Electronic, Optical, Thermoelectric and Elastic Properties of RbxCs1−xPbBr3 Perovskite

Inorganic halide perovskites of the type AMX3, 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, CsPbBr3, and Rb-doped CsPbBr3, 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 RbxCs1−xPbBr3 increases (x = 0, 0.25, 0.50 and 1). Rb0.75Cs0.25PbBr3 (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, CsPbBr3 (x = 0) and RbPbBr3 (x = 1), have a narrow and direct band gap and outstanding absorption power in the visible ultraviolet energy range.


Introduction
Material scientists have discovered that energy collection from low-cost sources utilizing the most effective methods has gained significant importance. As a result, it is necessary to define the materials' fundamental characteristics, in order to understand how they function in real-world working devices. Solar energy and unused heat can both be harvested with a great potential. Therefore, the goal of fundamental material research is to investigate innovative materials with optimum optical and thermoelectric properties. Generally, halide perovskite compounds with the general formula AMX 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 quantum dots shifted to the shorter wavelength, and with the increase of Rb + doping concentration, the lifetime of CsPbCl 3 quantum dots was prolonged.
Due to the suppressed luminescence from the deep-level defects, CsPbX 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 k σ remains at a minimum. When the mechanical and thermodynamic stabilities decrease from CsPbF 3 to CsPbBr 3 , the structural stability increases.
All-inorganic lead APbI 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.
In the cubic perovskite structure with space group P m3m , cesium-lead halides have been observed experimentally [52,53]. While the lattice constants of CsPbCl 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].
The aforementioned discussion makes it clear that there is only a limited amount of theoretical research on the optical, elastic, thermoelectric, and electronic properties of Rb-doped cesium lead bromide compounds. However, to our knowledge, no research has been reported on the optoelectronic and thermoelectric properties of this perovskite by density functional theory (DFT).
The aim of the present work was to investigate the electronic, optical, elastic and thermoelectric properties of CsPbBr 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.

Structural Properties
According to earlier research, AMX 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.
The cubic structure phase of the perovskite CsPbBr 3 compound has a space group P m3m (221) and lattice parameter a = 5.605 Å. Figure 1 shows the crystal structure of CsPbBr 3 . The structure of the cubic CsPbBr 3 compound was optimized. The optimization results for the lattice parameter are shown in Table 1 and are proved to be in reasonable agreement with experimental and theoretical values.
been observed experimentally [52,53]. While the lattice constants of CsPbCl3 and C were predicted using the ionic radii of the respective ions, the structural, electrica modynamic, and optical aspects of these compounds were experimentally exa [39,54]. Using the first-principles pseudopotential method with a local density ap mation and an empirical tight binding scheme, the structural and electrical characte of these compounds were also computed [15].
The aforementioned discussion makes it clear that there is only a limited amo theoretical research on the optical, elastic, thermoelectric, and electronic properties doped cesium lead bromide compounds. However, to our knowledge, no resear been reported on the optoelectronic and thermoelectric properties of this perovsk density functional theory (DFT).
The aim of the present work was to investigate the electronic, optical, elast thermoelectric properties of CsPbBr3 (CPB) and Rb-doped CsPbBr3, using density tional theory and the Boltzmann Transport Equation (BTE) simulations. The diffe between a pure CsPbBr3 and doped CPB (RbxCs1-xPbBr3), as well as the influence doping on these properties, are also discussed.

Structural Properties
According to earlier research, AMX3-type compounds display several phases ious temperatures, although at high temperatures, they all take on a cubic pero structure, where a three-dimensional framework of MX6 octahedrons with shared c is provided.
The cubic structure phase of the perovskite CsPbBr3 compound has a space P (221) and lattice parameter a = 5.605 Å. Figure 1 shows the crystal structure of CsPbBr3. The structure of the cubic C compound was optimized. The optimization results for the lattice parameter are sho Table 1 and are proved to be in reasonable agreement with experimental and theo values.

