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. 2026 Jul 8;16(36):38725–38739. doi: 10.1039/d6ra03926j

First-principles investigation of the mechanically and thermodynamically stable K2TlXCl6 (X = Sb or Sc) compounds for energy harvesting and photocatalytic applications

Mubarra Javed a, Muhammad Yaseen a,✉, Nada Alfryyan b, Sidra Sarfraz a, Muhammad Adnan a, Saima Noreen c, Muhammad Zahid c, Imed Boukhris d,e
PMCID: PMC13344879  PMID: 42422297

Abstract

Herein, the physical characteristics of K2TlSbCl6 and K2TlScCl6 are explored using the modified Becke–Johnson potential. The structural, optical, and thermoelectric characteristics are substantially impacted by the B′ site ordering. Both compounds are mechanically, structurally and thermodynamically firm, as confirmed by Born's criteria, Goldschmidt's tolerance factor and formation enthalpy, respectively. Electronic structure properties indicate the semiconducting behavior of K2TlSbCl6/K2TlScCl6 having energy gaps values of 2.26/3.83 eV, respectively. Mechanical analysis indicates their ductile behavior, with a positive Cauchy pressure, Poisson/Pugh ratio higher than 0.26/1.75, correspondingly. Optical studies reveal high absorption coefficients (α(ω)) of ≈6.59 × 105 cm−1 for K2TlSbCl6 and 7.13 × 105 cm−1 for K2TlScCl6. The compounds also exhibit low reflectivity and transparency from the visible to ultraviolet regions with static refractive indices n(0) > 1.8. TE analysis further reveals a figure of merit (ZT) higher than 0.7 at 300 K. Moreover, band-edge alignment indicates that K2TlScCl6 is suitable for hydrogen generation. Overall, the calculated TE and optoelectronic features suggest that K2TlSbCl6 and K2TlScCl6 are useful for power generation applications.

Electronic structure calculations reveal the semiconducting behavior of K2TlSbCl6/K2TlScCl6 with band gap values of 2.26/3.83 eV, respectively.graphic file with name d6ra03926j-ga.jpg

1. Introduction

The rising demand for electricity worldwide, primarily met by fossil fuels, has heightened the concerns about environmental degradation and the need for clean alternatives. However, fossil fuel reserves are finite, and their ongoing consumption makes a substantial contribution to global warming and ecological degradation.1–3 Among renewable energy technologies, solar cells and TE generators have attracted considerable interest as effective and sustainable solutions to meet the increasing energy demand.4–6 In this context, perovskite materials exhibit exceptional physical properties and structural flexibility; a wide variety of elements from across the periodic table is capable of forming stable perovskite phases. Also, these materials possess properties like ferromagnetism, superconductivity, and ionic conductivity, which play significant roles in a wide range of advanced technologies.7–9 Moreover, perovskite (ABX3) materials exhibit a broad range of interesting physical properties due to their structural flexibility. In the formula ABX3, ‘A’ can be +1 organic or inorganic cations, ‘B’ can be any suitable small high valence metal cation, and ‘X’ is usually a oxide, halide, or any other chalcogenide anion.10,11 Furthermore, because of their tunable Eg, efficient charge extraction, and great optical absorption, these materials have received a lot of attention for photovoltaic and optoelectronic applications.12–14 Despite their tremendous performance, conventional Pb-based halide perovskites (HPs) cause major environmental problems because lead is poisonous. This has prompted extensive research into Pb-free alternatives, specifically halide double perovskite (HDP) materials having A2B′B″X6 formula. These materials retain desirable optoelectronic properties while eliminating the toxic components. HDPs have shown potential for applications in various technologies, such as solar absorbers, UV detectors, scintillators, and X-ray detectors, due to their better optical, structural, thermal, and electronic characteristics. Importantly, the selection of the B′ and B″ sites components is important in determining whether the material exhibits a direct/indirect Eg.15,16 While the A-site cations and X-site halides exert minor influence on the Eg nature, the B-site configuration, particularly involving B3+ cations, is pivotal to achieve a direct and non-toxic semiconductor.17–20 Inspired by these fascinating attributes, first-principles computational methods are widely employed to reveal the geometrical, optoelectronic, and TE features of various HDPs, providing valuable guidance for designing reliable and efficient Pb-free HDPs.21–24

