A DFT exploration of the optoelectronic and thermoelectric features of a novel halide double perovskite A₂YAuI₆ (A = Rb, Cs) for solar cell and renewable energy applications

Abstract

Double-perovskite halides are potentially useful materials for producing renewable energy and are thought to meet the necessary criteria for addressing energy scarcity issues. Consequently, investigations into these halides have potential uses in the fields of thermoelectric and solar cell gadgets. The physical properties of A₂YAuI₆ (A = Rb, Cs) double perovskite halides were also investigated in the current work using DFT calculations based on the FP-LAPW approach for applications in renewable energy devices. The calculated Goldschmidt's tolerance factor and formation energy demonstrate that the explored halides are structurally and thermodynamically stable in the cubic phase. By analyzing mechanical properties, the measured Pugh and Poisson ratios demonstrate a ductile nature. In addition, we used electrical characteristics to compute bandgaps both with and without spin orbit coupling (SOC). We calculated the bandgap values of Rb₂YAuI₆ (E_g = 1.73 eV) and Cs₂YAuI₆ (E_g = 1.70 eV) using modified Becke–Johnson with spin orbit coupling (mBJ + SOC) potentials in order to get corrected bandgap values with respect to experimental data. Additionally, the optical characteristics of the halides under study were analyzed in light of their complex dielectric functions. Our computed optical parameter results clearly show that these halides have the best light absorption in the UV-visible region, indicating their suitability for use in solar cell applications. The power factor (PF), thermal conductivity, figure of merit, electrical conductivity, and Seebeck coefficient were computed to analyze transport characteristics against chemical potential, carrier concentration and temperature. Our findings can prove beneficial for further experimental research aimed at assessing A₂YAuI₆ (A = Rb, Cs) in relation to applications using renewable energy devices.

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1. Introduction

The demand for alternatives to fossil fuels has dramatically increased as a result of the shortage of traditional energy sources. The techniques that researchers are developing can draw energy from renewable resources. One of the primary sources of energy from renewable sources is the sun. Researchers can turn the heat and light energy from the sun into electrical energy using solar cells and thermoelectric generators. However, recent studies have shown that lead-involved hybrid perovskites can be created by mixing organic and inorganic components. Amongst the different material types, perovskites are one of the most popular materials that have been extensively explored for thermoelectric and photovoltaic applications, particularly in the field of perovskite solar cells. Perovskites are well-known for their cost-effectiveness, high absorption coefficient, adjustable bandgap, low effective mass, free availability, and stability in their structure. Moreover, they are simple to synthesize and low in weight.^1–10 However, because of their low stability and toxicity, as seen in CH₃NH₃PbX₃ (where X = Cl, Br and I) perovskites, they are hazardous to the environment and are of little utility in the solar cell system.^1–3 Lead-free material manufacturing is therefore urgently needed. Using lead-free products can help to protect the environment as well as reduce the need for fossil fuels. Consequently, lead-free double perovskites have drawn much interest from the scientific community.

Among the subclasses of perovskite structure, double perovskites has properties that combine the chemical and physical characteristics, whereas we cannot see these properties in a simple perovskite structure. These materials come in a greater variety of metals and oxidation states. This diversity makes it possible to enhance the optoelectronic characteristics that are beneficial for solar energy technologies. Shi et al.¹¹ created lead-free double perovskites that relax ferroelectrics in order to investigate the piezoelectric behavior. These rare-earth perovskites have great qualities that make them appropriate for solar energy applications. Still, a large number of researchers substituted tin-based halide perovskites for lead-based perovskites, specifically CH₃NH₃PbX₃.^12–14 Researchers are searching for various perovskites that could be used as solar absorber materials because of the breakdown and instability of these perovskites.^15–17 Lead-free double perovskite Cs₂AgBiX₆ has been the subject of recent investigations, and considered as an environmentally friendly substitute for CH₃NH₃PbI₃.¹⁸ Wang et al. have proposed the use of double perovskite Cs₂AgBiBr₆ in conjunction with reduced graphene oxide to produce photocatalytic hydrogen.¹⁹ The optical, thermoelectric, magnetic and electronic characteristics of the Ca₂Cr_1−xNi_xOsO₆ double perovskites were observed by Bhandari et al.²⁰ using density functional theory. The results demonstrated that the double perovskite that was produced had a bandgap of around (0.6 eV), and was optically isotropic and ferromagnetic. The fact that compounds with narrow bandgaps, such as PbTe and Bi₂Te₃,²¹ are thought to be appropriate for thermoelectric generators is highly interesting. According to a study carried out by Haque et al.,²² Cs₂InAgCl₆ has a direct band gap of 3.3 eV with a ZT value of 0.94 that makes it appropriate for thermoelectric appliances.

