Determination of the light matter interaction and thermal heat conversion efficiency of K₂TlAsZ₆ (Z = F, Cl, Br, and I) using a DFT approach

Abstract

Double perovskites have garnered significant attention as promising alternatives for sustainable energy solutions owing to their structural versatility and potential for integration into optoelectronic technologies. Halide double perovskites K₂TlAsZ₆ (Z = F, Cl, Br, and I) were systematically studied using density functional theory to assess their potential for optoelectronic and thermoelectric applications. Calculations were performed using the FP-LAPW + lo method, confirming structural and thermodynamic stability via formation energy, and the Goldschmidt's tolerance factor. The electronic structure calculations using the TB-mBJ + SOC potential revealed direct bandgaps ranging from 3.25 eV (Z = F) to 0.37 eV (Z = I), with significant UV absorption observed in the optical spectra. Thermoelectric performance versus chemical potential, evaluated via the Boltzmann transport theory, showed promising ZT values approaching 1.0 at 1000 K. Additionally, negative Gibbs free energy and increasing entropy with temperature indicate good thermal stability. These results suggest that K₂TlAsZ₆ compounds are promising materials for next-generation optoelectronic and thermoelectric devices.

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1 Introduction

Perovskite-based materials have attracted extensive attention in recent years due to their diverse physicochemical properties and applicability in photocatalysis, optoelectronics, thermoelectric, and spintronics.^1,2 Lead-based halide perovskites have achieved power conversion efficiencies above 25%, surpassing thin-film silicon and copper indium gallium selenide solar cells.³ Nevertheless, environmental concerns regarding lead toxicity have driven the search for non-toxic alternatives, leading to the development of lead-free double perovskite halides with competitive optoelectronic performance.⁴

Halide perovskites crystallize in the ABX₃ form, while double perovskites adopt A₂BB′X₆, where A is a monovalent cation, B and B′ are metal cations of different oxidation states, and X is a halide anion.⁵ Lead-free double perovskites (DPs) display distinctive electronic and optical characteristics, making them promising for renewable energy applications.⁶ For instance, Rb₂AgAsX₆ (X = Br, I, and Cl) shows tunable band gaps via halide substitution,⁷ and Cs₂ScTlX₆ (X = Cl, Br, and I) exhibits mechanical stability in cubic symmetry as confirmed by elastic constants and tolerance factors.⁸ Cs₂LiYCl₆, synthesized for thermal neutron detection, demonstrates potential as a scintillator material.⁹ Computational investigations reveal favorable optoelectronic behavior in Rb₂AgBiI₆,¹⁰ while X₂AgBiI₆ (X = Cs, K, and Rb) offers high thermoelectric figures of merit with advantageous optical properties.¹¹ Cs₂SnI₆ has been examined as a lead-free absorber for solar applications,¹² and Cs₂BiAgX₆ (X = Br and Cl) prepared via solid-solution synthesis exhibits direct band-gap behavior with modified transport characteristics.¹³

Within this class, potassium-based perovskites are emerging as particularly promising due to their stability and favorable electromagnetic, thermoelectric, and optoelectronic properties.¹⁴ Potassium doping has enabled the fabrication of hysteresis-free solar cells,¹⁵ while high-efficiency, low-hysteresis devices have been reported using potassium-based perovskites.¹⁶ Additionally, potassium thiocyanate interlayers have been proposed for eco-friendly indoor and outdoor photovoltaics.¹⁷ These findings underscore the potential of K-based halides for large-scale deployment in sustainable energy technologies,¹⁸ motivating continued research into novel potassium-based perovskite compositions for advanced solar energy conversion and storage applications.

Munir et al. (2025) have thoroughly examined the structural, electronic, mechanical, optical, and thermoelectric properties of K₂TlAsX₆ (X = Cl and Br) double perovskites, employing first-principles calculations within the framework of density functional theory (DFT). The electronic band structure was computed using the mBJ exchange potential, both with and without the inclusion of spin orbit coupling (SOC) effects, to account for relativistic interactions. Structural stability was confirmed through negative formation energies, optimized lattice parameters, positive phonon frequencies, and tolerance and octahedral factors within the permissible range for stable perovskite formation. Elastic constants were determined and the associated mechanical parameters, including elastic anisotropy, Pugh's ratio, Poisson's ratio, and Cauchy pressure, were derived to assess the ductility and mechanical anisotropy. Both compounds exhibit direct band gaps, with K₂TlAsCl₆ showing gaps of 2.22 eV (modified Becke–Johnson, mBJ) and 1.59 eV (mBJ + SOC), while K₂TlAsBr₆ presents gaps of 1.97 eV and 1.51 eV, respectively. The optical properties reveal strong absorption in the visible and ultraviolet regions, suggesting potential for optoelectronic integration. Moreover, electronic transport analysis indicates high electrical conductivity, favorable thermoelectric figure of merit (ZT) and low lattice thermal conductivity, making these halide perovskites promising candidates for energy-efficient and green optoelectronic applications.¹⁹ Although K₂TlAsZ₆ are designed as lead-free alternatives, it should be noted that these systems contain thallium (Tl) and arsenic (As), both of which are toxic and pose environmental and health risks if improperly handled. Therefore, the term “lead-free” in this context refers specifically to the absence of lead, a well-known hazardous element in conventional perovskites, rather than indicating complete non-toxicity.

