1 Introduction

As living standards grow and the world’s population increases, technological innovation will play a major role in the requirement of the world’s energy demand in the twenty-first century [1, 2]. The most sustainable, renewable, and environmentally friendly method of energy generation for meeting energy demands the appropriate approach is solar energy harvesting [3, 4]. Due to their fascinating optoelectronic features and high power conversion efficiency (PCE), halide perovskite-based solar cells have attracted the attention of scientists during the past decade. The halide perovskite solar cell is a highly efficient solar cell type owing to its greater absorption coefficient, low excitation binding energy, high charge carrier mobility, and longer charge carrier diffusion length [5,6,7]. Hybrid halide perovskites typically have the crystal structure ABX3, where A stands for an organic or inorganic cation, B for a metal cation, and X for an halogen family anion. Because these halide perovskites are naturally abundant and reasonably priced, they are widely used in a variety of industrial fields [8,9,10,11]. Numerous characteristics, including low exciton binding energy, low non-radiative recombination, long charge diffusion length, excellent optical absorption coefficient, and effective device functionality contribute to their efficiency [12,13,14,15].

There is evidence in the literature that the metal or cation choice in the metal halide influences the halide perovskites’ electrical properties, durability, and absorption capacity [16, 17]. Nevertheless, due to the fact that the other perovskite materials are toxic and may pollute the environment or degrade the ecosystem, their use has raised concerns about the environment [18, 19]. Different components have been substituted in the development of higher-performing solar cells based on perovskite. Tin iodide (SnI), is one of the more notable replacements. Due to their improved electrical characteristics reduced toxicity, and good optoelectronic capabilities related to the active layer’s tunable bandgap feature, SnI-based halide perovskites have emerged as the most promising substitute for solar cell applications [20,21,22,23]. The smallest bandgap of Sn-based halide PSCs is found to be between 1.31 and 1.60 eV, according to theoretical simulations. Improved optical and electrical characteristics are displayed by Sn-based PSCs, including greater charge mobility and a power conversion efficiency of almost 30% [23,24,25,26,27].

Using a quantum mechanical technique, Paschal and colleagues have investigated the electrical, thermodynamic, and structural properties of the guanidinium tin halide perovskite SnX3 (where X = Cl, Br, and I) via DFT [28,29,30,31,32]. According to their findings, the band gaps of the materials were 2.47, 1.78, and 3.0 eV, respectively. The narrowest bandgap is found in C(NH2)3SnI3 at 1.78 eV. The structural and electrical characteristics of a CsXCl3 perovskite photovoltaic solar cell (where X = Sn, Pb, or Ge) were determined by Idrissi et al. [33,34,35] using Quantum Expresso software. The CsSnCl3 material was determined to have the lowest bandgap among the three configurations after analysis, making it suitable for solar cell deployment [36, 37]. When compared to Cs-based perovskites, their investigation showed Thus, Sn-based PSCs had better optical conductivity, better light absorption, and more flexibility. Sn-based perovskite solar cells have the potential to be a workable replacement for lead-based perovskite, which is toxic, and other types of solar cells. It has been demonstrated that changing X to a different metal atom improves the physical and chemical characteristics of the perovskite structures, which may enable their use in a variety of solar systems. In this study, we examine how specific metals dopants affect the optoelectronic, optical, and structural characteristics of the perovskite structures XSnI3 (X = Rb, K, Tl, and Cs). The primary objective of the current work is to investigate the structural, and optical properties of XSnI3 materials using computational modeling with the CASTEP software. To the best of our knowledge, not much research has been published on the study of the XSnI3 structure, where X is cesium, thallium, rubidium, and potassium (Cs, Tl, Rb, and K). It is envisaged that such studies will lead to future experimental studies to exploit the significant potential of such materials in Sn-based PSCs application.