The optoelectronic and thermoelectric features of Cs₂ScAgY₆ (Y = Cl, Br and I) have been studied by Khan et al.²³ These compounds have demonstrated indirect bandgap values of 1.55 eV and 1.9 eV at 300 K, as well as a figure of merit value of 0.74. X₂ScInI₆ (X = Rb, Cs) was examined using DFT by Noor et al.,²⁴ and they were shown to be useful for optoelectronic devices. Iqbal et al. previously studied the opto-electronic and transport characteristics of Rb₂AlInX₆ (X = Cl, Br, I).²⁵ By applying first-principle computations, Anbarasan et al.²⁶ reported the mechanical, structural, and opto-electronic features of Cs₂AgInY₆ (Y = Cl, Br and I). Nazir et al.²⁷ has examined the double perovskite X₂LiInBr₆ (X = Rb, Cs) for solar cell applications. Due to their absorption and optical conductivity in the visible region of the electromagnetic spectrum, they are suitable candidates for solar cell applications.

In the present study, we have computed the structural, electronic, optical, and thermoelectric properties of novel double perovskite A₂YAuI₆ (A = Rb, Cs) compounds. As per our knowledge, neither experimental nor theoretical study has been performed earlier for these compounds. Our results would hopefully be helpful for future experimental and theoretical research studies in evaluating the A₂YAuI₆ (A = Rb, Cs) double perovskite for applications related to energy harvesting technologies.

2. Methodology

In our study, we used FP-LAPW^28,29 method-based WIEN2k code²⁸ to compute the optoelectronic and transport characteristics of Rb₂YAuI₆ and Cs₂YAuI₆via DFT calculations. Using Murnaghan equations of states for the examined compounds, the PBE-GGA functional was utilized to precisely calculate the lattice constants, ground state energy, and bulk modulus.^30–32 Although it facilitates the computation of structural parameters, this approximation underestimates electronic parameters, particularly the band gap. Despite having comparable accuracy, the TB-mBJ³³ potential is more flexible and requires less computation time than the PBE-GGA.³⁴ Therefore, the TB-mBJ potential is applied over PBE-GGA in order to accurately measure the band gap. Additionally, since the SOC coupling influences the band gap materials of heavy elements, it has also been included. While the electronic system solution is plane wave-like in the interstitial area, it is believed to be spherically harmonic within the muffin-tin sphere. The structure is continuously optimized until the strain-force for every atom is zero. The starting parameters have been adjusted by setting the angular momentum [small script l]_max = 10 in the reciprocal lattice, Gaussian parameter G_max = 14, and the product of the wave vector and muffin radius K_max × R_MT = 8. When the energy released from the system becomes constant, the 2000 k-point is used to select the k-mesh of the order of 12 × 12 × 12.³⁵ The optical properties were calculated by employing Kramers–Kronig equations. This is thought to be the convergence criterion's threshold limit in terms of the precision of the results acquired. The charge convergence is taken into account at 0.01 mRy. For the separation of the core and valence states, the cut-off energy was selected as −9 Ry. The core state is explained by the lower energy range compared to those equipped states. The thermoelectric parameters, such as the Seebeck coefficient, electrical conductivity, thermal conductivity, figure of merit (ZT) and power factor, were calculated in the temperature range of 100–1200 K against chemical potential and carrier concentration. The Boltztrap code,³⁶ which is based on the classical Boltzmann transport theory, was used to calculate the transport features by utilizing the optimized electrical structures via Tb-mBJ and the convergent energy. For the thermoelectric characteristics calculation, a fine grid mesh has been utilized (44 × 44 × 44).

3. Structural and mechanical properties

For all computations, we used the crystal structures of the Rb₂YAuI₆ and Cs₂YAuI₆ compounds with the space group of Fm3̄m #225. The crystal structure of A₂YAuI₆ (A = Rb, Cs) is presented in Fig. 1. Using the Murnaghan equation³⁷ of states, we optimized both compounds using PBE-GGA to obtain the ground state lattice parameters.

Fig. 1 Crystal structure of A₂YAuI₆ (A = Rb, Cs).