The aim of the present work is to assess the structural, optoelectronic, thermoelectric, and thermodynamic properties of K₂TlAsZ₆ (Z = F, Cl, Br, and I). Our findings are intended to support further theoretical and practical studies of K₂TlAsZ₆ (Z = F, Cl, Br, and I) halide double perovskites, with an emphasis on their possible applications in thermoelectric and optoelectronic devices.

2 Methods of simulations

Utilizing the WIEN2K algorithm,²⁰ a density functional theory (DFT) method based on the FP-APW + lo methodology was used to study the optoelectronic and transport characteristics of K₂TlAsZ₆ (Z = F, Cl, Br, and I). PBEsol-GGA, a generalized gradient approximation, was used in particular to optimize the structural parameters.²¹ To assess the structural and dynamic stability of the compounds being studied, we computed the Goldschmidt tolerance factor, also known as the enthalpy of formation. We used PBEsol-GGA, in combination with the Trans and Blaha modified Becke–Johnson techniques (TB-mBJ), along with the SOC to precisely determine the required band gaps.²² The electrical system's solution is regarded as spherically harmonic inside the muffin-tin sphere, but it exhibits plane wave-like behavior in the interstitial area. The initial parameters evaluated were the Gaussian parameter G_max = 16, the product of the wave vector and muffin radius K_max × R_MT = 8, and the angular momentum I = 10 in the reciprocal lattice. In a k-mesh of order 10 × 10 × 10, the correctness of the convergence criterion was tested using the 1000 k-point, where the energy emitted from the system remains constant.²³ At 0.01 mRy, charge convergence is attained. Furthermore, we used the convergent energy and optimum electronic structures using the TB-mBJ method to calculate the optical and photovoltaic characteristics. The BoltzTraP algorithm²⁴ with the combination of DFT and semiclassical transport theory is implemented in WIEN2k to compute the electronic transport coefficients of the studied compounds.

3 Results and discussion

3.1 Stability criteria for K₂TlAsZ₆ (Z = F, Cl, Br, and I)

To assess a structure's stability, we can compute Goldsmith's tolerance factor using the formula:²⁵where R_(K), R_Tl, R_As, and R_(F/Cl/Br/I) are the Shannon ionic radii for the potassium, thallium, arsenic, and halide atoms, respectively, in this context. Under ideal conditions, Goldsmith's criteria for stable cubic perovskites often approaches unity. Subsequently, the range of T_F values among 0.8 and 1.0 is decisive for stable cubic arrangements. As seen in Table 1, our calculated values fall within this steady range. Consequently, the halides under investigation exhibit stability in the cubic phase. K₂TlAsF₆ is at the optimum limit compared to K₂TlAsCl₆, K₂TlAsBr₆, and K₂TlAsI₆, according to a comparison analysis (see Fig. 1a). In order to show the materials' thermodynamic stability, the subsequent chemical expression is used to compute the formation enthalpy:

ΔH_F = E_{(K)aTlbAsc(F/Cl/Br/I)d} − aE_Kb − bE_Tl − cE_As − dE_(F/Cl/Br/I) (2)

E_Coh = [aE_K + bE_Tl + cE_As + dE_(F/Cl/Br/I)] − E_{(K)aTlbAsc(F/Cl/Br/I)d} (3)

where E_K, E_Tl, E_As and E_(F/Cl/Br/I) signify the energies for the individual atoms K, Tl, As, and Z, respectively, and E_{(K)aTlbAsc(F/Cl/Br/I)d} represents the total energy of the compounds under investigation. The conforming atoms is indicated by the subscripts a, b, c, and d in the equation above. Essentially, the negative value of ΔH_F indicates the amount of energy squandered throughout the material creation process. The materials' stability is essential. Table 1 displays the calculated values and shows the thermodynamic stabilities of the compounds under study.^1,26 The conclusions evidently indicate that K₂TlAsF₆ exhibits more stability than K₂TlAsCl₆, K₂TlAsBr₆, and K₂TlAsI₆ (see Fig. 1 (b)).

Table 1 Structural parameters and stability criteria for K₂TlAsZ₆ (Z = F, Cl, Br, and I)^a

Compounds	K2TlAsF6	K2TlAsCl6	K2TlAsBr6	K2TlAsI6
a Ref^A.^19
a0 (Å)	9.38	10.83	11.31	12.04
		11.00^A	11.53^A
V0 [a.u.]^3	1394.21	2144.88	2443.12	2945.71
		2586.71^A	2588.00^A
B (GPa)	43.48	26.96	23.51	19.09
		24.00^A	20.31^A
B'	5.00	5.00	5.00	5.00
		4.86^A	4.83^A
E0 [Ry.]	−48707.17	−53021.15	−78750.39	−132884.23
		−53047.39^A	−78789.00^A
τF	1.0	0.96	0.95	0.93
		0.88^A	0.87^A
ΔHF (eV/atom)	−1.90	−1.41	−1.10	−0.67
		−1.92^A	−1.61^A
ECohesive	1.90	1.41	1.10	0.67
ρ (g cm^−3)	3.79	2.98	3.84	4.26

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Fig. 1 (aand b) Tolerance factor (T_F), formation (ΔH_F) and cohesive energies of K₂TlAsZ₆ (Z = F, Cl, Br, and I).