Several stable and non-toxic lead-free HDPs have been identified for applications in green energy technologies, including solar absorbers, LEDs, and other photovoltaic applications.25,26 For instance, the non-magnetic nature of Rb2NaCoF6 makes it suitable for thin-films and high-performance ultraviolet (UV) optoelectronic devices. Additionally, Rb2TlInX6 (X = halogen) HDPs have been identified as potential contenders for future photo-electronic systems due to their suitable structural and electronic characteristics.26 HDPs such as K2TlSbY6 (Y = I, Cl, and Br) exhibit a direct Eg of 0.89, 1.35, and 1.05 eV, respectively, which reveal their optimal range for photovoltaic applications.27 Similarly, A2TlSbX6 (A = alkali metal) HDPs were theoretically predicted to possess a direct Eg within 1.82–2.76 eV, reflecting their potential use in photovoltaics.28–30 These works established the structural firmness, direct Eg nature, and promising TE operation of the Tl–Sb-based halide double perovskites and positioned them as potential contenders for solar systems and TE conversion systems. However, compositional variations involving trivalent cations beyond Sb3+, such as Sc3+, remain unexplored, particularly with regard to their influence on band structure engineering, lattice dynamics, and charge transport. In this context, the present work provides new insights by systematically comparing K2TlSbCl6 and K2TlScCl6 to understand how substituting Sb with Sc, which has a lower ionic radius and different electronegativity, modifies the Eg, effective masses, and optical absorption. Furthermore, their geometry and optoelectronic and TE characteristics are calculated for their prospective application in multifunctional energy devices.

2. Computational work

DFT calculations were executed to determine the physical features of K2TlSbCl6 and K2TlScCl6 using the WIEN2k code, which is based on the FP-LAPW technique.31 The volume of the unit cell is centered at the Wyckoff sites: K; 8c (0.25, 0.25, 0.75), Tl: 4b (0.5, 0, 0), Sb (0, 0, 0), Sc: (0, 0, 0), and Cl: (0.23400, 0, 0). The employed FP-LAPW scheme divides the region of the unit cells into two: interstitial and muffin tin. Spherical harmonics identical to atomic orbitals are found in the muffin tin zone, while plane wave subshells expand the potential in the interstitial region.27,28

2. 1

We used RMT × Kmax = 7, cut off energy = −0.6 Ry and k-points = 10 × 10 × 10 as input for the convergence of calculations in the first Brillouin zone. Optical characteristics were computed with a denser mesh of 2000 K-points. The self-consisted field were converged up to 10−5 Ry, and angular momentum lmax = 7 was used.27 The core and valence basis set were utilized as K (4s1), Tl (4f14 5d10 6s2 6p1), Sb (4d10 5s2 5p3), Sc (4s2 3d1), and Cl (3 s2 3p5). Optical features were determined using the Kramers–Kronig equations, and TE features were found using the Boltz-Trap algorithm based on the semi-classical theory.32

3. Results and discussions

3.1. Structural properties

K2TlSbCl6 and K2TlScCl6 stabilize in a perovskite geometry belonging to the Fm3̄m space group (Fig. 1 and Fig. 2).33 The optimized crystal structures and corresponding energy–volume (E–V) plots for K2TlSbCl6 and K2TlScCl6 are presented in Fig. 2. Both nonmagnetic (NM) and ferromagnetic (FM) orderings were considered during structural optimization (Fig. 2). ΔE = ENM − EFM derived from the E–V plots confirms the thermodynamic preference for the NM phase in both compounds. Ground-state geometrical attributes were computed by fitting the total energy to the Birch–Murnaghan isothermal equation of state (EOS).34,35

3.1. 2

Fig. 1. Octahedral configuration of the K2TlSbCl6 and K2TlScCl6 halide double perovskites.

Fig. 1

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Fig. 2. Optimized energy–volume curves of the (a and b) K2TlSbCl6 and (c and d) K2TlScCl6 halide double perovskites.

Fig. 2

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The values of the structurally optimized variables, including the lattice parameter (a0), volume (V0), bulk-moduli (B0) and its pressure derivatives Inline graphic, and ground state equilibrium energy (E0), are tabulated in Table 1. To analyze the phase stability of K2TlSbCl6 and K2TlScCl6, the key structural parameters were thoroughly investigated using Goldschmidt's empirical criteria. In addition, the octahedral mismatch (Δµ) and Bartel's tolerance factor (τ) are examined to obtain deeper insights into the lattice firmness.36 The Goldschmidt's tolerance factor (tG), octahedral factor ( Created by potrace 1.16, written by Peter Selinger 2001-2019 ), and other related structural firmness parameters are computed using the following equations to ascertain the geometry of the two HDPs.