2 Computational methodology

CASTEP simulation software is used for first-principles calculations [38]. The GGA (General Gradient Approximation) method was used to simulate the geometric structure and determine the electrical structure. The generalized gradient method of Perdew, Burke, and Ernzerhof (PBE) is frequently used to compute the exchange-correlation between energy and electrons. USP (ultra-soft pseudo-potential) was used to determine the electrostatic interactions between the valence electron and the ionic core [39]. An elongation of the wave function results from the accretion of plane waves with a cut-off energy of 365.5 eV. In structural analysis, the total energy convergence is less than −1.01 × 105 eV/atom and the self-consistent convergence value is assumed to be (4.67 × 10−4) eV/atom. The electrical configurations of tin, rubidium, potassium, thallium, and cesium are \(\left[{\rm{Kr}}\right]{4d}^{10}{5s}^{2}{5p}^{2}\), \(\left[{\rm{Xe}}\right]{5s}^{1}\), \(\left[{\rm{Ar}}\right]{4s}^{1}\), \([{\rm{Xe}}]{4{\rm{f}}}^{13}{6{\rm{S}}}^{2}\) and \(\left[{\rm{Xe}}\right]{6s}^{1}\) respectively. It is determined that the maximum stress is less than 5.20 × 10−2 and the maximum Hellmann-Feynman force is \(0.0004{eV}/\dot{A}.\) The highest atomic displacement that might occur is less than \(1.22\times {10}^{-2}{\text{\AA }}\). The Monkhorst Pack grid, which is composed of 16 × 16 × 4 k-points, was sampled using First Brillouin for structural optimization and electronic property computations. In the non-periodic direction, we used a vacuum of 35 \({\text{\AA }}\) is along the lattice vector to prevent undesired interactions. Fig. 1a–d presents the 2D supercell (8 × 8 × 2) of XSnI3 (X = Rb, K, Tl, and Cs) halide perovskite materials.

Fig. 1
figure 1

a–d Supercell of hybrid halide perovskite material XSnI3 (where X = Rb, K, Tl, and Cs)

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3 Results and discussion

3.1 Structural study

This section presents an analysis and presentation of the structural characteristics of the hybrid halide perovskites XSnI3 (where X = Rb, K, Tl, and Cs). The three crystal lattice parameters were initially used for the unit cell geometry optimization. The x-alkali dopants (X = Rb, k, Tl, and Cs) are uniformly distributed along the matrix of the SnI3. In the PBE-GGA approach the XSnI3 (where X = Rb, K, Tl, and Cs) is simulated using the CASTEP simulation software. The atomic positions and space groups are displayed in Table 1. In the hybrid halide perovskite, the dopants are positioned at various corners of the orthorhombic crystal structure as shown in Fig. 1a–d. Our observed structural properties showed that the lattice parameters have an impact on the atomic locations of atoms in XSnI3.

Table. 1 The XSnI3 supercell of structure materials’ space groups, atomic locations, and lattice characteristics
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3.2 Electronic properties

Figure 2a–d displays the band structures and corresponding density of states graphs for XSnI3 (where X = Rb, K, Tl, and Cs). The primary electronic structural factors that are essential for deriving out the materials’ orbital overlaps (or energy levels) and their relaxation influences on the different energy levels are the total density of states (TDOS) and partial density of states (PDOS). The energy range in which an electron can exist (conduction band, C.B.) and the locations where electron availability is zero are both depicted by the electronic band structure. The fermi level (Ef) is set to zero eV since all calculations were performed at 0 K without considering the effects of the finite temperature. The fermi level categorizes the conduction band and the valence band. The valance band (V.B.) is located below the energy Fermi level (Ef) and the C.B. is positioned above Ef. The bandgap can be found by comparing the valance band maxima (VBM) and conduction band minima CBM. Understanding material behavior and differentiating between conductors, insulators, and semiconductors will be made simpler by examining the presence or absence of a bandgap.