3.2 Structural properties

All calculations were conducted utilizing the crystal arrangements of K₂TlAsZ₆ (Z = F, Cl, Br, and I) halides, which are categorized within the space group Fm3̄m#225 (see Fig. 2a). We utilized PBEsol-GGA to optimize halide DPs (see Fig. 2b–f) to determine the fundamental state lattice parameters, applying Birch Murnaghan's equation of states.²⁷ We determined the fundamental state lattice constant a₀ (Å) and bulk modulus B (GPa) by volume optimization, as demonstrated in the following Table 1. The values of a₀ show an upsurge from K₂TlAsF₆ to K₂TlAsI₆, which can be explained by the increase in atomic radii from F to I. The increase in distance between the atoms is a result of the larger cation size, which subsequently leads to a decrease in the density and strength of the compounds. Therefore, a drop in the bulk modulus B₀ is observed by substituting F with Cl, Br and I.²⁸

Fig. 2 (a–f) Crystal structure, E–V parabolic curves, and unit cell volume and energy curves with percentage change in its volume for K₂TlAsZ₆ (Z = F, Cl, Br, and I).

3.3 Electronic properties

Using the modified Becke–Johnson (TB-mBJ) potential, the band structures of K₂TlAsZ₆ (Z = F, Cl, Br, and I) are illustrated in Fig. 3a–d. It was essential to use the mBJ + SOC method to get precise band gap metrics that corresponded to the empirically observed outcomes.²⁹ The band gap values were assessed for K₂TlAsZ₆ (Z = F, Cl, Br, and I) from 3.25 eV to 0.37 eV using mBJ + SOC methods as compared to the bandgap data of (2.22 eV) for K₂TlAsCl₆ and (1.97 eV) for K₂TlAsBr₆ reported by Junaid Munir et al. (2025).¹⁹ With the valence band maximum (VBM) and conduction band minimum (CBM) located at the Γ high-symmetry point within the Brillouin zone, the compounds exhibit a direct band gap nature. Our findings suggest that these halides may find use in opto-electronic devices since their band-gap measurements lie within the IR to visible (VIS) and smaller UV spectrum range.

Fig. 3 (a–d) Band structure via mBJ + SOC for K₂TlAsZ₆ (Z = F, Cl, Br, and I).

For evaluating the electronic characteristics of the compounds being studied, the TDOS and PDOS, utilizing TB-mBJ, are shown in Fig. 4a–d. The total contribution of each state is shown as TDOS throughout the occupied and unoccupied states, much like the band structure. The interband transitions and hybridization are entirely the result of the occupied state electrons, whose roles in the valence bands are seen in Fig. 4a–d. (F/Cl/Br/I)-5p makes the largest contribution, with As-3p making a smaller one in the semi-core regions of the occupied state, where electrons migrate to the unoccupied state. In the conduction band As-3p states make a larger contribution. The band gap values for K₂TlAsZ₆ (Z = F, Cl, Br, I) are determined in the range 3.25 eV to 0.37 eV using TB-mBJ methods. The investigated materials exhibit a direct band gap, with both the valence band maximum (VBM) and conduction band minimum (CBM) located at the Γ high-symmetry point in the Brillouin zone. These materials are therefore ideally suited for photovoltaic and optoelectronic applications, including tandem solar cells,³⁰ photodetectors that detect UV light,³¹ sensors that detect infrared,³² as well as LEDs.³³

Fig. 4 (a–d) TDOS and PDOS using mBJ + SOC for K₂TlAsZ₆ (Z = F, Cl, Br, and I).

3.4 Optical properties

It is common practice to use the dispersion of both real and imaginary components of dielectric characteristics to derive crucial optical variables such as reflectivity R(ω) and refractive index n(ω), respectively. The absorption coefficient and optical conductivity can also be assessed using these metrics. The dielectric constant ε(ω), reflectivity R(ω), optical conductivity σ(ω), extinction coefficient K(ω), absorption coefficient α(ω), refractive index n(ω), and other optical characteristics of K₂TlAsZ₆ (Z = F, Cl, Br, and I) were all calculated and presented using the TB-mBJ + SOC potential,²⁶ as revealed in Fig. 5a–h. Our halides crystallize in a cubic form; therefore, the mathematical expression that follows³⁴ from dispersion band theory (DBT) validates the close relationship between the band structures and the imaginary part of the dielectric tensor.

Fig. 5 (a–h) Optical properties of K₂TlAsZ₆ (Z = F, Cl, Br, and I) using the mBJ + SOC method.