3.1. 3
3.1. 4
3.1. 5
3.1. 6

In these expressions, R denotes the radius of the respective ionic systems, and nA represents the oxidation state of the A-site element. RA, RX, RB′ and RB″ are the radii of the K, Cl, Tl, and Sb/Sc ions, respectively. The tG values for K2TlSbCl6 and K2TlScCl6 are determined to be 0.829 and 0.831, respectively, agreeing well with the cubic structure stability criteria (0.8 < tG < 1.4).12 The Created by potrace 1.16, written by Peter Selinger 2001-2019 , Δµ, and τ values also fall within the cubic stability criteria. In addition to the geometrical stability analysis, thermodynamic stability analysis is presented, which can be linked with the negative value of formation energies, as computed using the equation given below.

ΔHf = Etotal − 2EK + ETi + ESb/Sc + 6ECl. 7

Here, Etotal is the total energy of the K2TlXCl6 (X = Sb or Sc) HDPs.24 The computed values of ΔHf are −2.573 and −1.929 eV for K2TlSbCl6 and K2TlScCl6, respectively (Table 1). The negative values demonstrate that the studied materials are thermodynamically stable, indicating their experimental synthesizability and potential for optoelectronic and TE applications.

Table 1. Calculated structural parameters, including the tolerance factor, octahedral factor, octahedral mismatch, and Bartel's tolerance factor, and thermodynamic parameter such as formation energy of the K2TlXCl6 (X = Sb or Sc) compounds.

Parameter K2TlSbCl6 K2TlScCl6
Lattice constant a0 (Å) 11.19 10.93
Bulk modulus B0 (GPa) 22.89 26.26
graphic file with name d6ra03926j-t27.jpg 0.476 2.158
Equilibrium volume V0 (a.u.)3 2361.9 2203.7
Ground state energy E0 (Ryd.) −61464.46 −50029.89
Tolerance factor (tG) 0.829 0.832
Octahedral factor ( Created by potrace 1.16, written by Peter Selinger 2001-2019 ) 0.624 0.620
Octahedral mismatch (Δµ) 0.204 0.209
Bartel's tolerance factor (τ) 1.193 1.206
Formation energy ΔHf (eV) −2.573 −1.929
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3.2. Electronic properties

The potential usage of a HDPs is strongly prompted by its electronic attributes, particularly the distribution of electrons within its band structure (BS).37 The generalized gradient approximation revealed a direct Eg of 3.4/1.53 eV for K2TlScCl6/K2TlSbCl6, respectively; however, these values are underestimated (SI Fig. 1S). Therefore, the mBJ potential was applied to validate the accuracy of the Eg values (Fig. 3). K2TlSbCl6 exhibits a Γ-centered direct Eg of 2.26 eV, revealing its semiconductive nature, while K2TlScCl6 shows an L-centered direct Eg of 3.83 eV. Direct Eg substances are expected to be appropriate for photon-based electronic transition devices like commercial solar energy devices.30 The partial (P) and total (T) density of states (DOS) were computed to comprehend the underlying electronic mechanism. The TDOS plot (see Fig. 4(a and b)) indicating peaks at energies comparable to the band dispersion plots in Fig. 3. The PDOS plots (Fig. 4c–f) show quantum level interactions between numerous sub-shells. The VB, extending from −6.1 to −2 eV, mainly consists of Cl-3p, Tl-6s, and Sc-4s subshells with a negligible influence of the Tl-5d, Tl-dt2g, Sc-3d, and Sb-5p orbitals. The conduction band (CB), within 2–3.94 eV, shows significant dispersions, primarily due to contributions from the Cl-3p, Tl-6s, and Sb-5p states. In the higher energy region (5 to 8 eV), the primary dispersion originates from the K-3s, K-2p, and Tl-d-t2g orbitals. Interestingly, K atoms play a very small role in the shallow energy levels in the VB and CB regions. Furthermore, the PDOS plots in Fig. 4(d and f) clearly show a crystal field splitting of the transition of metal d-orbitals into t2g and eg manifolds. This splitting results from the octahedral coordination of B-site cations (e.g., Tl, Sb, or Sc) surrounded by halide anions. Specifically, the t2g orbitals are oriented between the ligand axes and experience weaker electrostatic repulsion from the halide ions, resulting in a lower energy. In contrast, the eg orbitals point directly toward the ligands and are destabilized to a higher energy.