The bulk modulus and ground state lattice constant (a₀) are determined via the volume optimization method, as depicted in Table 1. The ground state energy versus optimized volume graphs of studied compounds are shown in Fig. 2. The determined values of a₀ increase from Rb₂YAuI₆ to Cs₂YAuI₆, as shown in Table 1, because the atomic radii increase from 2.48 Å (Rb) to 2.65 Å (Cs). The increase in cationic size leads to a decrease in the density (and therefore the solidity of the materials), which is why the bulk modulus B₀ (20.33 GPa 19.87 GPa) drops when Rb replaces Cs. Applying the formula,³⁸ we determine Goldschmidt's tolerance factor (t_G) for the computation of structural stability.

here, r_Rb/Cs, r_Y, r_Au and r_I are the atomic radius of Rb/Cs, Y, Au and I atoms, respectively. Goldschmidt's criterion typically reaches unity for stable cubic perovskites in an ideal scenario. Because of this, the range of values between 0.8 and 1.4 is significant for stable cubic structures. Table 2 shows our computed values in the above range. As a result, the compounds under study are stable in their cubic state. A comparative investigation demonstrated that Cs₂YAuI₆ is closer to the ideal limit than Rb₂YAuI₆. Additionally, the compound's thermodynamic stability is demonstrated by the chemical equation which computes the formation energy as follows:

ΔH_f = E_Total (Rb/Cs_aY_bAu_cI_d) − aE_Rb/Cs − bE_Y − cE_Au − dE_I (3)

where E_Rb/Cs, E_Y, E_Au, and E_I denote the energies for the isolated Rb/Cs, Y, Au, and I atoms, respectively, and E_Total (Rb/Cs_aY_bAu_cI_d) reflects the entire sum of energy of the examined compounds. In the equation above, the numbers a, b, c, and d represent the number of corresponding atoms. The quantity of energy lost during the process of material production is primarily represented by the negative value of ΔH_f. In actuality, it is essential to the material's stability. As a result, the calculated values shown in Table 1 validate the synthesized compounds thermodynamic stability.³⁹ It is evident from the data once more that Cs₂YAuI₆ is more stable than Rb₂YAuI₆.

Table 1 Calculated lattice parameter a (Å), bulk modulus B, its derivative B_P, the minimum total energy E_tot, energy of cohesion E_coh, of A₂YAuI₆ (A = Rb, Cs)

XC	a (Å)	V (a.u^3)	B (GPa)	B P	E tot (Ry)	E c (eV per atom)
Rb2YAuI6	12.07	2966.75	21.96	2.63	−142 222.705247	1.40
Rb2ScInI6^24	12.12		18.85
Cs2YAuI6	12.11	3002.41	22.66	3.68	−161 457.950263	1.34
Cs2ScInI6^24	12.15		17.95

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Fig. 2 Energy versus volume optimization curves of A₂YAuI₆ (A = Rb, Cs).

Table 2 Calculated values of the Bulk modulus (B), Shear modulus (G), Young's modulus (E), Poisson's ratio (σ), Pugh ratio (B/G), Cauchy pressure C^P, sound velocities (m s⁻¹), Debye temperature Θ_D (K) of A₂YAuI₆ (A = Rb, Cs) compared with the previous literature

Material property	Rb2YAuI6	Cs2YAuI6	Rb2ScInI6^24	Cs2ScInI6^24	Rb2LiGaI6^27	Cs2LiGaI6^27
Bulk modulus, B (GPa)	20.33	19.87	19.42	18.93	14.3699	18.6031
Shear modulus, G (GPa)	1.41	1.16	6.55	3.64
t G	0.954	0.995	0.95	0.97	0.9021	0.8152
Formation energy ΔHf	−1.296	−1.335	−1.04	−1.16
Young modulus, E (GPa)	4.15	3.43	17.67	10.24	18.8063	25.3824
Poisson ratio, σ (GPa)	0.47	0.47			0.245	0.232
Pugh ratio, B/G (GPa)	2.01	1.86	2.96	5.2	1.626	1.532
Cauchy pressure C^P (GPa)	18	17.9
Transverse sound velocity (m s^−1)	2092	1774
Longitudinal sound velocity (m s^−1)	3203	2874
Average sound velocity (m s^−1)	2293	1956
Temperature ΘD (K)	193.4	164.3	197.72	148.81

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The compounds are classified as ductile (B/G > 1.75) and brittle (B/G < 1.75) based on the critical limit of 1.75 of Pugh's ratio (B/G). The ductile criterion for the materials is defined by the Poisson ratio (υ), which is defined as σ greater than 0.26.⁴⁰ It has been demonstrated that Cs₂YAuI₆ is more ductile than Rb₂YAuI₆ due to its greater B/G and υ values. Using the Navier equation of states, we can also compute the sound velocity as the mean of the transverse and longitudinal components of the velocities.^41,42 However, Rb₂YAuI₆ has a greater value than Cs₂YAuI₆. This directly affects the Debye temperature, which is determined using the following formula:

where ρ denotes material density, M denotes molecular mass, υ_m stands for average sound velocity, N_A stands for Avogadro number, and k_B represents the Boltzmann constant. Table 1 displays the computed value of the Debye temperature, which is greater for Rb₂YAuI₆ than Cs₂YAuI₆. The specific heat capacity is directly impacted by the Debye temperature. The calculated results clearly show that Rb₂YAuI₆ has a greater capacity to withstand the heat produced by the lattice vibration.^38,43 Moreover, the compounds' thermodynamic stability is dependent on an increasing temperature.