Fig. 2
figure 2

a–d Bandgap structures and TDOS of XSnI3 (where X = Rb, K, Tl, and Cs)

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The material could have an indirect bandgap or a direct bandgap, based on the band structure. A direct Eg is observed when the VBM and CBM match. An indirect Eg is indicated when these two points are precisely off from one another. As illustrated in Fig. 2a–d, it has been found that the Eg of the XSnI3 (where X = Rb, K, Tl, and Cs) shows a decreasing trend with values as 2.76, 2.01, 1.90, and 0.34 eV obtained for RbSnI3, KSnI3, TISnI3, and CsSnI3 respectively. According to Fig. 2a–d, the VBM and CBM of the XSnI3 materials under investigation are located at different symmetry locations, indicating both direct and indirect energy band gaps (Eg). In particular, VBM and CBM of RbSnI3 and KSnI3 halide perovskite material are situated at the same position M consistent with a direct Eg. However, the VBM and CBM in TISnI3 are positioned at distinct places, indicating that TISnI3 is an indirect semiconductor material. The VBM and CBM for CsSnI3 are shown to be positioned at the same G point in Fig. 2d, indicating a direct semi-conducting material. The absorption of light increases from 1.7 × 105 cm−1 to 2.50 × 105 cm−1, and optical conductivity increases from 1.6 to 2.4 eV, as the bandgap decreases from 1.76 to 0.34 eV as shown in Table 2. The TDOS is used to measure the electronic bandgap represented per unit of energy regardless of how the partial density of states is used to analyze the contribution of ions to different band structures. With an Eg value of 0.34 eV, Fig. 4d demonstrates that the s state for the CsSnI3 halide perovskite material contributes more to the conduction band than the p, d, and f states. This is explained by the higher sigma state contribution in the bandgap reduction. Because of its lower bandgap value, CsSnI3 halide perovskite could be a desirable choice for solar cell applications.

Table 2 The bandgap (Eg), absorption coefficient, and optical conductivity of halide perovskite materials
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The decrease in the bandgap is explained by the TDOS and PDOS. The dotted line represents the Ef fermi level, which is located at the valence band’s peak. These TDOS charts demonstrate that in RbSnI3, KSnI3, TISnI3, and CsSnI3, recently produced extra gamma states are responsible for the bandgap decrement as shown in Fig. 3a–d. The C.B. shifted towards the EF along the G positioned as a result of newly produced gamma states of XSnI3 materials is a significant factor in decreasing the Eg of these materials. In semiconductors, these materials exhibit a direct-to-indirect bandgap structure, as demonstrated by the bandgap structures.

Fig. 3
figure 3

a–d Partial density of states XSnI3 (where X = Rb, K, Tl, and Cs)

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The data show that the “s” states highlighted in red contribute the most to C.B. for each material. However, in the s states, there are more gamma states. Consequently, s states contribute greater as compared to p and d states as shown in Fig. 4a–d. Figure 4a shows that the “s” state (in red color) is contributing more in the conduction band for the RbSnI3 halide perovskite material with an Eg value of 2.76 eV. Figure 4b shows that for the KSnI3 halide perovskite material with an Eg value of 2.01 eV, the “s” and “p” states (shown in red and green respectively) contribute more than the “d” state (shown in blue) in the conduction band. Figure 4c illustrates how the “s” and “p” states contribute more to the conduction band than the d for the TISnI3 halide perovskite material, with an Eg value of 1.90 eV. The energy bandgap for TISnI3 and CsSnI3 decreases due to an increase in the hybridization of s, p, and d states. These findings indicate that halide perovskites are suitable for solar cell application.

Fig. 4
figure 4

a–d Partial density of states (PDOS) of RbSnI3, KSnI3, TISnI3, and CsSnI3 halide perovskite materials

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3.3 Optical properties

Halide perovskite materials exhibit remarkable optical characteristics and can be employed in solar cells and photocatalysis. The complex ε (ω) dielectric parameters can explain the behavior of halide perovskite materials such as RbSnI3, KSnI3, TISnI3, and CsSnI3 in an electric field. In particular ε(ω), is composed of two elements, the real dielectric function (RDF) and the imaginary dielectric function (IDF), and is based on a function of the optical band structure of the crystal. The electronic structure of halide perovskite materials such as RbSnI3, KSnI3, TISnI3, and CsSnI3 can be used to define their optical properties, along with other features including the dielectric function, refractive index, coefficient of absorption, reflectivity, energy loss function, and optical conductivity. These characteristics are beneficial in indicating the materials’ applicability and stability in solar cell applications.

