3.1. Structural Properties
The materials Cs2LiMoX6 (X = Cl, I) disclose a cubic form structure characterized by the space group Fm3m (#225). The Li and Mo cations are surrounded by six halide ions (I− and Cl−) in the center of octahedra, creating MoX6 octahedral complex. Fig. depicts the optimization and the organization of these substances in a crystalline configuration, wherein octahedral complexes have vertices with the Cs cation placed at the core of the cubeoctahedral [18]. In the unit cell, the Cs atom is located at (3/4, 0.25, 0.25), the Li atom at (0, 0, 0), the Mo atom at (0.5, 0, 0), and the X (I and Cl) anion at (0.75, 0, 0). For gauging the structural robustness or extent of alteration in Cs2LiMoX6 (X = Cl, I), Goldschmidt's tolerance factor (τ) is utilized, and it is determined using the following expression [19].
$${\tau }=\frac{0.707({r}_{Cs}+{r}_{Cl/I})}{({r}_{avg}+{r}_{Cl/I})}$$
3
In this context, the symbols rCs, and rX denote the respective atomic radii of Cs and X = Cl, I atoms and ravg is the average ionic radius of Li and Mo. Fedorovskiy et al. stated that a τ falling within the range of 0.7 ≪ 1.2 is permissible for the generation of stable crystal structures[20]. The calculated τ values are 0.85 and 0.88 for Cs2LiMoI6 and Cs2LiMoCl6, respectively. As such, both compounds exhibit stable cubic structures, with their tolerance factor (τ) values situated within the acceptable range (refer to Table 1). Additionally, we assessed the stability of the materials by computing the formation enthalpy (ΔH) using the expression[21]:
$$\varDelta H={E}_{\text{C}\text{s}2\text{L}\text{i}\text{M}\text{o}\text{X}6}a{E}_{Cs}b{E}_{Li}c{E}_{Mo}d{E}_{X}$$
1
Here, ECs2LiMoX6 represents the total formation energy of Cs2LiMoX6 (X = Cl, I), while ECs, ELi, EMo, and EX denote the formation energies of the individual element and a, b, c and d are the number of atoms of respective element. The ΔH values are calculated as 3.948 eV and − 2.168 eV for Cs2LiMoI6 and Cs2LiMoCl6, respectively. The indication of the thermodynamic steadiness of the understudy materials is underscored by the existence of values with minus sign for their ΔH[22], as outlined in Table 1.
Table 1
Calculated values of lattice constant (Å), bulk modulus B (GPa), pressure derivative of bulk modulus Bp (GPa), tolerance factor (τ), and ground state energies E0(Ry) of stable state of cubic Cs2LiMoX6 (X = Cl, I).
Parameters

Cs2LiMoI6

Cs2LiMoCl6

Lattice constant (Å)

11.7813

10.3649

Bulk modulus (GPa)

20.9476

33.7648

Pressure derivative (Bʹ)

5.1133

3.7694

Volume(a.u.)3

2758.7638

1878.5071

Ground state energy E0(Ry)

124704.7764

44814.4987

Tolerance factor \(\tau\)