In this equation, m is the mass, ‘e’ is the electron's charge, and “ω” is the electromagnetic radiation's angular frequency. Conversely, optical transitions are indicated by the dipole matrix element M_ck(k) = (u_ck|e∇|vk).²⁸ The real part of the dielectric function is found using the Kramers–Kronig equation, as explained in the next section:

The real component of the dielectric function is represented by ε₁(ω). The resonant frequencies for K₂TlAsZ₆ (Z = F, Cl, Br, and I) correspond to energy values ranging from 1.76 eV to 3.20 eV, where peak scattering of light occurs (Fig. 5a). The peaks exhibit a rapid decline to their lowest levels after achieving resonance. The zero-frequency limit of the real part of the dielectric function, ε₁(0), which corresponds to the electronic contribution to the static dielectric constant, is a critical parameter in the optical spectrum. The calculated ε₁(0) values for K₂TlAsZ₆ (Z = F, Cl, Br, and I) are summarized in Table 2.^35,36 An inverse dependence is observed between the band gap energy (E_g) and the static dielectric constant ε₁(0). For the investigated K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds, ε₁(0) exhibits an initial increase from zero frequency, attains a maximum, and then gradually decreases. With increasing photon energy, additional distinct peaks emerge, as depicted in Fig. 5a. These spectral features, or kinks, are primarily located within the photon energy range of 1.76 eV to 3.20 eV. The maximum values of the real part of the dielectric function ε₁(ω) are found to be 3.22 at 4.93 eV for K₂TlAsF₆, 4.80 at 2.87 eV for K₂TlAsCl₆, 6.42 at 2.35 eV for K₂TlAsBr₆, and 9.90 at 1.72 eV for K₂TlAsI₆. The Penn model³² has been employed to estimate ε₁(0) and E_g, subject to the criterion for semiconducting materials, i.e., ε₁(0) >1.

Table 2 Static dielectric constant for K₂TlAsZ₆ (Z = F, Cl, Br, and I)^a

Compounds	ε1(0)	n(0)	R(ω)
a Ref^A.^19
K2TlAsF6	2.01	1.42	24.40% (10.73 eV)
K2TlAsCl6	3.54	1.88	24.14% (12.15 eV)
	3.18^A	1.76^A
K2TlAsBr6	4.83	2.19	30.58% (12.12 eV)
	3.75^A	2.94^A
K2TlAsI6	7.01	2.64	40.12% (12.47 eV)

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The imaginary part of the dielectric function, ε₂(0), is strongly influenced by the electronic band structure and plays a critical role in determining the optical absorption characteristics of the material. It provides insight into the interband electronic transitions. As the photon energy increases, transitions occur from the valence band, predominantly composed of halogen (F/Cl/Br/I)-5p orbitals, to the conduction band, which mainly consists of As-3p states. This transition consistently gives rise to the first prominent absorption peak observed across all investigated compounds. As illustrated in Fig. 5b, the second peak is also attributed to similar interband transitions. The calculated value of ε₂(ω) is determined from Fig. 5b, showing that all photons with energy E (eV) beneath the matching energy gaps (E_g) of K₂TlAsZ₆ (Z = F, Cl, Br, and I) have a constant value of zero. The threshold points in the spectra of ε₂(ω) are 3.80 eV, 1.74 eV, 1.11 eV, and 0.67 eV for K₂TlAsF₆, K₂TlAsCl₆, K₂TlAsBr₆, and K₂TlAsI₆, respectively, as compared to the highest peak observed at 3.9 eV for K₂TlAsCl₆ and 3.1 eV for K₂TlAsBr₆.¹⁹ A strong correlation is observed between the electronic band gaps and the corresponding peaks in the optical spectra. The calculated maximum values of the imaginary part of the dielectric function, ε₂(ω), for the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds are 2.99 at 5.51 eV for K₂TlAsF₆, 3.73 at 3.30 eV for K₂TlAsCl₆, 5.07 at 3.14 eV for K₂TlAsBr₆, and 8.63 at 2.40 eV for K₂TlAsI₆. Notably, the most prominent absorption features appear near 3.1 eV and 2.7 eV for K₂TlAsCl₆ and K₂TlAsBr₆, respectively, indicating a direct relationship with their optical transition energies.¹⁹ According to our accurate observations, both halides have remarkable visible spectrum photon absorption properties, which are essential for the efficiency of opto-electronic systems.

For optical components to be successfully integrated into devices like photonic crystals, waveguides, solar cells, and detection devices, an extensive knowledge of the refractive index is essential. Fig. 5c depicts the variation in the refractive index for K₂TlAsZ₆ (Z = F, Cl, Br, and I) as a function of the photon energy. The calculated static refractive index values, n (0), for each composition are listed in Table 2 in comparison to the data reported by Junaid Munir et al. (2025).¹⁹ As illustrated in Fig. 5c, the refractive indices of the investigated compounds exhibit an increasing trend from zero photon energy (0 eV) up to their respective maximum values of 1.87 (5.04 eV) for K₂TlAsF₆, 2.25 (3.08 eV) for K₂TlAsCl₆, 2.62 (2.32 eV) for K₂TlAsBr₆, and 3.24 (1.78 eV) for K₂TlAsI₆ as compared to the highest peak observed at 3.38 eV for K₂TlAsCl₆ and 2.79 eV for K₂TlAsBr₆, respectively.¹⁹ Following the attainment of its peak value, the refractive index for each compound exhibits a decline across specific energy intervals. This behavior suggests a transition in the optical response of the medium from linear to nonlinear, thereby imparting superluminal characteristics to the material.³³ The calculated refractive index values for the double perovskites K₂TlAsZ₆ (Z = F, Cl, Br, and I) are found to be equal to or greater than unity, indicating that these materials exhibit linear optical behavior. As depicted in Fig. 5c (left panel), the group velocity (V_g = c/n) remains positive, with no indication of a transition toward negative energy. A refractive index greater than one implies a decrease in photon velocity upon entering the medium, attributed to the interaction between incident photons and the electronic structure of the material, thereby inducing a temporal delay in photon propagation. In addition, the absorption coefficient k(ω), shown in Fig. 5d, characterizes the material's capacity to attenuate incident light. The spectral profile of k(ω) follows a similar trend to that of the imaginary component of the dielectric function, ε₂(ω). The peak values of k(ω) are identified as 0.99 (5.59 eV), 1.06 (4.39 eV), 1.29 (3.33 eV), and 1.72 (2.54 eV) for K₂TlAsZ₆ with Z = F, Cl, Br, and I, respectively.

