskite solar cells, such as CsSnI

_{3}[37] and CsPbI

_{3}[38], have been used. Trots et al. [39] investigated the RbPbI

_{3}and CsPbI

_{3}compounds by the means of synchrotron powder diffraction experiments and Rietveld refinement technique. They have shown that both compounds crystallize in orthorhombic P

_{nma}symmetry, revealing almost the same relative change of the lattice parameters upon heating, with an expansion isotropically close to 600 K. Moreover, they have observed that CsPbI

_{3}undergoes first-order reversible phase transformation, whereas no transitions in RbPbI

_{3}were detected. Zhao et al. [40] incorporated the Rb

^{+}ions into the lattice of CsPbCl

_{3}quantum dots by partially substituting the sites of Cs

^{+}ions by the modified hot injection method. They observed that the high photoluminescence yields of CsPbCl

_{3}were improved from 5.7% to 13% with Rb

^{+}doping. It was observed that the emission and absorption peaks of CsPbCl

_{3}quantum dots shifted to the shorter wavelength, and with the increase of Rb

^{+}doping concentration, the lifetime of CsPbCl

_{3}quantum dots was prolonged.

_{3}(X = Br or Cl) perovskites produced with a single crystalline nature have been demonstrated to be desirable for high-resolution detection at room temperature (RT) [41]. Further, it has been reported that the bulk-recrystallized CsPbBr

_{3}emits bright green radiation at room temperature and provides a greater space for the free carriers, which lowers the recombination rates and, consequently, the poor quantum yield [42]. Fatty acids have been shown to inhibit the formation of CsPbBr

_{3}nanocrystals, providing a novel technique to adjust the visible optical properties [43]. The obvious function of CsPbBr

_{3}for optical devices is also covered in numerous additional experimental publications that are readily available [44,45]. Babu et al. [13] calculated the optical, electronic, structural and elastic properties of CsCaCl

_{3}using the full potential linearized augmented plane wave method in the density functional theory. They found that this compound has an indirect energy band gap with a mixed ionic–covalent bonding, optically isotropic and structurally anisotropic property. The values of the band gaps found with different methods are 5.29, 5.35, 5.43, and 6.93 eV using LDA, GGA-PBE, GGA-WC, and mBJ pseudopotentials, respectively. Chang and Park [15] explored the electronic and structural properties of an inorganic perovskite, CsPbX

_{3}(where X = Cl, Br and I), and the lead-halide-based inorganic–organic (CH

_{3}NH

_{3})PbX

_{3}perovskites, using the first-principles calculations within the local density approximation. They found that the lattice constants for the cubic structure of CsPbX

_{3}were smaller than the corresponding values for (CH

_{3}NH

_{3})PbX

_{3}; however, the electronic structures of both kinds of perovskites were found to be similar. Murtaza et al. [46] studied the optical, electronic and structural properties of cubic CsPbX

_{3}(X = Cl, Br and I) using DFT calculations. They found that all of these compounds are direct, with a wide bandgap located at the R-symmetry point, which decreases from Cl to I. The refractive index, reflectivity and zero frequency limits of dielectric function increase with the decrease in bandgap (from Cl to Br to I), while the absorption coefficient and maximum optical conductivity decrease. Duong et al. [47] have demonstrated a novel multiplication method with methylammonium (MA), formamidinium (FA), Cesium (Cs) and Rb, to achieve high efficiency 1.73 eV bandgap perovskite cells, with negligible hysteresis. Mahmood et al. [48] investigated the thermoelectric, optical and mechanical properties of CsPbX

_{3}(X = F, Cl, Br) using DFT calculations. They found that the thermal (k) and electrical (σ) conductivities increase with the increasing of temperature, and the ratio $\frac{k}{\sigma}$ remains at a minimum. When the mechanical and thermodynamic stabilities decrease from CsPbF

_{3}to CsPbBr

_{3}, the structural stability increases.

_{3}perovskites (with A = K, Li, Na or Cs cations), made via a self-organization process approach at room temperature, were experimentally explored by Dimesso et al. They discovered that the A cation size has a small impact on how these APbI

_{3}perovskites’ bandgap energies change. Rb and K atoms have similar atomic radii to Cs, thus the correlation effect may be minimal [49]. The bandgap energy for CsPbBr

_{3}with a cubic crystal structure was calculated by Qian et al., using the density functional theory (DFT) approach. They determined that this bandgap energy is 1.75 eV [50]. The anion electronegativity is a significant additional consequence of anion exchange. The calculations for CsPbX

_{3}have also been done by Castelli et al., where X is changed for every halide group [51]. Their calculations revealed that the bandgap energy increased as the electronegativity of the anions increased. However, it appeared that the lattice constant, rather than the electronegativity, had a greater impact on the bandgap energy in the case of organometal perovskites. This explains why a perovskite with an formamidinium cation has a higher bandgap energy than one with a methyl ammonium cation. It is crucial to look into the role of the A cations and the X anions in the formation of the electronic structure of APbX

_{3}perovskites, particularly the valence band and conduction band, as well as crystal binding properties, which are in charge of the processes of light absorption and photo generation of charge carriers.