The interaction between electromagnetic waves and valance electrons between the core electrons of XSnI3 materials are responsible of all of the optical conductivity features. Since these characteristics are all related, the complex dielectric function can be expressed as follows [40,41,42,43].

$$\varepsilon \left(\omega \right)={\varepsilon }_{1}\left(\omega \right)+{{\rm{i}}\varepsilon }_{2}\left(\omega \right)$$
(1)

The optical response to the impact of XSnI3 material hybridization on the dielectric function, which is quantifiable using the following formulas [44,45,46,47,48].

$$\begin{array}{l}{\varepsilon }_{2}\left(\omega \right)=-\frac{V{e}^{2}}{2\pi{{\hslash} m}^{2}{\omega }^{2}}\int {d}^{3}k\sum _{{{nn}}^{{\prime} }}1 \,<\, {kn}\left|P\right|k{{{\tilde{n}}}} \,>\, {I}^{2}f(k)\\\qquad\qquad\times \,\left(\right.1-f(k{{{\tilde{n}} }})\delta ({E}_{{kn}}-{E}_{k{{{\tilde{n}}}}}-{{\hslash} \omega} )\end{array}$$
(2)

The relative permittivity of XSnI3 materials is given by the dielectric constants. The interpretation of the term “dielectric” explicitly, indicates the extent to which an electric field can pass through atoms of XSnI3 materials. This primarily shows the maximum polarization that XSnI3 material can withstand at different X atoms. Since there can never be a field inside the confines of an electrical conductor, the ideal conductor should have zero value. The terms are related to real part \({[{\rm{\varepsilon }}}_{1}\left({\rm{\omega }}\right)]\), and the imaginary part \({[{\rm{\varepsilon }}}_{2}\left({\rm{\omega }}\right)]\) within the XSnI3 materials.

As shown in Fig. 5a, the main peaks of \([{\varepsilon }_{1}\left({\omega }\right)]\) have values of about 10, 4.3, and 2.4 respectively at 4 eV for CsSnI3, TISnI3, and RbSnI3. In the case of KsSnI3 the maximum value of >12 eV is observed at the 2 eV and then decreased to 2.3 near 4 eV. The peaks of the remaining halide perovskite materials started to drop after 5 eV as shown in Fig. 5b. With Kramer’s-Kronig mathematical formulation [49, 50], the RDF \([{\varepsilon }_{1}\left({\omega }\right)]\) is formed by the IDF (imaginary dielectric function) \({[{\rm{\varepsilon }}}_{2}\left({\rm{\omega }}\right)]\), as illustrated in Fig. 5b. For structures CsSnI3, TISnI3, KSnI3, and RbSnI3, the maximum values of 11.8,10, 4.5, and 2.5 respectively are observed at 2.5–3 eV. Above 3 eV, the values of \({\varepsilon }_{1}\left(\omega \right)\) are shifted to lower values. These calculations suggest that these XSnI3 halide perovskites are suitable for the solar cells industry.

Fig. 5
figure 5

a, b The a real and b imaginary part of the dielectric functions of RbSnI3, KSnI3, TISnI3, and CsSnI3halide perovskite materials Real (a) and imaginary (b) terms

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3.3.1 Refractive index

Figure 6a, b displays the extinction coefficient (k(w)) and refractive index (n(w)) of the XSnI3 halide perovskite material, which were carefully investigated. The results indicate optical transparency and capturing of electromagnetic (EM) wave radiations. The \({{\rm{\varepsilon }}}_{1}({\rm{\omega }})\) and \({{\rm{\varepsilon }}}_{2}({\rm{\omega }})\) dielectric functions dielectric vary with frequency in addition to providing a means of confirming the n(ω) complex refractive index provided by the following mathematical terms [51,52,53,54].