Electronic conduction is the cause of the optical conductivity σ(ω). As displayed in Fig. 5e, the first peak values of σ(ω) are 2679.7 [Ω cm]⁻¹ at 9.51 eV, 4578.4 [Ω cm]⁻¹ at 12.88 eV, 4768.5 [Ω cm]⁻¹ at 9.34 eV, and 5351.3 [Ω cm]⁻¹ at 7.66 eV for K₂TlAsZ₆ (Z = F, Cl, Br, and I), whereas the highest peak ever seen is 5351.3 [Ω cm]⁻¹ at 7.66 eV for K₂TlAsI₆. In the ultraviolet light spectrum, the σ(ω) is expected to exhibit a significant rise in the 7.6 to 13 eV region, which is essential for optoelectronic applications. Consequently, K₂TlAsI₆ is the best option for these applications because of its exceptional values.³⁷ Fig. 5f shows the absorption coefficient α(ω) of K₂TlAsZ₆ (Z = F, Cl, Br, and I). The extent to which light may enter a material before it is fully absorbed is indicated by the symbol α(ω). Each substance has a threshold at which it stops absorbing light. When photons collide with valence-state electrons, photon–electron interactions take place, which, once the threshold value is exceeded, produce very efficient light absorption. The formula shows a significant decrease in energy.^38–40

α = 4πk/λ (7)

The materials K₂TlAsZ₆ (Z = F, Cl, Br, and I) display an upsurge in absorbance from E_g (eV), reaching utmost intensities between 8 to 12 eV. The cutoff value of the absorption edge is established in the following manner: 3.60 eV for K₂TlAsF₆; 1.82 eV for K₂TlAsCl₆; 1.30 eV for K₂TlAsBr₆; and 0.99 eV for K₂TlAsI₆ as compared to the data at 2.08 eV for K₂TlAsCl₆ and 1.69 eV for K₂TlAsBr₆, respectively.¹⁹ The apex standards for the α(ω) are 85.26 cm⁻¹ (9.61 eV) for K₂TlAsF₆; 141.04 cm⁻¹ (12.01 eV) for K₂TlAsCl₆; 155.01 cm⁻¹ (11.90 eV) for K₂TlAsBr₆; and 140.59 cm⁻¹ (9.59 eV) for K₂TlAsI₆, as compared to the highest peak observed at 4.01 eV for K₂TlAsCl₆ and 1.87 eV for K₂TlAsBr₆, respectively.¹⁹ The double perovskites under analysis show a notable rise in optical absorption in the blue (UV) shift. The in-depth examination of the spectral features confirms that K₂TlAsZ₆ (Z = F, Cl, Br, and I) has excellent absorption qualities, making it ideal for optoelectronic applications.

The energy emitted per unit area may be expressed as L(ω). Dispersion, thermal impacts, and plasmonic interaction are the principal factors influencing the optical loss L(ω). Fig. 5g exemplifies the reaction of the energy loss function L(ω). The most significant optical loss for K₂TlAsZ₆ (Z = F, Cl, Br, and I) is documented as 1.98 (10.92 eV), 0.77 (7.79 eV), 0.62 (5.97 eV), and 0.63 (4.77 eV), as compared to the highest peak observed at 7.8 eV for K₂TlAsCl₆ and 5.8 eV for K₂TlAsBr₆, respectively.¹⁹ Fig. 5g illustrates that the L(ω) reaches its minimum as the reflectivity approaches its maximum, and vice versa. In the purview of visible light absorption, the L(ω) decreases. The findings demonstrate that materials K₂TlAsZ₆ (Z = F, Cl, Br, and I) are suitable for photonic and solar power applications.

The surface reflectivity of a solid can be assessed through ε(ω), which measures the quantity of incident sunlight that is reflected.⁴¹ It may be represented as:

Fig. 5h illustrates the depicted reflectivities R(ω) of DP K₂TlAsZ₆ (Z = F, Cl, Br, and I). The R(ω) of the DPs vary at diverse light energies (in eV): 24.10% (10.70 eV) for K₂TlAsF₆, 23.19% (12.23 eV) for K₂TlAsCl₆, 30.61% (12.09 eV) for K₂TlAsBr₆, and 39.65% (12.47 eV) for K₂TlAsI₆, respectively.

3.5 Thermoelectric properties

3.5.1 Thermoelectric properties vs. temperature. Thermopower can elucidate a material's thermoelectric characteristics. When a change in the internal temperature of a substance induces an electromotive force, the phenomenon occurs. Electrical conductivity EC (σ), power factor (PF), Seebeck coefficient (S), figure of merit (ZT), and electronic thermal conductivity (κ_e) serve as principal metrics for elucidating thermoelectric properties through band structure analysis. Consequently, the values for K₂TlAsZ₆ (Z = F, Cl, Br, and I) were determined over the T (K) of 100–1000 K, as illustrated in Fig. 6a–e.