4. Electronic properties

One of the most crucial electronic properties that aids in understanding a compound's device applications is its band gap, which offers a thorough explanation of its optical, electronic, and thermodynamic characteristics. We calculated the band structures of Rb₂YAuI₆ and Cs₂YAuI₆ by using PBE-GGA, TB-mBJ potential with and without spin orbit coupling (SOC) in order to analyze the electronic structure properties, as depicted in Fig. 3(a–f). The numerical values of the electronic band structure are presented in Table 3. Furthermore, as shown in Fig. 3(a–d), we identified substantially underestimated energy band gap values for the examined compounds utilizing the PBE-GGA technique. Based on this, the Tb-mBJ functional is a better exchange correlation functional for improving the optoelectronic features.⁴⁴ The energy band gap of 1.77 eV and 1.72 eV has been observed for Rb₂YAuI₆ and Cs₂YAuI₆, respectively, by the Tb-mBJ potential. Applying SOC is essential in order to acquire the right bandgap values in relation to the values derived from experiments.⁴⁵ Consequently, we employed mBJ + SOC potentials to explore the Cs₂YAuI₆ (E_g = 1.73 eV) and Rb₂YAuI₆ (E_g = 1.70 eV) bandgap values. The band gap decreased as the atomic numbers of A (Rb, Cs) increased. The well-known inter-site exchange interactions can be connected to this. This has a direct relationship with the coinciding orbital overlaps, which scale inversely with the lattice constants.⁴⁶ Furthermore, the bands have been divided by the spin–orbit coupling (SOC). It was found that the SOC splitting inside the valence/conduction bands was stronger, and decreased the band gap width with bigger elemental components.

Fig. 3 Representations of the band structures of A₂YAuI₆ (A = Rb, Cs) using (a, b) PBE-GGA, (c, d) TB-mBJ and (e, f) mBJ + SOC approximation.

Table 3 Calculated energy bandgap (in eV) of A₂YAuI₆ (A = Rb, Cs) by different potentials, PBE, mBJ and mBJ + SOC

Properties	PBE	mBJ	mBJ + SOC
Rb2YAuI6	1.07	1.77	1.73
Cs2YAuI6	1.06	1.72	1.70

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Total Density of States (TDOS) has been computed for the purpose of verifying the results of the energy band structure study. The Fermi level is shown by a vertical dashed line in all partial density of states (PDOS) and total density of states (TDOS) graphs. The energy plots range from −10 eV to 10 eV. Conduction band (CB) states appear at positive energy, whereas the valence band (VB) states are present at negative energy. The total of the individual states is shown by TDOS in both conduction and valence bands resembling the band structure. The total density of states TDOS results are depicted in Fig. 4(a and b).

Fig. 4 Representations of the TDOS of (a) Rb₂YAuI₆, (b) Cs₂YAuI₆ using Tb-mBJ approximation.

The results of PDOS for the studied compounds that describe the orbital contribution to the electron conduction mechanism are elaborated in Fig. 5(a and b). For Rb₂YAuI₆, the valence band formation is due to hybridization of the Au-5d and I-5p states. Meanwhile, the conduction band is occupied due to the hybridization of the Y-4d and Rb-5p states with a negligible contribution of the Au-5p states. In the case of Cs₂YAuI₆, the valence band is formed due to hybridization of electrons from I-5p and Au-5p and a nominal contribution of Cs-6s. However, the conduction band is occupied by hybridization of the Y-4d and Cs-4d states and a negligible contribution of the I-5p and Au-5p states.

Fig. 5 Representations of the PDOS of (a) Rb₂YAuI₆, (b) Cs₂YAuI₆ using Tb-mBJ approximation.

4.1 The ELF and bader charge analysis

To investigate the nature of the bond identified between the constituents of the examined material, the electron localization function (ELF) is determined,⁴⁷ as illustrated in Fig. 6(a and b). The ELF is based on the probability density of locating an electron at a certain position in space, and offers information about how localized or delocalized the electrons are. There is no charge between the components of an ionic connection, in contrast to atoms in covalent and metallic bonds sharing a charge. Fig. 6 shows the spatial charge configuration for the Rb₂YAuI₆ and Cs₂YAuI₆ perovskites.