Fig. 3. Calculated BS plots of the cubic (a) K2TlSbCl6 and (b) K2TlScCl6 halide double perovskites.

Fig. 3

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Fig. 4. TDOS plots of (a) K2TlSbCl6 and (b) K2TlScCl6. (c and d) PDOS plots of K2TlSbCl6 and (e and f) K2TlScCl6.

Fig. 4

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3.3. Optical properties

Based on the semiconducting nature of the K2TlSbCl6 and K2TlScCl6 HDPs, we proceeded further to compute their optical response to incident light. The values of the intended dielectric tensor (real part: ε1(ω) and imaginary part: ε2(ω)) are given in Fig. 5(a and b). The concept of light polarization is explained by ε1(ω), while electronic transitions are linked to ε2(ω), which represents the material's optical absorbance.38 An analysis of the ε1(ω) spectrum shows that the ε1(0) value for K2TlSbCl6 and K2TlScCl6 are 3.25 and 2.67, respectively (Table 2). As shown by the Penn model, ε1(0) ≈ 1 + (ħωp/Eg)2, the ε1(0) and the optical Eg are correlated.39

Fig. 5. (a) ε1(ω), (b) ε2(ω), (c) n(ω) and (d) k(ω) plots of K2TlSbCl6 and K2TlScCl6.

Fig. 5

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Table 2. Calculated static real parts of the dielectric function, reflectivity, and refractive index of the K2TlXCl6 (X = Sb or Sc) compounds.

Optical parameter K2TlSbCl6 K2TlScCl6
Current Previous
ε 1 (0) 3.25 3.59,27 3.18 (ref. 30) and 2.24 (ref. 31) 2.67
n (0) 1.82 1.89 (ref. 27) and 1.49 (ref. 31) 1.64
R (0) 0.08 0.09 (ref. 27) and 0.03 (ref. 31) 0.05
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The higher peaks of both the K2TlSbCl6 and K2TlScCl6 compounds were found to be at 2.87 and 4.40 eV, while lower peaks occurred at 2.87 and 4.40 eV, respectively. ε1(0) can be computed through the Kramers–Kronig relations.6,11

3.3. 8

In the ε2(ω) plot, the transitions of electrons from the VB to CB are illustrated, providing a measure of the materials' optical absorption, and it is entirely associated with the BS,40 (see Fig. 5(b)). The threshold energy values for investigated materials are revealed by the ε2(ω) spectrum, which corresponds to the band dispersion plots. The lower peaks of both the K2TlSbCl6 and K2TlScCl6 compounds were found at 8.10 and 8.70 eV, while the higher peaks occurred at 3.94 and 5.52 eV, respectively. The ε2(ω) part is computed by the following equation:31

3.3. 9

An essential physical parameter in optics is the refractive index n(ω), which can be computed using the following expression:41

3.3. 10

while the static n(0) can be computed as42

3.3. 11

n(ω) characterizes the ability of a substance to bend light, making it a critical factor for determining the desirable compounds for solar cells, optoelectronic detectors and waveguides.43 The values of the static refractive indices n(0) of both the K2TlSbCl6 and K2TlScCl6 compounds were 1.82 and 1.64, respectively (Table 2). A minimum of n(ω) was found at 7.68 for K2TlSbCl6 and at 7.54 eV for K2ScTlCl6 while the maxima appeared at 3.5 and 4.43 eV for K2TlSbCl6 and K2ScTlCl6, respectively, as illustrated in Fig. 5c. The k(ω) is the complex part of the index that tells us the feasibility of the electromagnetic waves to pass through any material. ε2(ω) is linked with k(ω), which regulates the transmission of electromagnetic waves through a substance. It can be obtained with the following equation:44

3.3. 12

The k(ω) plot shows the minima of the compounds at 8.15 and 8.72 eV, while the maxima of the two compounds occur at 4.02 and 7.12 eV for K2TlSbCl6 and K2TlScCl6, respectively (Fig. 5d). For solar cells and related technologies, reflectivity is an essential optical property. Both the examined HDP materials exhibit low R(ω) in the visible-UV region. Reflectivity R(ω) can be computed as follows:45

3.3. 13

The static reflectivity of K2TlSbCl6 and K2TlScCl6 were 0.08 and 0.05 (Table 2), respectively. The minima of R(ω) was found at 8.00 for K2TlSbCl6 and 7.74 eV for K2TlScCl6, while the maxima appeared at 3.94 and 4.54 eV for K2TlSbCl6 and K2TlScCl6, respectively (Fig. 6c). Another two important optical parameters are the coefficient of optical absorption α(ω) and optical conduction σ(ω). α(ω) is the measure of light absorbance per unit length deep inside the material that is directly linked with ε2(ω) and can be represented as follows:46

3.3. 14

Fig. 6. (a) α(ω), (b) σ(ω) and (c) R(ω) plots of K2TlSbCl6 and K2TlScCl6.