Elastic and Electronic Properties
For describing the mechanical properties of materials, the elastic constants are tial and basic. The elastic constants are significant factors that describe how macro stress is responded to. In addition to defining how a material is deformed under and subsequently recovered and returned to its original shape after tension is rem the elastic constants of solids also serve as a link between the mechanical and dyn behavior of crystals. The elastic constants are significant material characteristics th provide vital details about a material's structural stability, the nature of its atomic b and its anisotropic properties. There are three distinct elastic constants for a cubic s B11, B12, and B44.

Elastic and Electronic Properties
For describing the mechanical properties of materials, the elastic constants are essential and basic. The elastic constants are significant factors that describe how macroscopic stress is responded to. In addition to defining how a material is deformed under stress and subsequently recovered and returned to its original shape after tension is removed, the elastic constants of solids also serve as a link between the mechanical and dynamical behavior of crystals. The elastic constants are significant material characteristics that can provide vital details about a material's structural stability, the nature of its atomic bonds, and its anisotropic properties. There are three distinct elastic constants for a cubic system: B 11 , B 12 , and B 44 .
The state and behavior of the electrons in the material are completely described by a collection of characteristics and representations known as the electronic properties. Such a representation is, for instance, the electronic band structure, which characterizes the state of the electrons in terms of their energy, E, and momentum, k. The electric and optical characteristics, which define how a material reacts to electromagnetic radiation, are both closely related to the electronic properties. Examples of these are electrical conductivity and dielectric response.
The elastic and electronic properties of CsPbBr 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] [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

Optical Properties
Using solar cells and other optoelectronic devices, it is possible to directly convert the impinging photons into electricity. This capability has motivated scientists to look for materials with higher energy conversion efficiency. Direct inter-band transitions in direct band gap semiconductors are crucial, because indirect band gap semiconductors with intra-band transitions cause heating from phonons. This is because the optical properties of a typical semiconductor depend on the band gap, making transition or recombination rates crucial.
From the complex dielectric function, the optical properties of the halide perovskite materials CsPbBr 3 and RbPbBr 3 were theoretically studied. At a lower energy expression of complex, the dielectric function is: where ε 1 (ω) and ε 2 (ω) are the real and imaginary part of the dielectric function, respectively. The real and imaginary part of the dielectric tensor can be estimated using the Kramer-Kronig relation: The absorption coefficient α can be expressed as a function of the extinction coefficient k: The imaginary part of the complex dielectric function, ε 2 (ω), is related to the band structure of the material and describes its absorption behavior. From Figure 4a,b, the spectra of ε 2 (ω) for CsPbBr 3 and RbPbBr 3 had similar features: the critical points (onset) in the spectra of ε 2 (ω) 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(ω).
The real part of the complex dielectric function, ε 1 (ω), is shown in Figure 4a,b. The most important quantity in the spectra is the zero-frequency limit ε 1 (0), which is the electronic part part of the static dielectric constant. The value ε 1 (0), for CsPbBr 3 and RbPbBr 3 was 3.5. The ε 1 (ω), of CsPbBr 3 and RbPbBr 3 started to increase from the zerofrequency limit, reached its maximum value, then decreased, and in certain energy ranges, went below zero. The optical conductivity spectra, σ(ω) 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, σ(ω) reached its maxima and then, again, decreased gradually. These compounds had a similar highest σ(ω). Similar features were observed regarding the absorption coefficients α(ω) (Figure 4d) in the range 0-6 eV, but the highest peaks were observed in the absorption range 6-9 eV of α(ω). 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(ω) 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].

Thermoelectric Properties
The exponential growth of technology has led to enormous energy waste as a result of rising energy demands. Researchers have also been forced to create unique systems that can recycle waste heat into electrical energy due to the lower capacity of the available energy sources. One of the finest options is a thermoelectric generator, which can transform temperature gradients (phonons) directly into potential differences. The computed electrical conductivity measures the free carrier motion that results from temperature gradients that increase the carriers' kinetic energy. A higher electrical conductivity is required to realize the commercial uses of thermoelectric devices, since it reduces the joule heating effect.
By comparing the ratio of heat efflux per area per unit time to the temperature gradient, it is possible to assess the flow of thermal energy. The two categories of thermal conductivity are electronic and phononic. Due to the importance of intra-band transitions in metals, as compared to semiconductors, lattice vibrations (phonons) have a greater influence in metals than they do in semiconductors. Additionally, whereas the phonon energy has little significance for direct bandgap semiconductors, it is significant for indirect bandgap semiconductors. In order to increase the efficiency of thermoelectric devices, the Wiedemann-Franz law specifies the minimal thermal-to-electrical conductivity ratio [66].