Fig. 6 Representations of the ELF of A₂YAuI₆ (A = Rb, Cs) along the 101 plane direction using PBE-GGA approximation.

There are no charge contours that cross over to overlap with the I atoms in the perfectly spherical charge distribution between Rb and Y/Au. It was shown that there was an ionic connection between the I atom and the Cs, Rb atoms. In contrast, the Rb charge distribution varies from perfectly spherical to distorted, leading to a covalent interaction with the Cs atom (dumbbell type). It has been established that I accepts electrons from the Au network due to its covalent characteristics.

5. Optical properties

The optical properties were thoroughly explored in order to identify the importance of the computed material for solar cell applications. The optical performance of the transition study from the valence to the conduction band was examined. The optical properties are illustrated by the relative positions of light and materials. The intensity of light emission and absorption for optoelectronic devices is determined by the inter- and intra-band transitions.

The complex dielectric function (ε(ω)), absorption coefficient (α(ω)), complex refractive index (n(ω)), reflectivity R(ω), loss function L(ω) and optical conductivity σ(ω) have all been examined in relation to the optoelectronic features. The complex dielectric (CD) functions⁴⁸ are represented as follows:

ε(ω) = ε₁(ω) + iε₂(ω) (5)

Fig. 7(a and b) displays the energy value 0–12 eV of the CD function plot. The dielectric constant has two parts: the imaginary part is denoted by ε₂(ω), and the real part represents ε₁(ω). The real and imaginary components of the dielectric function are related by the most familiar Kramers–Kronig equations,⁴⁹ which are given as:

Fig. 7 The calculated energy-dependent optical parameters: (a) real part of the dielectric function, (b) imaginary part of the dielectric function, (c) refractive index coefficient, and (d) extinction coefficient of A₂YAuI₆ (A = Rb, Cs).

The imaginary portion is indicated as,

Here, e indicates the electric charge, u is the electric field, and E^c_k, E^ν_k and E stand for the energies of photons in the conduction band, valence band, and total photonic energy, respectively. The most significant portion of the ε₁(ω) spectrum is the 0 Hz limit ε₁(0), which only specifies the dielectric function's electronic factor. The static dielectric function values of 4.06 and 4.11 have been observed for Rb₂YAuI₆ and Cs₂YAuI₆, respectively. The maximum peaks examined for Rb₂YAuI₆ and Cs₂YAuI₆ are 3.98 and 3.88 eV, respectively. It is important to note that the static dielectric function and electronic energy band gap are associated with each other, as shown in Table 4, according to the Pens model.⁵⁰

Table 4 Calculated optical properties of A₂YAuI₆ (A = Rb, Cs) by mBJ-GGA compared with literature

	Material property	Rb2YAuI6	Cs2YAuI6	Rb2ScInI6^24	Cs2ScInI6^24	Rb2LiGaI6^27	Cs2LiGaI6^27
Optical properties	ε 1(0)	4.06	4.11	21.12	17.82	3.3	3.45
	n(0)	2.01	2.03	4.59	4.22	1.83	1.85
	R(0)	0.113	0.115

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Possible light-energy absorptions leading to an inter-band transition have been associated with ε₂(ω). This can be seen in Fig. 7(b), where the visible portions of the two computed compounds are the primary focus of the light absorptions. Cs₂YAuI₆ has a smaller bandgap that polarizes electron transfer in the conductance band and improves the photovoltaic outcomes. The methodical examination of ε₁(ω) and ε₂(ω) revealed that Rb₂YAuI₆ and Cs₂YAuI₆ have shown sufficient energy to absorb light in the broad range of wavelengths from 2–5 eV. Fig. 7(c and d) displays the complex refractive index. The real part of n(ω) in any medium of electromagnetic wave (EM) is the measurement of the phase velocity. Additionally, the imaginary component k(ω) arises as a result of an electromagnetic wave propagating through the same medium. The calculated static value n(0) has been observed at 2.01 and 2.03 for Rb₂YAuI₆ and Cs₂YAuI₆, respectively. The imaginary k(ω) and real n(ω) components of the maximum estimated values for Rb₂YAuI₆ are observed at 2.47 eV and 2.64 eV, respectively. Upon converting the composition to Cs₂YAuI₆, these values changed to 1.48 eV and 2.66 eV, respectively. When some halides are exposed to a specific frequency of photon radiation, the electrons begin to condense.