$${{{\tilde{n}}}}\left(\omega \right)=n\left(\omega \right)+{\rm{i}}k\left(\omega \right)={\epsilon }^{1/2}={({\varepsilon }_{1}+{\rm{i}}{\varepsilon }_{2})}^{1/2}$$
(3)
$${\rm{I}}\left({\rm{\omega }}\right)=\sqrt{2}{\rm{\omega }}{\left(\sqrt{{{\rm{\varepsilon }}}_{1}{({\rm{\omega }})}^{2}}+\sqrt{{{\rm{\varepsilon }}}_{2}{({\rm{\omega }})}^{2}}-{{\rm{\varepsilon }}}_{1}\left({\rm{\omega }}\right)\right)}^{1/2}$$
(4)
$$\sqrt{\varepsilon \left({\rm{\omega }}\right)}={\rm{n}}\left({\rm{\omega }}\right)+{\rm{iK}}\left({\rm{\omega }}\right)={\rm{N}}\left({\rm{\omega }}\right)$$
(5)
Fig. 6
figure 6

a, b Extinction coefficient and refractive index of CsSnI3, TISnI3, KSnI3, and RbSnI3

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In this case, the imaginary extinction coefficient equation is presented by k(ω), whereas the real refractive index is commonly measured by n(ω) [5, 8, 55,56,57].

$${\rm{r}}\left({\rm{\omega }}\right)=\frac{{\rm{n}}+i{\rm{K}}-1}{{\rm{n}}+i{\rm{K}}+1}$$
(6)
$${\rm{K}}\left({\rm{\omega }}\right)=\frac{{\rm{I}}({\rm{\omega }})}{2{\rm{\omega }}}$$
(7)
$${{\rm{\varepsilon }}}_{2}\left({\rm{\omega }}\right)=2{\rm{nK}}$$
(8)
$${{\rm{\varepsilon }}}_{1}\left({\rm{\omega }}\right)={{\rm{n}}}^{2}-{{\rm{K}}}^{2}$$
(9)

The refractive index (n) at zero eV of the XSnI3 samples is 1.7, 2.2, 3.4, and 3.5 for RbSnI3, KSnI3, TISnI3, and CsSnI3 respectively. The refractive index peaks shifted toward a sharp decline with energy up to 5 eV. The refractive index (n) ranges from 0 to 5 in the energy range of 0–40 eV because of the different frequencies of the inner-transition band. In the prominent energy range, there is less polarization as indicated by the lower refractive index.

3.3.2 Absorbance and energy loss

The absorption quality of a material is directly related to its ability to absorb luminous electromagnetic radiation as opposed to photons with the appropriate energy, (E = ħω). Additionally, the energy loss function represented by L(\(\omega\)) in Fig. 7b, describes the dissipation of incident photons energy in the materials. The following expression can be employed to address the absorption coefficient “\({a}\left(\omega \right)\)” [58,59,60,61].

$$a(\omega)=2\omega k(\omega)=\sqrt{2}[\{{\varepsilon }_{1}({\omega})^{2}+{\varepsilon }_{2}({\omega})^{2}\}^{1/2}-{\varepsilon }_{1}(\omega)]^{1/2}$$
(9)
$$a\left(\omega \right)=\frac{4k\pi }{\lambda }=\frac{\omega }{{nc}}{\varepsilon }_{2}\left(\omega \right)$$
(10)
$${\rm{L}}\left({\rm{\omega }}\right)=\frac{{{\rm{\varepsilon }}}_{2}}{{{\rm{\varepsilon }}}_{1}{({\rm{\omega }})}^{2}+{{\rm{\varepsilon }}}_{2}{({\rm{\omega }})}^{2}}$$
(11)

The absorbance of RbSnI3, KSnI3, TISnI3, and CsSnI3 is shown in Fig. 7a. Investigations have shown that light absorption is lowest in areas with roughly equal energy and maximum reflection. The capacity to absorb according to Fig. 7a, of halide perovskite materials containing RbSnI3, KSnI3, TISnI3, and CsSnI3 is sharply rising. A slight shift toward higher energy values is observed in all of the absorption peaks. As illustrated in Fig. 7a, the absorption coefficient α(w) values are 2.4 × 105 cm−1, 2.2 × 105 cm−1, 1.9 × 105 cm−1, and 1.7 × 105 cm−1 for CsSnI3, TISnI3, KSnI3, and RbSnI3 respectively. When compared to other materials CsSnI3, and TISnI3, with absorption coefficients of 2.4 × 105 cm−1 and 2.2 × 105 cm−1, respectively, exhibit the maximum absorption. The absorption coefficient α(w) indicates a shift toward high absorbance in the range of 5 eV to 20 due to the notable decrease in energy bandgap. These absorption results indicate that the halide perovskite materials CsSnI3, TISnI3, KSnI3, and RbSnI3 are suitable for solar cell applications.