Fig. 6

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Calculated absorption coefficients demonstrate minimal peaks at 8.13/5.19 eV, while maximal absorption peaks are observed at 6.59/7.13 eV for K2TlSbCl6 and K2TlScCl6 (Fig. 6a), respectively. Optically active free electrons are produced in a material when it absorbs incident light. These electrons enhance the conduction of a given substance, and this phenomenon is called optical conductivity σ(ω), which can be obtained using the following relation:47

3.3. 15

The minima of σ(ω) were found at 7.95 for K2TlSbCl6 and 5.16 eV for K2TlScCl6 while the maxima appeared at 6.50 and 7.02 eV for K2TlSbCl6 and K2TlScCl6, respectively (Fig. 6b). The maximum absorption coefficient and σ(ω) of both the compounds and the lower reflectivity confirm the use of the titled materials in UV optoelectronic devices.

3.4. Elastic properties

Elastic properties describe the mechanical behavior of a material, including its stiffness, ductility, strength, flexibility and resistance to deformation. These properties determine the material's mechanical response to applied strain and stress. In cubic crystal systems, mechanical stability is determined by three elastic constants: C11, C12, and C44.48C11 reflects the resistance of a material to longitudinal strain, C12 represents the elastic coupling between stresses along different axes, and C44 measures the resistance against shape distortion. The stability of K2TlSbCl6 and K2TlScCl6 is assessed using Born and Huang stability criteria: C44 > 0, C11 − C12 > 0, C11 + 2C12 > 0, C11 > 0, and C12 < B < C11.49

The computed elastic constants enable the determination of several mechanical properties, including the Pugh's ratio (B/G), Cauchy pressure (CP), bulk modulus (B), shear modulus (G), anisotropic factor (A), Poisson's ratio (ν) and Young's modulus (E). These parameters are evaluated through the Voigt–Reuss–Hill (VRH) scheme.26,50

3.4. 16
3.4. 17
3.4. 18
3.4. 19
3.4. 20
3.4. 21

B reflects the ability of a material to resist uniform compression. The calculated results suggest that the compounds can tolerate both volume and shape distortions while maintaining compressibility. The B of K2TlScCl6 (25.07) exceeds that of K2TlSbCl6 (22.12), indicating that K2TlScCl6 is more resistant to volume distortion. E describes the stiffness of a material under applied stress. The computed values of E are 30.43 for K2TlSbCl6 and 25.34 for K2TlScCl6, depicting that K2TlSbCl6 has greater stiffness than K2TlScCl6. G illustrates a material's response to shape distortion; the B/G ratio provides the evidence for whether a material is brittle/ductile. The critical value of the B/G ratio is 1.75; values lower than this threshold indicate brittle behavior while higher values suggest ductility.51 Both compounds exhibit ductile behavior (see Table 3). Furthermore, Cauchy pressure (CP = C12 − C44) and Poisson's ratio (ν) provide insights into the ductile or brittle nature of materials. A positive CP value suggests ductility, while a negative CP value suggests brittleness. The positive CP values of both materials further support their ductility. According to the criterion proposed by I. N. Frantsevich, materials with ν > 0.26 exhibit a ductile behavior while those with ν < 0.26 reveal ductile feature.52 The calculated ν ratios reported in Table 3 further indicate the ductile character of the investigated materials. The anisotropy factor describes the directional variation in the elastic properties of a material. The value of A = 1 reveals the isotropic behavior and any deviation indicates anisotropy.53 The computed values are 0.27 for K2TlSbCl6 and 0.33 for K2TlScCl6, signifying elastic anisotropy. The anisotropic behavior of G, linear compressibility, ν and E are illustrated in Fig. 7(a–d) and 8(a–d) through both 2D and 3D representations. Linear compressibility (β) describes the directional strain response of a material under uniaxial stress, indicating expansion or contraction along specific crystallographic directions. Such directional mechanical characteristics are important for assessing the material's firmness and performance of solar cell materials.54 In an ideal isotropic system, elastic properties exhibit spherical symmetry, reflecting invariance with respect to direction. Deviations from perfect symmetry are indicative of elastic anisotropy.55

Table 3. Calculated elastic parameters of the K2TlXCl6 (X = Sb or Sc) compounds.