The electronic conduction causes this optical conductivity (σ(ω)), which is depicted in Fig. 8(a). The first peak of σ(ω) has been observed for Rb₂YAuI₆ at 718 (ω cm)⁻¹ at 2.54 eV, and the highest value of conductivity is found at 4772 (ω cm)⁻¹ at 7.68 eV. On the other hand, the first peak of 721 at 2.21 eV with a maximum conductivity is 4533 at 9.12 eV for Cs₂YAuI₆.

Fig. 8 The calculated energy-dependent optical parameters: (a) optical conductivity, (b) absorption coefficient, (c) optical reflectivity, and (d) energy loss function of A₂YAuI₆ (A = Rb, Cs).

The distance is measured prior to the absorption of light at a particular frequency. It is clear that the absorption coefficient spectrum α(ω) for Rb₂YAuI₆ begins at 2.21 eV and peaks at 10.54 eV as s displaced in Fig. 8(b). However, for Cs₂YAuI₆, α(ω) begins at 2.20 eV and reaches its maximum value at 10.20 eV. When the approaching photons reach the absorption edge, a dramatic increase in the absorption coefficient values is also noticed. Meanwhile, the optical absorptions of Cs₂LiGaI₆ and Rb₂LiGaI₆ have their initial maximum absorption bands and conductivity peaks in the visible energy band of (2.8–3.0 eV) and (2.65–2.8 eV), respectively.²⁷ It has been observed that our examined compounds can absorb electromagnetic radiation from the (UV-visible) region with wavelengths of 119–562 nm for Rb₂YAuI₆ and 122–565 nm for Cs₂YAuI₆, as shown in Fig. 8(b). This further supports the validity of these materials in the solar energy industry. The results of the optical absorption show that both compounds can absorb a wide UV-visible range of electromagnetic radiation in accordance with their electronic behavior, making it a potential candidate for optoelectronic and solar cell applications.

Similar to the previous discussion of optical parameters, the reflectivity R(ω) and optical loss L(ω) are significant variables, and their values should be low for the best optical materials. The surface morphology of the studied compounds is depicted by R(ω). The computed R(ω) values are displayed in Fig. 8(c). For Rb₂YAuI₆ and Cs₂YAuI₆, their values at zero energy are 0.113 and 0.115, respectively. The maximum value of reflectivity has been observed at 10.62 eV (0.34) and 10.38 eV. The light scattering and dispersion are displaced in Fig. 8(d), as illustrated by L(ω). It is worth mentioning that minimal energy loss has been observed in the visible region, which shows that these materials are suitable for solar cell applications.

6. Thermoelectric properties

Based on the figure of merit (ZT) provided by ,^51–53 we have evaluated the thermoelectric efficiency of our examined compounds. The enhanced Seebeck coefficient (S), higher electrical conductivity (σ), and lowered thermal conductivity (κ) describe an effective thermoelectric compound. For Rb₂YAuI₆ and Cs₂YAuI₆, we have calculated their thermoelectric properties in relation to the temperature T (K), chemical potential μ (eV) and carrier concentration N (e/u.c) using the Boltztrap code.³⁶ All calculated transport parameters at room temperature, i.e., 300 K, are tabulated in Table 5.

Table 5 Calculated transport properties of A₂YAuI₆ (A = Rb, Cs) by mBJ-GGA

	Material property	Rb2YAuI6	Cs2YAuI6
Transport properties (300 K)	σ/τ (10^18 Ω^−1 m^−1 s^−1)	9.14	8.68
	S (μV K^−1)	207	212
	R H (10^−8 m^3 C^−1)	1.96	2.04
	κ e (10^14 W m^−1 K^−1 s^−1)	1.57	1.53
	S ^2 σ/τ (10^11 W K^−2 m^−1 s^−1)	3.93	3.89

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The carrier's concentration has been used to classify the p and n-type features. The energy required to add or remove electrons from the material is indicated by μ. When electrons are introduced into the system, μ is positive. When they are taken out of the system, the μ is negative. The system operates as an n-type material when electrons are introduced. When electrons are removed, the system becomes more hole-filled, indicating that the materials under study are p-type.⁵⁴ Additionally, Fig. 4 shows that the DOS are more heavily occupied in the valence band than in the conduction band, confirming the p-type character of holes and their function as dominating charge carriers. The possibility that carriers will exist is represented by the density of states. Compared to the conduction band, the valence band contains a higher density of closely spaced states, indicating that there are more carriers available there. In semiconductors, the behavior becomes p-type if there are more holes (VB) than electrons (CB).