_{3}and CsPbBr

_{3}were predicted using the ionic radii of the respective ions, the structural, electrical, thermodynamic, and optical aspects of these compounds were experimentally examined [39,54]. Using the first-principles pseudopotential method with a local density approximation and an empirical tight binding scheme, the structural and electrical characteristics of these compounds were also computed [15].

_{3}(CPB) and Rb-doped CsPbBr

_{3}, using density functional theory and the Boltzmann Transport Equation (BTE) simulations. The differences between a pure CsPbBr

_{3}and doped CPB (Rb

_{x}Cs

_{1−x}PbBr

_{3}), as well as the influence of Rb doping on these properties, are also discussed.

## 2. Results and Discussion

#### 2.1. Structural Properties

_{3}-type compounds display several phases at various temperatures, although at high temperatures, they all take on a cubic perovskite structure, where a three-dimensional framework of MX

_{6}octahedrons with shared corners is provided.

_{3}compound has a space group ${\mathrm{P}}_{\mathrm{m}\overline{3}\mathrm{m}}$ (221) and lattice parameter a = 5.605 Å.

#### 2.2. Elastic and Electronic Properties

_{11}, B

_{12}, and B

_{44}.

_{3}, and Rb-doped CsPbBr

_{3}, including density of states and band structures, are calculated after the optimization of the lattice parameters. Table 2 lists the values of the elastic constants calculated via DFT calculations. Figure 2 shows the calculated electronic band structures of CsPbBr

_{3}and Rb-doped CsPbBr

_{3}along the higher symmetry directions G, R, X, and M. From the investigation results of CsPbBr

_{3}, Castelli et al. [51] reported that the bandgap energy for this perovskite is 1.63 eV, while Qian et al. [55] reported 1.75 eV. The present calculation results for CsPbBr

_{3}(Figure 2a), which was 1.70 eV, gave a closer value to their computational results. Experimentally, Kulbak et al. [56] reported 2.32 eV regarding the bandgap energy of CsPbBr

_{3}, whereas Stoumpos et al. [45] reported 2.25 eV. On the other hand, the band structures of Rb

_{x}Cs

_{1−x}PbBr

_{3}(x = 0.75 and 1) were calculated and shown in Figure 2b,c, respectively. It is clearly seen that these perovskites exhibited a direct band gap (Figure 2a–c), and achieved around 1.70, 3.76, and 1.71 eV, respectively, upon an increase in Rb content. Distinctions regarding the band gap energies are attributed to the atomic level and size. Furthermore, the gap energy observed in Figure 2c is lowered compared to the previous figures after an increase in the Rb-doping (x = 1), due to several mechanisms, such as the size of Rb. The results for CsPbBr

_{3}(Figure 2a) are in good agreement with the experimental values [45,56,57]. The increase in the energy band gap, followed by a decrease as the doping content increases, is due to either octahedron tilting or a decrease in the overlap of the electron wave function, due to crystal structure contraction and distortion caused by the doping rubidium (Rb) atom [58,59]. To understand the electronic band gap nature, the densities of states (DOSs) of Rb

_{x}Cs

_{1−x}PbBr

_{3}(x = 0, 0.75 and 1) were calculated and displayed in Figure 3. As can be appreciated, the valence bands in CsPbBr

_{3}(x = 0) and RbPbBr3 (x = 1) are mostly composed of Brs, Brp, Pbs, Pbp, and Pbd orbitals, with a small contribution from Css, Csp, and Csd states. The conduction bands in both systems are mainly dominated by Pbp and Css orbitals, with small contributions of Brd and Brp. While in Rb

_{0.75}Cs

_{0.25}PbBr

_{3}(x = 0.75), the valence band consists mainly of the orbital contributions Br-s, Pb-s, and Pb-d. The conduction band consists of the Pbp, Brp, and Brd orbitals, with small Css, Csp, and Csd states. It has been noted that the p orbitals of Pb and Br in the conduction band maximum have an effect on increasing the band gap in the 0.75 doping of Rb.