0.85

0.88

Utilizing the Murnaghan equation of state[23], the structural parameters comprising the equilibrium lattice parameter, bulk modulus (B), bulk modulus derivative (B′), volume at the ground state (V0), and ground state energy (E0) are optimized.
$$E\left(V\right)=\frac{9{B}_{0}}{16}{V}_{0}\left[{\left\{{\left(\frac{{V}_{0}}{V}\right)}^{2/3}1\right\}}^{2}{Bꞌ}_{0}+{\left\{{\left(\frac{{V}_{0}}{V}\right)}^{\frac{2}{3}}1\right\}}^{2}{\left\{64{\left(\frac{{V}_{0}}{V}\right)}^{2}\right\}}^{\frac{1}{3}}\right]{+E}_{0}$$
2
Through a volume optimization procedure for each compound, illustrated in Fig. 1 the total energy versus the volume of the unit cell. By incorporating this information into the equation, the structural factors (a, B, B′, E0, and V0) are determined, as presented in Table 1.
3.2. Electronic Properties
The understanding of carrier transport mechanisms is greatly prejudiced by the electronic properties of perovskites, which facilitates the differentiation between metals, semiconductors, and insulators through the analysis of Eg[24]. These electronic characteristics have farreaching implications on other properties, including optical and thermoelectric aspects, shaping the suitability of materials for diverse industrial and commercial applications.
Utilizing the PBEGGA approach, we have computed the electronic BS of Cs2LiMoI6 and Cs2LiMoCl6 along the highly symmetrical path WLΓXWK, as depicted in Fig. 2. In this study, the calculated Eg values for Cs2LiMoI6 are determined to be 0.96 eV in the spindown channel and 1.76 eV in the spinup channel. Correspondingly, for Cs2LiMoCl6, the Eg is found to be 3.35 eV in the spindown channel and 1.63 eV in the spinup channel. Notably, the Eg for these compounds are direct (at the Xpoint), enhancing their appeal for potential use in solar cells. The other similar studies also reported Eg in this range like Cs2GeVM6 (M = Cl, Br, I) which exhibit minimal bandgaps, falling within the range of 2 to 3 eV, and suggested suitable for optoelectronic applications in both the visible and UV regions[25].
Furthermore, we conducted calculations for DOS and projected DOS (PDOS) to offer a comprehensive understanding of the orbital and elemental contributions to the electronic structures. In this study, Fig. 3–5 display the total DOS and PDOS for elapsolites Cs2LiMoCl6 and Cs2LiMo6, respectively, presenting total, elementwise, and orbitalwise dispersions. In Fig. 5, Css, Lip and Csp orbitals notably shows dispersion in the valence band (VB) in the energy range − 5 eV to 2 eV. However, larger influence is seen near the EF from Modt2g. In conduction band (CB), less dispersion is seen for both compounds with large involvement from Lip and Lis orbitals near 3 eV.
Conversely, for Cs2LiMoI6, orbitals actively participate in the conduction band. In Fig. 4, it is evident that Modt2g and Modeg, make significant contributions near the EF in the VB, and Is and Ip show dispersion in energy range of 4 eV to 1 eV. While Lip and Lis are prominently involved in the conduction band for Cs2LiMoI6. The fundamental physics responsible for the narrow Eg can be elucidated by considering the presence of dispersed spherical orbitals (Modt2g and Modeg) at the conduction band minimum. Due to overlapping of halide orbits caused by this presence, extremely dispersive bands are produced. The close proximity of the valence band maxima and conduction band minima is a consequence of the vastly diffused nature of these bands, ultimately creating slight Eg.
The bonding nature becomes evident through electron density plots, as illustrated in Fig. 6. In Cs2LiMoCl6, covalent bonding is observed among Cs, Li, and Mo atoms. Similarly, in Cs2LiMoI6, the same covalent character is observed between Cs, Li, and Mo atoms. In general, both Cs2LiMoCl6 and Cs2LiMoI6 exhibit a covalent bonding nature among their constituent atoms, with specific ionic contributions in interactions involving Cs, Li, Mo, and other atoms as identified.
3.3. Magnetic Properties
Magnetic characteristics have been explored through spin dependent computations. The total magnetic moment (µB) of Cs2LiMoCl6 is found as 3.02436 µB which is slightly higher than that of Cs2LiMoI6 which is 3.00332 µB, indicating a more pronounced magnetic behavior in Cs2LiMoCl6. The confirmation of the ferromagnetic nature in both substances is evidenced by values that represent integer multiples of the total µB[26]. The Modt2g orbitals are the key factors determining the magnetic behavior in both perovskites. The sign of µB functions as an indicator for the magnetic orientation of atoms. A negative µB implies ferrimagnetic or antiferromagnetic characteristics, whereas a positive µB defines the arrangement of atoms in the respective direction[27]. Detailed information on total, partial, and interstitial µB for both perovskites is provided in Table 2.
Table 2
Computed interstitial, individual and total magnetic moments of Cs2LiMoX6 (X = Cl, I)
Compounds

Cs (µB)

Li (µB)

Mo (µB)