The Seebeck coefficient (S) may be expressed as S = ΔV/ΔT, where ΔT denotes the temperature gradient. The Seebeck coefficient is the material's capacity to produce induced emf when a temperature differential is applied. Fig. 9(a) displays the computation of the Seebeck coefficient (S) alongside T. As the temperature rises, the value of S decreases. For two complexes, the value of S is positive, indicating that the majority of the charge carriers in those materials are holes. The calculated values of S at room temperature are 207.3 μV K⁻¹ (Rb₂YAuI₆) and 211.6 μV K⁻¹ (Cs₂YAuI₆). Fig. 9(b) shows the Seebeck coefficient for the given material at different Kelvin temperatures versus chemical potential. It is clear that the Seebeck coefficient shows distinct peaks and slopes over the whole chemical potential range.

Fig. 9 Variation of the band structures of A₂YAuI₆ (A = Rb, Cs) with temperature using TB-mBJ approximation: (a) Seebeck coefficients, (b) and (c) Seebeck coefficients against chemical potential, (d) electrical conductivity, (e) and (f) electrical conductivity versus chemical potential.

The chemical potential of the p-type region has the highest value, 2925 μV K⁻¹ for (Rb₂YAuI₆) and 2897 μV K⁻¹ for (Cs₂YAuI₆) at room temperature. The chemical in consideration exhibits a greater Seebeck coefficient value due to its large band gap and restricted carrier mobility. However, at higher temperatures, S is somewhat dropped to 719 μV K⁻¹ and 708 μV K⁻¹ for Rb₂YAuI₆ and Cs₂YAuI₆, respectively. The bipolar effect, which continues when bound electrons are propelled by thermal energy and create electron–hole pairs, is responsible for the drop in the Seebeck coefficient at higher temperatures. These findings entirely agree with the Mott relationship, which measures how much temperature and chemical potential influence the Seebeck coefficient.⁵⁵ The current material may find application in temperature sensors, thermocouples, thermoelectric coolers, and generators due to its non-zero value of S at high temperatures.

The material flow is caused by electronic movement (k_el) and lattice vibration (k_ph), which also results in thermal conductivity (k_e/τ).⁵⁶ Electrons are triggered by elastic waves produced by lattice vibration. Therefore, when estimating the electronic component of k_e/τ, we ignored the phononic element of the thermal conductivity. Fig. 9(c) shows the thermal conductivity against the chemical potential, while Fig. 9(d) shows the behavior of the thermal conductivity against temperature. The k_e/τ rises as the temperature rises because a higher temperature increases the carrier mobility, which in turn improves the electronic movement and raises the electronic thermal conductivity.

The electrical conductivity (σ/τ) versus chemical potential m (eV) is represented in Fig. 10(b and c), whereas the electrical conductivity versus temperature (K) is shown in Fig. 10(a). The chemical potential, whose value is 0 at the Fermi level, is the necessary amount of energy to get the electrons included in the circuit for overcoming the Coulomb potential. The chemical potential could either have a positive or negative value. The p-type behavior is expressed by the negative value, and the n-type reaction is represented by the positive value.^57,58 A greater number of conduction electrons are present at high energies, as indicated by the high values of σ/τ between 0 and −0.7 and −0.8 eV for Rb₂YAuI₆ and Cs₂YAuI₆, respectively. The σ/τ peak for Rb₂YAuI₆ indicates a large intensity (which means it includes more conduction electrons than Cs₂YAuI₆), which is in accordance to previous studies.²⁴

Fig. 10 Variation of the band structures of A₂YAuI₆ (A = Rb, Cs) with temperature using TB-mBJ approximation: (a) electronic thermal conductivity, (b) and (c) electronic thermal conductivity against chemical potential, (d) power factor, (e) and (f) power factor versus chemical potential.

The power factor (S²σ/τ) computed to guess the thermoelectric strength of the examined compounds versus temperature and chemical potential is displayed in Fig. 10(a and b). As the temperature rises, the power factor increases linearly, as demonstrated by the sharp decline in the value of the Seebeck coefficient depicted in Fig. 9(a).

A factor known as the figure of merit is in charge of the compound thermoelectric activity. Furthermore, by dividing the power factor by thermal conductivity, this factor is used to calculate the true thermoelectric efficiency.⁵¹ZT alongside temperature (T) is displayed in Fig. 11(a), and also against the chemical potential, as shown in Fig. 11(b and c) respectively. The maximum value of ZT is 0.98 at the chemical potential value of 0.48 eV for Rb₂YAuI₆ and 0.99 at 0.50 eV for Cs₂YAuI₆ at room temperature, 300 K. In other studies, the value of ZT decreased with the rise in temperature.²⁴ As a result, the compounds under study exhibit a high ZT at room temperature. The description above makes clear how crucial the analysis is for thermoelectric generators and other thermoelectric functions.