Fig. 7
figure 7

a Absorption coefficient b energy loss function for CsSnI3, TISnI3, KSnI3, and RbSnI3

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3.3.3 Optical conductivity and reflectivity

The conductance of photo-generated electrons caused by the photoelectric process is characterized by optical conductivity. Particle bonding is broken by electromagnetic radiation. The optical conductance of CsSnI3, TISnI3, KSnI3, and RbSnI3 is displayed in Fig. 8a–d, covering the 0–40 eV range. The real peaks of optical conductance σ1(w) for RbSnI3, KSnI3, TISnI3, and CsSnI3 are in the energy range of 0 to 20 eV and originate from the origin point and touch their maximum conductivity values of 2.3, 2.8, 4.5, and 5.2 eV respectively. The real component of their optical conductivities decreases steadily 40 eV reaching a maximum around at 5 eV. Conversely, the imaginary optical conductivity σ2(w) for the RbSnI3, KSnI3, TISnI3, and CsSnI3 have maximum values at 5 eV of 2.1, 2.4, 2, and 2.6 cm−1 respectively. The optical conductivity results suggest that RbSnI3, KSnI3, TISnI3, and CsSnI3 are suitable materials for solar cell applications.

Fig. 8
figure 8

a, d Optical conductivity of a RbSnI3 b KSnI3 cTISnI3, and d CsSnI3 halide perovskite materials

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Any material’s reflectivity can be utilized to analyze how its surface behaves. Figure 9 displays the surface reflectivity behavior of TISnI3, CsSnI3, KSnI3, and RbSnI3. From 0 to 15 eV, the reflectivity peaks increased; however, they began to decrease at 16 eV as shown in Fig. 9. The reflectivity peaks of the RbSnI3, KSnI3, TISnI3, and CsSnI3 halide perovskite materials are 0.15, 0.38, 0.42, and 0.62 respectively. The maximum reflectance peaks of 0.62 and 0.42 are observed in these two CsSnI3 and TISnI3 materials when compared to other halide perovskite materials, KSnI3 and RbSnI3. CsSnI3 and TISnI3 show improved absorption and optical conductivity compared to other materials. According to the results in Fig. 9, there is a slight shift in the reflectivity peaks toward higher energy levels.

Fig. 9
figure 9

Optical reflectivity of RbSnI3, KSnI3, TISnI3, and CsSnI3

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4 Conclusion

Utilizing the Perdew Burke-Ernzerhof Generalized Gradient Approximation (PBE-GGA) for the exchange-correlation potentials, a density functional theory-based (DFT) study is used to investigate the structural, optical, and electrical aspects of XSnI3 (X = Rb, K, Tl, and Cs) materials. According to the DFT calculation, the energy band gaps (Eg) of XSnI3 (X = Rb, K, Tl, and Cs) materials are 2.76, 2.01, 1.90, and 0.34 eV respectively. The direct energy bandgap (Eg) indicates that these materials are appropriate semiconductors for solar cell application. A thorough analysis of optical conductivity indicates that the optical conductance peaks of XSnI3 (X = Rb, K, Tl, and Cs) have maxima of 2.3, 2.8, 4.5, and 5.2 eV in the ultraviolet spectrum and shift slightly at higher energy bands. The maximal absorbance of XSnI3 (X = Rb, K, Tl, and Cs) materials were (1.4 × 105 Ω−1 cm−1), (1.8 × 105 Ω−1 cm−1), (2.2 × 105 Ω−1 cm−1) and (2.4 × 105 Ω−1 cm−1) respectively. The XSnI3 (X = Rb, K, Tl, and Cs) is a group of materials with enhanced surface area for light photon absorption. XSnI3 (X = Rb, K, Tl, and Cs) are suitable halide perovskites for solar cell applications due to their enhanced optical conductivity, energy absorption, and refractive index. The halide family has enormous promise for energy generation application. Additionally, current modeling studies are waiting experimental validation.