Cl/I

Interstitial µB

Total µB

Cs2LiMoI6

0.00263

0.00890

2.26185

0.01214

0.67228

3.00332

Cs2LiMoCl6

0.00106

0.00059

2.18821

0.02714

0.67481

3.02436

Table 3
Computed optical parameters for Cs2LiMoX6 (X = Cl, I)
Compounds

ε1(0)

n(0)

R(0)

Cs2LiMoI6

4.84

7.33

0.14

Cs2LiMoCl6

3.20

5.39

0.08

3.4. Optical Properties
The efficiency of photovoltaic devices is a crucial condition in optoelectronics, assessing the suitability of these gadgets for use in photovoltaic cell production dedicated to energy scavenging applications. In this context, the dielectric function ɛ(ω), as defined by ɛ(ω) = ɛ1(ω)+ ɛ2(ω)[19, 28], holds significance as a complex function determining how a photovoltaic material responds to an incoming electromagnetic (EM) radiation. Various other optical parameters, including the absorption coefficient α(ω), refractive index n(ω), reflectivity R(ω), extinction coefficient k(ω), optical conductivity σ(ω), and energy loss function L(ω) can be derived from the real ɛ1(ω) and imaginary ɛ2(ω) parts of the ɛ(ω).
The graphed figures in the energy range of 0–8 eV illustrate the computed optical parameters mentioned earlier. Within these figures (Figs. 7 and 8), ɛ1(ω) conveys details related to dispersion, while ɛ2(ω) provides insights into absorption characteristics. Both ɛ1(ω) and ɛ2(ω) exhibit nearly identical patterns for Cs2LiMoI6 and Cs2LiMoCl6, as depicted in Fig. 7(a) and 7(b), owing to their analogous electronic BS.
The ɛ1(ω) parameter serves to describe the extent of Photon scattering and the propagation speed, which, in turn, relies on the maximum dispersal of light[29]. As ω approaches 0, the values of ɛ1(ω) are approximately 4.84 and 3.20 for Cs2LiMoI6 and Cs2LiMoCl6, correspondingly, as indicated in Fig. 7(a). Various investigators have delved into the static dielectric features of analogous compounds. HDP composites, namely M2NaInI6 (where M = Rb, Cs), exhibit ε1(0) values of approximately 3.8 and 4.03, respectively[30]. Similarly, the M2AgCrBr6 compounds (where M = Cs, K, Rb) demonstrate ε1(0) values of around 6.1, 5.4, and 5.8 [31].Notably, Penn's model is affirmed in this context, confirming the contrary relationship between the static dielectric constant, represented by ɛ1(0) and the band gap Eg.[32]
The ɛ2(ω) spectrum shows linear increase initially with initial peaks at 2.52 eV and 4.27 eV for Cs2LiMoI6 and Cs2LiMoCl6, respectively and then fluctuation is seen. The ɛ2(ω) component, indicative of electronic shifts from the VB to the CB, exhibits protuberant peaks at 6.21 eV for Cs2LiMoI6 and 8 eV for Cs2LiMoCl6, as demonstrated in Fig. 7(b). Analyzing the α(ω), a key parameter for discerning a material's suitability for solar cell applications, reveals that both compounds have substantial values across a wide range of energies. Notably, Fig. 8(a) highlights significant optical transitions occurring at approximately 8 eV for both Cs2LiMoI6 and Cs2LiMoCl6, corresponding to the Ultraviolet (UV) spectrum.
The refractive index n(ω), an essential parameter for determining material transparency or opacity in optical investigations, follows a movement akin to that of ɛ2(ω), as depicted in Fig. 8(d). The expression for the complex refractive index, denoted as n’(ω), is as follows [19]
$$n{\prime }\left({\omega }\right)= \text{n}\left({\omega }\right)+i\text{k}\left({\omega }\right)$$
3