Fig. 11 The computed results of figure of merit by Tb-mBJ (a) ZT against temperature, (b) and (c) ZT versus chemical potential.

The Seebeck coefficients S and σ/τ versus N (e/u.c) with positive and negative doping in the materials are shown in Fig. 12(a–d). Increasing the doping and temperature (K) and σ/τ do not alter the graphs of the Seebeck coefficient. Thus, as illustrated in Fig. 12(a and b), we found that the Cs₂YAuI₆ molecule exhibits a significant variance in p-type doping from 0 to 0.2. Furthermore, for both compounds under study, a maximum accomplishment in σ/τ is observed at 1.1 N (e/u.c). The electronic thermal conductivity (k_e/τ) of the explored composites has been calculated in order to investigate the heat energy created by the materials. We investigated these materials against n-type and p-type doping at temperatures of 300, 700, and 1200 K. The electronic conductivity is presented in Fig. 13(a and b) against the carrier concentration. For the Rb₂YAuI₆ compound, we managed to achieve the maximum value at 300, 700, and 1200 K, which is 0.51 × 10¹⁵ W m⁻¹ K⁻¹ s⁻¹, 0.90 × 10¹⁵ W m⁻¹ K⁻¹ s⁻¹ at −1.1 N (e/u.c) and 1.05 × 10¹⁵ at −1.86 N (e/u.c). Similarly, Cs₂YAuI₆ has values at 300, 700, and 1200 with 0.50 × 10¹⁵ W m⁻¹ K⁻¹ s⁻¹, 0.87 × 10¹⁵ W m⁻¹ K⁻¹ s⁻¹ at −1.1 N (e/u.c), and 1.02 × 10¹⁵ at −1.86 N (e/u.c).

Fig. 12 Accomplished thermoelectric properties (a, b) S and (c, d) σ/τ against the carrier concentration of A₂YAuI₆ (A = Rb, Cs) using the TB-mBJ approximation.

Fig. 13 Accomplished thermoelectric properties (a, b) κe/τ and (c, d) PF against the carrier concentration of A₂YAuI₆ (A = Rb, Cs) using the TB-mBJ approximation.

The PF for both double perovskites is shown in Fig. 13(c and d) at various temperatures, including 300, 700, and 1200 K at 1 N (e/u.c). For n-type doping, there is no PF, but both materials achieve values close to the Fermi level at 1200 K, which is 7.44 × 10¹¹ and 7.40 × 10¹¹ W m⁻¹ K⁻² s⁻¹ for Rb₂YAuI₆ and Cs₂YAuI₆, respectively.

Fig. 14(a and b) shows that for both compounds under study, the ZT values at 0 N (e/u.c) are 1, but that the ZT values sharply decline with increasing p-type doping and linearly decrease with increasing n-type doping.

Fig. 14 Calculated thermoelectric properties figure of merit ZT against the carrier concentration of (a) Rb₂YAuI₆, (b) Cs₂YAuI₆ using TB-mBJ approximation.

7. Conclusion

In our study, we computed the opto-electronic and thermoelectric features of A₂YAuI₆ (A = Rb, Cs) using DFT calculations. The PBE generalized gradient approximation has been implemented to optimize the structural parameters. In order to acquire appropriate energy band gaps, Tb-mBJ with and without SOC schemes were used instead of PBE-GGA. It has been established that Rb₂YAuI₆ is more ductile than Cs₂YAuI₆ due to its higher Pugh's and Poisson's ratio values. The materials have indirect bandgaps of 1.73 and 1.70 eV for Rb₂YAuI₆ and Cs₂YAuI₆, respectively, which are potential materials for the manufacturing of solar cell devices and lie in the UV-visible region of the EM spectrum. The negative values of the formation energy and tolerance factor show that the examined materials are thermodynamically stable. The ELF plots demonstrate that there is an ionic connection between Rb/CS, I, and Au. The solar cells are favored by the lowest values of optical loss and reflection, combined with significant visible light attenuation. Thermoelectric generator applications benefit greatly from the huge values of ZT, Seebeck coefficient, and PF at low temperatures of 300 K, as well as the ultralow value of the lattice vibration. As a result, these compounds will provide the experimental community with profound insight into how to implement them to advance the solar cell and thermoelectric generator industries.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

M. M. is supported by APVV-21-0272, VEGA-2/0070/21 Project, Slovakia.

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