Parameter K2TlSbCl6 K2TlScCl6
G (GPa) 11.98 9.51
B (GPa) 22.13 25.07
E (GPa) 30.43 25.34
B/G 1.847 2.634
N 0.272 0.33
A 0.473 0.134
C 11 (GPa) 46.94 62.11
C 12 (GPa) 9.72 6.55
C 44 (GPa) 8.84 3.72
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Fig. 7. 2D and 3D representations of the elastic moduli of K2TlSbCl6.

Fig. 7

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Fig. 8. 2D and 3D representations of the elastic moduli of K2TlScCl6.

Fig. 8

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As observed in Fig. 7 and 8, both K2TlScCl6 and K2TlSbCl6 display noticeable deviations from spherical symmetry, confirming their anisotropic nature. In contrast, linear compressibility retains a circular distribution, indicating isotropic behavior. This suggests that despite significant directional dependence in most elastic properties, linear compressibility remains isotropic in these cubic systems. The degree of anisotropy is further confirmed by the extreme values of ν and G, as summarized in Table 4.

Table 4. Minimum and maximum values of the elastic moduli, along with their corresponding anisotropy ratios, for the K2TlXCl6 (X = Sb or Sc) compounds.

Compound Linear compressibility (GPa−1) Young's modulus (GPa) Shear modulus (GPa) Poisson's ratio
β min β max A E min E max A G min G max A ν min ν max A
K2TlSbCl6 0.01505 0.0150 1 28.5 43.6 1.53 8.85 17.8 2.01 0.17 0.33 1.94
K2TlScCl6 0.01329 0.0132 1 21.7 60.8 2.80 3.72 7.41 1.99 0.096 0.42 4.37
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Quantitative analysis of the elastic parameters reveals distinct differences between the two compounds. K2TlSbCl6 exhibits moderate anisotropy, as evidenced by the Young's modulus ranging from 28.5 to 43.6 GPa (A ≈ 1.53) and shear modulus from 8.85 to 17.8 GPa (≈2.01). In comparison, K2TlScCl6 shows a significantly broader variation in its Young's modulus (21.7–60.8 GPa), corresponding to a higher anisotropy ratio (≈2.80), indicative of enhanced stiffness anisotropy. However, its shear modulus variation (3.72–7.41 GPa; ≈1.99) remains comparable to that of K2TlSbCl6. Poisson's ratio exhibits substantial directional variation, ranging from 0.17 to 0.33 (≈1.94) for K2TlSbCl6 and from 0.096 to 0.42 (≈4.37) for K2TlScCl6, reflecting stronger anisotropic lateral deformation in the latter. Meanwhile, linear compressibility remains isotropic (A = 1) in both compounds, consistent with their cubic symmetry (Table 5).

Table 5. Calculated thermoelectric parameters: electrical conductivity, thermal conductivity, Seebeck coefficient, power factor, and figure of merit of the K2TlXCl6 (X = Sb or Sc) compounds.

Thermoelectric parameter K2TlSbCl6 (K) K2TlScCl6 (K)
300 800 300 800
Electrical conductivity (σ/τ) × 1019 (Ω−1 m−1 s−1) 0.023 0.09 0.58 1.02
Thermal conductivity (ke/τ) × 1014 (W m−1 K−1 s−1) 0.049 0.570 0.739 2.885
Seebeck coefficient (S) × 10−5 (µV K−1) 236.90 242.96 150.76 159.58
Power factor (S2σ/τ) × 1011 (µW K−2 cm−1) 0.128 0.534 1.329 2.616
ZT = ((S2 × σ/κ) × T) 0.78 0.75 0.54 0.73
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3.5. Thermoelectric (TE) properties

As elaborated in the introduction section, fossil fuel consumption has increased dramatically in last few decades, mainly due to the outstanding growth of the global economy, resulting in significantly severe environmental damage and energy shortages.56–58 Since TE materials have the ability to directly convert heat into electrical power, they have attracted significant research interest.59 We computed the TE parameters between 300 and 800 K using the BoltzTraP algorithm.60 The BS performs a key role in determining the TE parameters as these are primarily influenced by the effective mass, concentration, and type of charge carriers. Moreover, the BS around the Ef are closely associated with the material's transport behavior.