The variation in n(ω) is directly associated with bonding characteristics of perovskites. As a general trend, ionic compounds typically demonstrate minor values of n(ω) in contrast to covalently bonded perovskites. This discrepancy arises from the distinctive electronsharing characteristics of covalent bonding, where more electrons are shared. In covalent compounds, the greater electron distribution increases the probability of photonelectron interactions. As a result, photons experience a more pronounced slowing down within covalently bonded structures, distinguishing them from their ionic counterparts [33]. When ω approaches 0, the static refractive index values, denoted as n(0), are 7.33 and 5.39 for Cs2LiMoI6 and Cs2LiMoCl6, respectively. These n(0) values conform to the relation \(n\left(0\right)= \sqrt{\epsilon \left(0\right)}\). Additional prominent peaks for n(ω) occur at 2.38, 4.74 and 5.84 eV for Cs2LiMoI6 and at 3.96 eV, 5.48 eV, and 6.25 eV for Cs2LiMoCl6.
An additional optical parameter is the extinction coefficient, denoted as k(ω), which is directly associated with α(ω). This connection is expressed through the relationship[34] α(ω) = \(\frac{4\pi k\left(\omega \right)}{{\lambda }}\), as illustrated in Fig. 7(d). k(ω) serves as a measure of the attenuation of incident photons (k ˃ 0) within the studied alloys, attributed to absorption and scattering. Importantly, it is analogous to the imaginary part because both are interconnected through Kramers–Kronig relations [52]. The behavior of k(ω) can be elucidated using the following formula:
$$k\left(\omega \right)=\frac{1}{\sqrt{2}}{\left.\left({\left.\left({\epsilon }_{1}^{2}\left(\omega \right)+{\epsilon }_{2}^{2}\left(\omega \right) \right.\right)}^{\frac{1}{2}} {\epsilon }_{1}\left(\omega \right)\right.\right)}^{\frac{1}{2}}$$
4
The k(ω) spectrum shows fluctuating trend and the maximum values appeared at 6.97 eV and 8 eV for Cs2LiMoI6 and Cs2LiMoCl6, correspondingly. The optical conductivity, denoted as σ(ω), delineates the passage of photons released via photoelectric effect in a material. In the existence of intense electromagnetic (EM) rays, it elucidates the rupture of bonds [49]. Given that σ(ω) discloses free carriers produced upon the absorption of EM light and there is a close interconnection between σ(ω) and α(ω). Notably, the peak values of σ(ω) reach 4631 (Ω cm)−1 and 3618 (Ω cm)−1 at 6.24 eV and 8 eV for Cs2LiMoI6 and Cs2LiMoCl6, respectively (refer to Fig. 7(c)). σ(ω) can be computed as
$$\sigma \left({\omega }\right)=\frac{2{W}_{cv}ħ{\omega }}{{\underset{E}{\to }}^{2}}$$
5
Figure 8(b) portrays the reflectivity R(ω), a significant parameter indicating the proportion of EM light reflected at a specific energy[35]. Primarily, the R(ω) values were significant, specifically 0.14 and 0.08 at ω = 0 for Cs2LiMoI6 and Cs2LiMoCl6, respectively. As the energy rises, R(ω) experiences an upward trend. The R(ω) value is present at 0 eV due to the lattice vibrations occurring within the unit cell. The maximum reflectance is shown at 6.26 eV and 4.13 eV for Cs2LiMoI6 and Cs2LiMoCl6, respectively. R (ω) can be computed by using the given relation [35, 36]
$$R=\frac{{(n1)}^{2}+{k}^{2}}{{(n+1)}^{2}+{k}^{2}}=\left\frac{nˊ1}{nˊ+1}\right$$
6
The L(ω), depicted in Fig. 7(d), measures the reduction in photon energy as it traverses the material. The peaks are computed at 6.71 eV and 7.64 eV for Cs2LiMoI6 and Cs2LiMoCl6, correspondingly (see Fig. 8(c)). The Eg of a perovskite influences the energy levels of electronic shifts, impacting the absorption of light and resulting dielectric attenuation in the energy range. Compounds with larger Eg in the relevant energy range may exhibit larger energy electronic shifts and potentially higher L(ω). Notably, the compounds under investigation exhibit small Eg, which are associated with reduced Eg. The Eg can also be determined using ɛ2(ω) and L(ω), and parallelized with those calculated from the BS[37].