Fig. 6 (a–e) Thermoelectric properties vs. temperature of K₂TlAsZ₆ (Z = F, Cl, Br, and I) using the TB-mBJ method.

The generated electric potential in a compound when exposed to a temperature differential is predicted by the Seebeck coefficient (S), a thermoelectric characteristic. Fig. 6(a) for K₂TlAsZ₆ (Z = F, Cl, Br, and I) DPs displays the computed S versus temperature. All of the temperature ranges under consideration have positive values for S. Consequently, like the majority of DPs, K₂TlAsZ₆ (Z = F, Cl, Br, and I) DPs are p-type semiconductors.^42–44 Moreover, the S-value for DPs falls as the temperature escalates. Table 3 displays, accordingly, the calculated values of S for K₂TlAsZ₆ (Z = F, Cl, Br, and I). Fig. 6(a) shows that at 300 K, S values for K₂TlAsZ₆ (Z = F, Cl, Br, and I); at higher temperatures, the S values for the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds decline. There have also been reports of this kind of fluctuation in S for other DPs, when F is replaced at the X-site by Cl, Br or I, respectively.⁴⁵

Table 3 Thermoelectric data of the K₂TlAsZ₆ (Z = F, Cl, Br, and I) DPs^a

Compounds	K2TlAsF6		K2TlAsCl6		K2TlAsBr6		K2TlAsI6
Temperature	300 K	1000 K	300 K	1000 K	300 K	1000 K	300 K	1000 K
a Ref^A.^19
S (μV K^−1)	234.17	239.54	231.63	238.21	243.89	244.42	232.50	246.74
			238^A		254^A
σ (S m^−1)	38.77	213.50	24.77	131.59	16.28	98.17	15.52	87.30
κe (Wm^−1 K^−1)	85.93	1630.35	53.96	1000.00	38.19	780.64	34.22	756.44
PF	0.21	1.23	0.13	0.75	0.10	0.59	0.08	053
ZT	0.74	0.75	0.73	0.74	0.76	0.75	0.73	0.70
			0.75^A		0.99^A

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A crucial thermoelectric characteristic, σ/τ, measures the amount of charge carriers that may be transported through a material. Fig. 6(b) displays the computed σ/τ that is dependent on temperature for the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds. DPs follow the same trend as the other semiconductors, in that σ/τ grows linearly with temperature. This is because a greater number of charge carriers become accessible for conductivity as temperatures rise. Table 3 presents the σ/τ values calculated for K₂TlAsZ₆ (Z = F, Cl, Br, and I). The greater band gap is the main reason why K₂TlAsF₆ has greater σ values compared to other compounds. The relationship between the σ and the carrier concentration (N) may be expressed as σ = Neμ.

The temperature-dependent κ_e is shown in Fig. 6(c) as having greater values for the K₂TlAsF₆ compound compared to other compounds. One possible explanation is that K₂TlAsF₆ has a greater band gap than other compounds. The band gap has also been shown to cause this kind of κ_e change in other DPs.^45,46 On the other hand, similar to other types of semiconductors, the values of κ_e DPs grow as the temperature rises.^13,47 The highest κ_e values for K₂TlAsZ₆ (Z = F, Cl, Br, and I) at the temperatures of 300 K and 1000 K are summarized in Table 3.

The power factor (PF) precisely computes the thermoelectric functionality of a material, which is expressed by:

PF = S² × σ (9)

The relationship among the Seebeck coefficients (S) and electrical conductivity (σ) is described in eqn (9). Fig. 6d shows the PF vs. T (K) data for the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds, depicting the potential use of these compounds in thermoelectric devices with a PF of one or greater than unity.^48,49 The PF values increase linearly with the upturn in temperature from 300 K to 1000 K. The decrease in the (S) and surge in the EC (σ) may be due to the variation in the temperature, affecting the PF metrics for the K₂TlAsZ₆ (Z = F, Cl, Br, and I) DPs. The PF values at different temperatures are given in Table 3.

A material's figure of merit (ZT) is a compilation of its thermoelectric efficiency, which is determined as follows:⁵⁰

For the ZT values to be high, the denominator (κ_e) must be low, while S and σ must be high. Fig. 6(e) displays the computed ZT for K₂TlAsZ₆ (Z = F, Cl, Br, and I). When the temperature is elevated to 1000 K, the ZT standards for K₂TlAsZ₆ (Z = F, Cl, Br, and I) reach 0.75, 0.74, 0.75, and 0.70, respectively. A prominent aspect of DPs is this rising tendency in ZT with temperature.⁵¹ Although some predictable DPs have lower ZT standards, K₂TlAsZ₆ (Z = F, Cl, Br, and I) DPs have maximum ZT standards.⁵² Thus, K₂TlAsZ₆ (Z = F, Cl, Br, and I) are promising materials for thermoelectric power generation.