#### 2.3. Optical Properties

_{3}and RbPbBr

_{3}were theoretically studied. At a lower energy expression of complex, the dielectric function is:

_{3}and RbPbBr

_{3}had similar features: the critical points (onset) in the spectra of ${\epsilon}_{2}\left(\omega \right)$ were found at 1.66 eV for CsPbBr

_{3}and RbPbBr

_{3}. These points are closely related to the band gap 1.70 eV for CsPbBr

_{3}and RbPbBr

_{3}. Different characteristic peaks, beyond the critical points, could be identified by the density of states (Figure 3). The first peaks were due to the transition of electrons from Br

_{p}states of the VB to the Pb

_{p}states in the CB. The other peaks originated because of the electronic transition from Br

_{p}states of VB to the unoccupied Cs

_{(s;d)}and Rb

_{(s;d)}states, and its mixed states with Pb

_{p}states in CB. Interestingly, similar features were found in the spectra (Figure 4e) of the extinction coefficients,$k\left(\omega \right)$.

_{3}and RbPbBr

_{3}was 3.5. The ${\epsilon}_{1}\left(\omega \right)$, of CsPbBr

_{3}and RbPbBr

_{3}started to increase from the zero-frequency limit, reached its maximum value, then decreased, and in certain energy ranges, went below zero. The optical conductivity spectra, $\sigma \left(\omega \right)$ presented in Figure 4c, showed that the optical conductance started at around 1.52 and 1.58 eV for CsPbBr

_{3}and RbPbBr

_{3}, respectively. Beyond these points, $\sigma \left(\omega \right)$ reached its maxima and then, again, decreased gradually. These compounds had a similar highest $\sigma \left(\omega \right)$. Similar features were observed regarding the absorption coefficients $\alpha \left(\omega \right)$ (Figure 4d) in the range 0–6 eV, but the highest peaks were observed in the absorption range 6–9 eV of $\alpha \left(\omega \right)$. Furthermore, the absorption range 2–8 eV showed the usefulness of CsPbBr

_{3}and RbPbBr

_{3}for various optical and optoelectronic devices working in this range. For an optical material to be used in optical devices, such photonic crystals, waveguides, solar cells, and detectors, it is crucial to understand the refractive index it has. The variation in the refractive indexes (n) for CsPbBr

_{3}and RbPbBr

_{3}, as a function of incident photon energy, is shown in Figure 4f. The most important quantity in the spectra is the zero-frequency limit n(0), and its value is 2 for both CsPbBr

_{3}and RbPbBr

_{3}. The $n\left(\omega \right)$ for these compounds increased gradually from the zero-frequency limit, reaching its maximal value, before decreasing. The theoretical analysis of CsPbBr

_{3}′s optical characteristics were equivalent to experimental analysis [63,64,65].

#### 2.4. Thermoelectric Properties

_{3}and Rb-doped CsPbBr

_{3}compounds, the thermal $\frac{k}{\tau}$ and electrical $\frac{\sigma}{\tau}$ conductivities were calculated in the temperature range of 400–800 K, as displayed in Figure 5. It was observed that the electrical and thermal conductivities increased with increasing temperature until 800 K for pure and Rb-doped CPB. The decreasing slope of the electrical and thermal conductivity curves corresponding to Rb

_{0.25}Cs

_{0.75}PbBr

_{3}and Rb

_{0.5}Cs

_{0.5}PbBr

_{3}could be related to the increase of the band gap at T = 800 K, hence, the electrical conductivities at this temperature were 0.25 × 10

^{17}, 1.9 × 10

^{17}, 3.37 × 10

^{17}, and 3.75 × 10

^{17}Ω

^{−1}m

^{−1}s

^{−1}for Rb

_{0.25}Cs

_{0.75}PbBr

_{3}, Rb

_{0.5}Cs

_{0.5}PbBr

_{3}, RbPbBr

_{3}, and CsPbBr

_{3}, respectively. At T = 800 K the thermal conductivities were 1.5 × 10

^{13}, 2.4 × 10

^{13}, 4.4 × 10

^{13}, and 9 × 10

^{13}WK

^{−1}m

^{−1}s

^{−1}for Rb

_{0.25}Cs

_{0.75}PbBr

_{3}, Rb

_{0.5}Cs

_{0.5}PbBr

_{3}, RbPbBr

_{3}, and CsPbBr

_{3}, respectively. Our results are in good agreement with the experimental and theoretical studies reported in by [48,67,68,69], notably electrical conductivity. The increasing slope of the electrical conductivity curves from Rb

_{0.25}Cs

_{0.75}PbBr

_{3}to RbPbBr

_{3}is justified by the variation in the size of the atomic by effect of doping, which varies the free charge carriers.