Thermoelectric Properties
The exponential growth of technology has led to enormous energy waste as a result of rising energy demands. Researchers have also been forced to create unique systems that can recycle waste heat into electrical energy due to the lower capacity of the available energy sources. One of the finest options is a thermoelectric generator, which can transform temperature gradients (phonons) directly into potential differences. The computed electrical conductivity measures the free carrier motion that results from temperature gradients that increase the carriers' kinetic energy. A higher electrical conductivity is required to realize the commercial uses of thermoelectric devices, since it reduces the joule heating effect.
By comparing the ratio of heat efflux per area per unit time to the temperature gradient, it is possible to assess the flow of thermal energy. The two categories of thermal conductivity are electronic and phononic. Due to the importance of intra-band transitions in metals, as compared to semiconductors, lattice vibrations (phonons) have a greater influence in metals than they do in semiconductors. Additionally, whereas the phonon energy has little significance for direct bandgap semiconductors, it is significant for indirect bandgap semiconductors. In order to increase the efficiency of thermoelectric devices, the Wiedemann-Franz law specifies the minimal thermal-to-electrical conductivity ratio [66].
In the present work, to discuss the transport behavior of CsPbBr 3 and Rb-doped CsPbBr 3 compounds, the thermal k τ and electrical σ τ 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.

Materials and Methods
New perovskite materials and their properties must be efficiently discovered using computational tools. Considering the wealthy amount of data, it is reasonable that this trend of employing computational methods will continue. Calculating the characteristics of materials is now possible without using experimental methods, because of the density functional theory (DFT). In physics and material science, DFT is a quantum mechanical modeling technique, that is employed to look into the electronic structure of many-body systems. In DFT, the exchange-correlation function, which is a mathematical approximation of the many-body effects of electron correlation, is used to treat electron correlation. In contrast, electron correlation is not included in HF, which reduces accuracy, but simplifies computation. As a result, it is probable that DFT is generally more accurate for many different calculations than HF, especially for systems with strong electron correlations [70]. DFT was realized in the 1980s by Pierre Hohenberg and Walter Kohn [71]. It is a commonly used computational method in material science for quickening the development of new materials and performing high-throughput simulations [72].
The electronic and optical properties of CsPbBr3 and Rb-doped CsPbBr3 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 CsPbBr3 according to the reported data in the literature [79]. The thermoelectric properties were calculated using BoltzTraP code [80].

Conclusions
In this work, a systematic investigation of the electronic, optical, thermoelectric, and elastic properties of cesium lead bromide CsPbBr3 and RbxCs1-xPbBr3 (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 Rb0.75Cs0.25PbBr3 is a wide band gap material, which indicates that it is a better candidate for high-frequency UV device applications.

Materials and Methods
New perovskite materials and their properties must be efficiently discovered using computational tools. Considering the wealthy amount of data, it is reasonable that this trend of employing computational methods will continue. Calculating the characteristics of materials is now possible without using experimental methods, because of the density functional theory (DFT). In physics and material science, DFT is a quantum mechanical modeling technique, that is employed to look into the electronic structure of many-body systems. In DFT, the exchange-correlation function, which is a mathematical approximation of the many-body effects of electron correlation, is used to treat electron correlation. In contrast, electron correlation is not included in HF, which reduces accuracy, but simplifies computation. As a result, it is probable that DFT is generally more accurate for many different calculations than HF, especially for systems with strong electron correlations [70]. DFT was realized in the 1980s by Pierre Hohenberg and Walter Kohn [71]. It is a commonly used computational method in material science for quickening the development of new materials and performing high-throughput simulations [72].
The electronic and optical properties of CsPbBr 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].

Conclusions
In this work, a systematic investigation of the electronic, optical, thermoelectric, and elastic properties of cesium lead bromide CsPbBr 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. CsPbBr 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.