3.5.2 Thermoelectric properties vs. chemical potential. The transport properties of K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds were systematically investigated over a broad temperature range to assess their thermoelectric performance. The chemical potential (μ − E_F) was analyzed at fixed temperatures of 300 K, 500 K, 700 K, and 1000 K. In this context, the transport behavior associated with the heavy valence bands was evaluated within the framework of Boltzmann transport theory. The Boltz-Trap⁵³ computational tools were employed to determine the Seebeck coefficient, as well as the electrical and thermal conductivities, aiming to evaluate the power conversion efficiency of the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds. The Seebeck coefficient of the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds was calculated as a function of the chemical potential at fixed temperatures of 300 K, 500 K, 700 K, and 1000 K, as presented in Fig. 7a–d. A systematic decrease in the Seebeck coefficient is observed with increasing temperature.⁵⁴ The Seebeck coefficient is found to be notably higher for p-type carriers across the investigated compounds. For the series K₂TlAsZ₆ (Z = F, Cl, Br, and I), the maximum values of the Seebeck coefficient generally appear away from the Fermi level, except in the case of K₂TlAsF₆, where the peak is observed in closer proximity to the Fermi level. At 300 K, the highest Seebeck coefficients for the K₂TlAsZ₆ compounds are calculated as 265.20 μV K⁻¹, 254.63 μV K⁻¹, 182.39 μV K⁻¹, and 126.03 μV K⁻¹, for Z = F, Cl, Br, and I, respectively.⁵⁵

Fig. 7 Thermoelectric properties vs. chemical potential of K₂TlAsZ₆ (Z = F, Cl, Br, and I) using the TB-mBJ method.

Semiconducting compounds exhibit pronounced sensitivity to temperature gradients; whereby elevated temperatures promote the excitation of charge carriers from the valence band (VB) to the conduction band (CB). This thermal activation results in an increased carrier concentration across the VB and CB gap. The dependence of electrical conductivity (σ) on the chemical potential (μ − E_F) is presented in Fig. 7a–d.

As illustrated in Fig. 7a–d, the σ/τ ratio exhibits an increasing trend with temperature at lower chemical potentials. Conversely, at higher chemical potentials, σ/τ decreases as the temperature rises. This behavior is in agreement with the theoretical expectations for intrinsic semiconductors. For the investigated K₂TlAsZ₆ (Z = F, Cl, Br, and I) double perovskite compounds, the calculated results reveal that the electrical conductivity is more significantly enhanced under negative (μ − E_F) conditions than under positive (μ − E_F) shifts relative to the Fermi level.⁵⁶

The dependence of κ_e/τ on the chemical potential exhibits a trend analogous to that of σ/τ, albeit with a more pronounced temperature dependence. This discrepancy can be attributed to the Wiedemann–Franz law (κ_e = L_σT), which relates the electronic thermal conductivity to the electrical conductivity by means of the Lorentz number (L).⁵⁷ This trend further reveals that the κ_e to τ ratio increases with temperature, as clearly depicted in Fig. 7a–d, thereby validating the reliability of the computed results.

To assess the performance of the compounds under study, the power factor (PF) was calculated using the relation (PF = S²σ), where S is the Seebeck coefficient and σ is the electrical conductivity. This parameter reflects the efficiency of a material in converting thermal energy into electrical power. Fig. 7a–d illustrates the variation of the chemical potential with respect to PF for the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds. The results indicate that with increasing temperature from 300 K to 1000 K, the peak positions of PF shift away from the Fermi level and exhibit an increase in magnitude. This trend suggests that elevated temperatures and moderate carrier concentrations contribute to an enhancement in the power factor. Additionally, it is observed that the maximum PF values are located in the p-type region, indicating that these compounds possess significant potential for application as p-type thermoelectric materials. The relationship between ZT and chemical potential is also depicted in Fig. 7a–d. Fig. 7a–d depicts the decrease in the thermoelectric efficiency (Figure of Merit) ZT value for K₂TlAsZ₆ (Z = F, Cl, Br, and I), respectively. The peak nearly reaches unity. The gradual decrease in the ZT metrics with increasing temperature indicates the peaks approaching unity (1.0) within the lower temperature limits (300 K).^55,58

3.6 Thermodynamic calculations

3.6.1 Gibbs free energy (G*). The Gibbs free energy (G*) is specified by the relation G* = H −TS, where H denotes the enthalpy, T is the absolute temperature, and S represents the entropy, and is a crucial parameter for assessing the stability of materials under varying temperature and pressure (see Table 4). For the K₂TlAsZ₆ (Z = F, Cl, Br, and I) series, analyzing the Gibbs free energy (G*) as a function of these variables provides critical insights into their thermal and mechanical stability. Fig. 8(a and b demonstrates that at both pressure levels, the Gibbs free energy (G*) falls as the temperature rises from 200–600 K. Since the entropy component (−TS) decreases the free energy as the temperature increases, this pattern is consistent with theory. At all temperatures, the Gibbs free energy values are somewhat less negative at 10 GPa than they are at 0 GPa, indicating a minor reduction in thermodynamic stability under higher pressure (see Table 4). Nevertheless, the small differences between the two pressure conditions suggest that the K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds maintain thermodynamic stability over the entire range of temperatures and pressures examined.^59,60

Table 4 Gibbs free energy G* (×10⁷) [kJ mol⁻¹] and Entropy (S) [J mol⁻¹ K⁻¹] of K₂TlAsZ₆ (Z = F, Cl, Br, and I)