The number of accessible electrons via conduction is calculated by the ratio σ/τ. It is found that σ/τ rises with temperature. Its value at 300 K is 0.023 × 10−19 for K2TlSbCl6 and 0.584 × 10−19 (Ω−1 m−1 s−1) for K2TlScCl6. It is observed that when Sc is used in place of Sb, σ/τ increases. At 800 K, it is noted that K2TlSbCl6 has a peak value of 0.09 × 10−19, while K2TlScCl6 has the highest value of 1.03 × 10−19 (Ω−1 m−1 s−1) (Fig. 9a). The investigated materials are perfect for TE appliances since both offer the high electrical conductivities. The ratio k/τ is the amount of heat transference from one point to another in a material. It has two parts: (i) electronic conductivity (Ke) and lattice conductivity (Kl). Both electrical and photonic components are involved in the thermal conductivity (ke/τ) in semiconductor substances. However, at temperatures greater than 300 K, Kl values diminish nearly to zero. The predicted ke/τ values for K2TlSbCl6 and K2TlScCl6 at 300 K are 0.049 and 0.74, while the values at 800 K are 0.57 and 2.88 (Wm−1 K−1 s−1), respectively (Fig. 9d), which represents an increasing trend with temperature. This shows good correspondence with the Wiedemann–Franz law, which correlates electrical and thermal conductivities and is expressed as κ = σLT.61

Fig. 9. σ/τ (a), PF (b), S (c) and k/τ (d) plots of K2TlSbCl6 and K2TlScCl6.

Fig. 9

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A temperature difference generates thermal electromotive forces that produce a potential difference of several microvolts, known as the Seebeck effect. The formula S = ΔV/ΔT is used to assess the Seebeck coefficient.62 Band dispersions, particularly near the Ef, are strongly linked to the magnitude of S, whose dependence on effective mass and carrier concentration is given by Inline graphic. Here, m* denotes the effective mass and n denotes the carrier concentration.63 The S values for K2TlSbCl6 and K2TlScCl6 are 236.9 and 150.7 µV K−1 at 300 K and 242.9 and 159.6 µV K−1 at 800 K (Fig. 9c). The value of S decreases with an increase in temperature and is proportional to σ/τ for both materials. Such large S values are achievable only when the material possesses a relatively high carrier concentration. Moreover, the consistently positive S values indicate that the studied materials retain the p-type semiconductor behavior across the entire temperature range.

PF serves as a key indicator of efficiency as it essentially predicts the power output of TE devices, and it can be computed as PF = σS2. The maximum values of PF for K2TlSbCl6 and K2TlScCl6 at 800 K are 0.53 × 1011 and 2.62 × 1011 W K−2 s−1, respectively (Fig. 9b). Owing to their high S values, the investigated materials exhibit a significantly enhanced PF.

The ZT parameter is one of the most crucial factors in forecasting the operation of TE devices and can be computed as follows:64,65

3.5. 22

The lower σ/τ of the Sb-based HDP and the higher σ/τ of the Sc-based HDP influence the efficiency of TE devices. The ZT values for K2TlSbCl6 and K2TlScCl6 at 800 K are 0.75 and 0.73, respectively (Fig. 10). Overall, K2TlSbCl6 and K2TlScCl6 exhibit remarkable TE properties, indicating their potential as promising materials for TE coolers and generators.

Fig. 10. ZT plots of K2TlSbCl6 and K2TlScCl6.

Fig. 10

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3.6. Photocatalytic properties

The global energy crisis and increasing environmental concerns have accelerated the research on efficient photocatalytic materials for sustainable energy applications. Photocatalysis has emerged as an effective pathway for renewable energy production by converting sunlight directly into hydrogen. In this regard, perovskite-based materials have gained substantial interest because of their tunable lattice, adjustable band gaps, variable oxidation and valence states, and versatile compositions.66