## 3. Materials and Methods

_{3}and Rb-doped CsPbBr

_{3}perovskite were studied using DFT calculations, implemented in the ABINIT software package [73,74], with generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof function, proposed in [75], using the plane wave pseudo-potential formalism, in order to obtain the response function calculations [76,77]. An energy cut-off of 45 Ha was used for the plane wave expansion, which are well converged. The Monkhorst Pack Mesh scheme [78] k-point grid sampling was set at 5 × 5 × 5, to perform the irreducible Brillouin zone integrations. We use a starting point for CsPbBr

_{3}according to the reported data in the literature [79]. The thermoelectric properties were calculated using BoltzTraP code [80].

## 4. Conclusions

_{3}and Rb

_{x}Cs

_{1−x}PbBr

_{3}(x = 0, 0.25, 0.50, 0.75, and 1) was carried out, using the density functional theory within the generalized gradient approximation and the Boltzmann transport equation simulations. The optical properties, such as dielectric function, optical conductivity, absorption coefficient, refractive index, and extinction coefficient, were studied in the energy range of 0–10 eV. The calculated band gap energy agrees well with the available theoretical and experimental values, and it increased then decreased as the Rb doping content increased. Our calculations revealed that Rb

_{0.75}Cs

_{0.25}PbBr

_{3}is a wide band gap material, which indicates that it is a better candidate for high-frequency UV device applications.

_{3}(x = 0) and RbPbBr

_{3}(x = 1), which have excellent absorption powers in the visible ultraviolet energy range and a short and direct band gap, could be used in solar cells.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

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**Figure 2.**Band structures of CsPbBr

_{3}and Rb

_{x}Cs

_{1−x}PbBr

_{3}(x = 0 (

**a**), 0.75 (

**b**), 1 (

**c**)). Valence bands (blue lines) and conduction bands (red lines).

**Figure 3.**Partial density of states (PDOSs) of RbxCs

_{1−x}PbBr

_{3}(x = 0, 0.75 and 1). (

**a**) Br

_{s}orbitals; (

**b**) Br

_{p}orbitals; (

**c**) Br

_{d}orbitals; (

**d**) Cs

_{s}orbitals; (

**e**) Cs

_{p}orbitals; (

**f**) Cs

_{d}orbitals; (

**g**) Pb

_{s}orbitals; (

**h**) Pb

_{p}orbitals; (

**i**) Pb

_{d}orbitals vs. energy.

**Figure 4.**Optical properties of CsPbBr

_{3}and RbPbBr

_{3}: (

**a**,

**b**) Dielectric function (${\epsilon}_{r}\left(\omega \right)$ and ${\epsilon}_{i}\left(\omega \right)$); (

**c**) optical conductivity $\sigma \left(\omega \right)$; (

**d**) absorption coefficient $\alpha \left(\omega \right)$; (

**e**) extinction coefficient $k\left(\omega \right)$; (

**f**) refaction index $n\left(\omega \right)$.

**Figure 5.**Thermoelectric properties of Rb

_{x}Cs

_{1−x}PbBr

_{3}(x = 0, 0.25, 0.5, 1). (

**a**) Electrical conductivity ($\frac{\sigma}{\tau}$) vs. temperature; (

**b**) thermal conductivity ($\frac{k}{\tau}$) vs. temperature.

Present Work a (Å) | Other Works a (Å) | Experimental a (Å) |
---|---|---|

5.93 | 5.74 [15] 5.84 [29] 5.94 [60] 5.99 [58,60] | 5.87 [60] 5.90 [57] |

**Table 2.**Elastic properties of CsPbBr

_{3}and gap energies of Rb

_{x}Cs

_{1−x}PbBr

_{3}(x = 0, 0.75 and 1).

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**MDPI and ACS Style**

Ouaaka, E.; Aazza, M.; Bouymajane, A.; Cacciola, F.
Electronic, Optical, Thermoelectric and Elastic Properties of Rb_{x}Cs_{1−x}PbBr_{3} Perovskite. *Molecules* **2023**, *28*, 2880.
https://doi.org/10.3390/molecules28072880

**AMA Style**

Ouaaka E, Aazza M, Bouymajane A, Cacciola F.
Electronic, Optical, Thermoelectric and Elastic Properties of Rb_{x}Cs_{1−x}PbBr_{3} Perovskite. *Molecules*. 2023; 28(7):2880.
https://doi.org/10.3390/molecules28072880

**Chicago/Turabian Style**

Ouaaka, Elmustafa, Mustapha Aazza, Aziz Bouymajane, and Francesco Cacciola.
2023. "Electronic, Optical, Thermoelectric and Elastic Properties of Rb_{x}Cs_{1−x}PbBr_{3} Perovskite" *Molecules* 28, no. 7: 2880.
https://doi.org/10.3390/molecules28072880