Compounds	Pressure	Gibbs free energy G* (×10^7) [kJ mol^−1]				Entropy (S) [J mol^−1 K^−1]
		300 K	500 K	700 K	1000 K	300 K	500 K	700 K	1000 K
K2TlAsF6	0 GPa	−6.394	−6.394	−6.394	−6.394	346.946	482.670	578.845	689.461
	10 GPa	−6.393	−6.394	−6.394	−6.394	259.326	385.177	473.546	571.786
K2TlAsCl6	0 GPa	−6.960	−6.960	−6.960	−6.960	410.583	548.675	646.129	758.499
	10 GPa	−6.960	−6.960	−6.960	−6.960	287.857	414.489	502.298	599.054
K2TlAsBr6	0 GPa	−1.034	−1.034	−1.034	−1.034	468.737	607.868	705.701	818.426
	10 GPa	−1.034	−1.034	−1.034	−1.034	333.206	461.502	549.534	646.041
K2TlAsI6	0 GPa	−1.744	−1.744	−1.744	−1.744	522.761	662.775	761.152	874.695
	10 GPa	−1.744	−1.744	−1.744	−1.744	368.341	497.301	585.192	681.141

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Fig. 8 (a and b) Gibbs free energy (G*) (kJ mol⁻¹) of K₂TlAsZ₆ (Z = F, Cl, Br, and I).

3.6.2 Entropy. Fig. 9a and b depicts the temperature dependence of entropy for K₂TlAsZ₆ (Z = F, Cl, Br, and I) under pressures of 0 GPa and 10 GPa, highlighting the essential thermodynamic characteristics (see Table 4). For all compositions and pressures, entropy increases consistently with temperature. This behavior is attributed to enhanced kinetic energy at elevated temperatures, which allows access to a greater number of microstates, thereby increasing the system's disorder. Conversely, entropy exhibits a clear negative correlation with pressure. At a fixed temperature, increasing the pressure from 0 GPa to 10 GPa results in a reduction of entropy (see Table 4). This effect arises from the material's compression under pressure, which constrains atomic and molecular movement, reduces the available phase space for configurations, and thus lowers the overall disorder.⁶¹

Fig. 9 (a and b) Entropy S (J mol⁻¹ K⁻¹) of K₂TlAsZ₆ (Z = F, Cl, Br, and I).

The equation for the entropy using the Debye–Slater model is expressed as:

S = −3nk_B ln(1 − e^−θ_D/_T) + 4nk_BD(θ_D/T) (11)

3.7 Phonon dispersion

By utilizing the density functional perturbation theory (DFPT), the phonon dispersion band structures associated with high symmetry points in the 1st Brillouin zone for K₂TlAsF_6, K₂TlAsCl₆, K₂TlAsBr₆, and K₂TlAsI₆ are computed and displayed in Fig. 10. For this purpose, Phonopy code is used, which is employed through Wien2K.⁶¹ Due to the presence of 10 atoms per unit cell, there are three acoustic modes, i.e., one longitudinal and two transverse, and 27 optical modes, which are classified as low frequency and high frequency phonons, respectively. Positive phonon frequencies, as depicted from the curves, have been used to confirm the thermodynamic stability of K₂TlAsF_6, K₂TlAsCl₆, K₂TlAsBr₆, and K₂TlAsI₆ at higher frequencies, reaching 11 and 9 THz. The phonon gap is visible in optical modes for all double perovskites.¹⁹

Fig. 10 (a–d) Phonon curves of K₂TlAsZ₆ (Z = F, Cl, Br, and I).

4 Conclusion

Using density functional theory (DFT) computations, this work examines the optoelectronic and photovoltaic characteristics of K₂TlAsZ₆ (Z = F, Cl, Br, and I). Calculated tolerance factors and formation energies were used to assess the structural stability of both compounds; these values together show their dynamic stability. The generalized gradient approximation (GGA) using the PBE for solids (GGA-PBEsol) functional were used to optimize physical parameters. The Trans and Blaha modified Becke–Johnson (mBJ) and spin orbit coupling (SOC) potentials were then used to compute precise band gap values. Both substances have K₂TlAsZ₆ (Z = F, Cl, Br, and I) values between 3.25 eV and 0.37 eV, indicating that they are direct band gap semiconductors. These values have significant potential for use in optoelectronic devices since they fall between the infrared to ultraviolet area of the electromagnetic spectrum. The thermodynamic study demonstrates that entropy and the Gibbs free energy decrease with increasing pressure and increase with rising temperature. The K₂TlAsZ₆ (Z = F, Cl, Br, and I) compounds are interesting prospects for upcoming optical and photovoltaic applications because of their straight band-gap, enhanced optical conductivity, and prominent absorption coefficients in the UV-region of the EM spectrum. The high ZT for the compounds K₂TlAsBr₆ and K₂TlAsI₆, except K₂TlAsZ₆ (Z = F and Cl), indicates the use of these materials as thermoelectric devices.

Conflicts of interest

There are no conflicts to declare.

Data availability

Data can be made available on reasonable request.

Supplementary information is available. See DOI: https://doi.org/10.1039/d5ra05932a.

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Footnote

† Murtaza Shahab and Muhammad Awais Jehangir made equal contributions.

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