First-principles insights into double perovskite fluorides for energy harvesting applications

Abstract

The DFT calculations utilizing the FP-LAPW approach were conducted for a detailed study of structural, electronic, optical, elastic, and thermoelectric properties of cubic K₂TiHgF₆ and Na₂TiHgF₆ Heusler fluorides. It was found that both materials have a cubic crystal structure of type Fm3̄m and display negative values of formation and cohesion energy, which indicates good thermodynamic stability, with the elastic parameters being consistent with the Born stability criteria. Calculations within the framework of the TB-mBJ approach reveal semiconductor nature, the band gap value for which amounts to 2.14 eV for K₂TiHgF₆ and 1.88 eV for Na₂TiHgF₆, with valence and conduction bands being composed of F 2p/Hg 5d and Ti 3d levels, respectively. Based on the optical computations, there are very low values of the static dielectric constant, good transparency below the absorption edge, high ultraviolet absorbance, high optical conductivity, and strong plasmon resonances at 6.5 to 8.0 eV, which suggest good ultraviolet photonic activity. The elastic moduli show moderate stiffness, low compressibility, low elastic anisotropy, and mechanically stable structures, with Na₂TiHgF₆ being more rigid compared to K₂TiHgF₆. From the thermoelectric analysis, it is evident that these compounds have with negative Seebeck coefficients, high electrical conductivity, and low thermal conductivity at high temperatures.

Based on the calculations, both materials are stable mechanically and satisfy the Born stability criteria. High ultraviolet light absorption capacity, significant optical conductivity and plasmonic effects reveal good optoelectronic nature.