The Eg of a semiconducting material plays a crucial part in finding its photocatalytic activity. An optimal photocatalyst possesses a Eg greater than 1.23 eV to facilitate both the oxygen and hydrogen evolution reactions but smaller than 3.0 eV to allow efficient solar light absorption. Furthermore, the band edges must align with the redox potentials of water. On the vacuum energy scale, the conventional water oxidation and reduction potentials, corresponding to O2/H2O and H+/H2, are −5.64 eV and −4.44 eV, respectively. The CB edge (ECBM) must exceed the H+/H2 reduction potential (−4.44 eV) to derive photo-excited electrons from VB to produce H2. In contrast, the VB edge (EVBM) should remain below the O2/H2O (−5.64 eV) oxidation potential to ensure smooth hole transfer for oxygen evolution. For the studied materials, the CB condition is satisfied, while the VB requirement is assessed via Mulliken's electronegativity.67

3.6. 23
ECBM = EVBM − Eg 24

The standard electrode potential on the hydrogen scale is taken as Eel = 4.5 eV. Here, χ denotes Mulliken's electronegativity, calculated as Inline graphic and a, b, c, and d are the number of individual atoms present in the compound. χ(x) is computed using the electron affinity (EEA) and first ionization potential (EIE) of the corresponding atom Inline graphic. Fig. 11 illustrates the band-edge alignment of the investigated K2TlXCl6 (X = Sc and Sb) compounds. For K2TlSbCl6, the CBM is positioned at −3.10 eV, which is slightly more negative than the H+/H2 reduction potential on the vacuum scale. Consequently, the photogenerated electrons possess insufficient reducing power to drive hydrogen evolution. Similarly, its VBM lies at −5.27 eV, which is less negative than the H2O/O2 oxidation potential (−5.64 eV), indicating its insufficient oxidizing ability for oxygen evolution. Therefore, K2TlSbCl6 is not thermodynamically promising for overall water splitting.

Fig. 11. Computed band-edge alignment of the K2TlSbCl6 and K2TlScCl6 halide double perovskites.

Fig. 11

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In contrast, K2TlScCl6 indicates the CBM at −0.66 eV, which provides a substantial thermodynamic driving force for proton reduction and hydrogen evolution. However, its VBM is positioned at −4.56 eV, which is significantly less negative than the H2O/O2 oxidation potential, indicating its inadequate oxidizing power for water oxidation. The analysis of band-edge alignment indicates that K2TlScCl6 and K2TlSbCl6 are not potentially suitable for overall water splitting as they both fail to meet the valence-band requirement for the oxygen evolution reaction. However, K2TlScCl6 fulfills the conduction-band criterion and hence is suitable for hydrogen generation.

4. Conclusion

Based on the first-principles calculations, K2TlXCl6 (X = Sb or Sc) are predicted to be wide-Eg nonmagnetic semiconductors. The calculated tG and formation enthalpy values verify the lattice and thermodynamic firmness of both HDPs in cubic phases. Both HDPs fulfill the Born benchmark. Notably, K2TlScCl6 exhibits a greater stiffness with C11 = 62.11 GPa as compared with K2TlSbCl6, which has C11 = 46.94 GPa. Electronic properties revealed that K2TlScCl6 and K2TlSbCl6 exhibit a direct Eg of 3.83 and 2.26 eV, respectively, which highlight the significance of these HDPs for a range of high-frequency photonic applications. The optical features show a high α(ω) of ≈6.59 × 105 cm−1 for K2TlSbCl6 and 7.13 × 105 cm−1 for K2TlScCl6, paired with low reflectivity. Moreover, temperature-dependent TE characteristics reveal a higher TE efficiency with ZT values higher than 0.7. The present first-principles characterization identifies K2TlXCl6 (X = Sb or Sc) as highly stable double perovskites with strong potential for further experimental investigations, which may ultimately enable advanced technological applications.

Conflicts of interest

There are no conflicts to declare.

Supplementary Material

RA-016-D6RA03926J-s001

Acknowledgments

The authors extend their appreciation to University Higher Education Fund for funding this research work under Research Support Program for Central labs at King Khalid University through the project number CL/RP/3. The authors also express their gratitude to the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R291), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data availability

Data will be available from the authors upon reasonable request.

Supplementary information (SI): calculated BS for cubic (a) K2TlSbCl6 and (b) K2TlScCl6 compounds with GGA method. See DOI: https://doi.org/10.1039/d6ra03926j.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

RA-016-D6RA03926J-s001
RA-016-D6RA03926J-s001.pdf (323.9KB, pdf)

Data Availability Statement

Data will be available from the authors upon reasonable request.

Supplementary information (SI): calculated BS for cubic (a) K2TlSbCl6 and (b) K2TlScCl6 compounds with GGA method. See DOI: https://doi.org/10.1039/d6ra03926j.

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