1. Introduction

Perovskite solar cells have been gaining much attention in the past two decades due to their outstanding power conversion efficiency and cost-effective manufacturing process. The laboratory efficiency of the solar cell has been greatly enhanced from 3.8% in 2009 to 25.5% in 2021.^1,2 As a result of their outstanding properties, which include a favourable band gap, high absorption coefficients, and similar effective masses of electrons and holes, they have beaten traditional thin film solar cells such as a-Si: H, CdTe, and CIGS.^3,4 However, even though they possess all of those wonderful qualities, the addition of lead poses toxicity problems and low stability in the presence of moisture, heat, and light.^5,6 Toxicity-free double halide perovskites of the formula A₂M′M″X₆ (A refers to a monovalent cation, M′ and M″ refer to divalent cations, and X refers to a halide anion) have gained great interest in use as solar cells,^7,8 optoelectronic devices,⁹ and thermoelectric devices.¹⁰ Therefore, extensive experimental and theoretical research has been done on Cs-based inorganic lead-free double perovskites with formula Cs₂M⁺M³⁺X₆, where M⁺ = Ag⁺, Ga⁺, Cu⁺ and others, M³⁺ = Bi³⁺, In³⁺ and X = Cl, Br, I. The band gap energy of Cs₂AgBiCl₆ and Cs₂AgBiBr₆ was investigated by McClure et al., and its value was about 2.77 and 1.95 eV, respectively. Still, these energy gap values are larger than the optimum Shockley–Queisser limit for the highest efficiency in solar cells, leading to losses of about 2.5%.¹¹ Besides, Cs₂AgBiBr₆ shows poor stability in long-term exposure to air and light. Due to the significant potential of double perovskites, investigation of new compounds among this class of materials becomes increasingly relevant.^12–14 Cs₂TlAsX₆ (X = I, Br, Cl) with indirect band gaps of 1.476 eV, 1.874 eV and 2.19 eV, small carrier effective masses, high optical absorption and low reflectivity is a good compound for photovoltaics and optoelectronics.¹⁵ On the other hand, since Rb₂CuBiX₆ (X = Br, Cl) has an indirect band gap,¹⁶ thermodynamically and mechanically stable materials can be seen as more appropriate for flexible solar cells. Likewise, lead-free double-halide perovskites are viewed as promising thermoelectric materials due to their high Seebeck coefficient,^17,18 high electrical conductivity,¹⁹ and low thermal conductivity. Usually, the figure of merit (ZT) of a thermoelectric material is regarded as 1.0. Among the available double perovskites, the ZT value of Cs₂AgBiX₆ (X = Cl, Br) is 0.75,²⁰ and Rb₂ScTiX₆ (X = Cl, Br, I) also shows high thermoelectric behaviour.²¹ Notably, the ZT values of Rb₂InScCl₆ and Rb₂ScTiBr₆ are 0.77 and 0.82, respectively; Rb₂ScTiI₆ also has a significantly high ZT of 0.95.²¹ Moreover, some potential photovoltaic candidates such as Cs₂AlBiCl₆ (ref. 22) and Cs₂AgSbCl₆ (ref. 23) have been prepared and studied recently. The fast development of computational tools has made DFT very useful in predicting the structure and other properties of lead-free perovskites. As a result, several double perovskites based on Na have been explored using this method and include Na₂CuMX₆ (M = Sb, Bi; X = Cl, Br)²⁴ and Na₂AgSbX₆ (X = F, Cl, Br, I),²⁵ but more experimental work is needed to test their photovoltaic and device properties. Qaid et al.²⁶ showed that Rb₂CuSbX₆ (X = Cl, I) is a promising thermoelectric compound with ZT equal to 0.79 and 0.75 for Rb₂CuSbCl₆ and Rb₂CuSbI₆, respectively. Also, Nazir et al.²⁷ found that Rb₂AgAsX₆ (X = Cl, Br, I) is characterized by good figure of merit values and a wide optical band gap, which gives prospects for photovoltaic and thermoelectric devices. Behera et al.²⁸ work on Rb₂InSbX₆ (X = Cl, Br) showed positive p-type Seebeck coefficients and high-power factors of these compounds suitable for thermoelectric devices. Hnuna et al.²⁹ reported high Seebeck coefficients together with high concentrations of holes of 3.08 × 10²⁰ cm⁻³, 1.72 × 10²⁰ cm⁻³, and 8.37 × 10¹⁹ cm⁻³ in Rb₂AgInCl₆, Rb₂AgInBr₆, and Rb₂AgInI₆, respectively. Paiva et al.³⁰ have studied the electronic structure and optical properties of the photosensitive materials Rb₂CuBiX₆ (X = Br, Cl), indicating its potential as a solar cell material. In another study based on DFT calculations, M. A. Razzaq et al.³¹ reported on the structural, mechanical, electronic, optical, and transport properties of the materials Rb₂TlGaX₆ (X = F, I) double perovskites. The author has found that Rb₂TlGaI₆ is a ductile material, whereas Rb₂TlGaF₆ is brittle. The calculated band gap for Rb₂TlGaI₆ is 1.06 eV and 6.05 eV for Rb₂TlGaF₆. Moreover, these materials exhibit high optical conductivity and low reflectivity (approximately 20% and 40%), as well as good photon absorption in the visible and ultraviolet regions. Also, they have high values of ZT, that is, 0.61 for Rb₂TlGaF₆ and 0.81. Crystal structure, stability, morphology, and electronic, optical, and thermoelectric properties of Na₂AgBiBr₆ have been studied by Ayyaz et al.³² Structural stability of the double perovskite has been proved using the X-ray diffraction (XRD) technique, which has also successfully synthesized the compound.³² Furthermore, the negative formation energy and favourable tolerance factor are responsible for the high stability of the compound through the results of DFT calculations.³² Lattice parameter and band gap have been reported theoretically, which agree well with the experimental XRD and UV-visible spectroscopic data. In addition, elastic calculations have been used to confirm the mechanical stability, anisotropy, and ductility of the compound. Strong absorption and low reflectance have been observed using optical calculations in the 0–6 eV energy range, whereas improved electrical conductivity, reduced thermal conductivity, and a ZT value of 0.77 have been obtained using thermoelectric calculations.³² M. Basit Shakir et al.³³ have performed detailed DFT studies for K₂AuBiX₆ (X = F, Cl, Br, I) and determined their structural, elastic, electronic, optical, and thermoelectric properties. They have confirmed good mechanical stability along with ductile, brittle, and anisotropic behaviour of these compounds. The use of the TB-mBJ potential gave band gap energies of 2.88, 2.02, 1.38, and 0.95 eV for K₂AuBiF₆, K₂AuBiCl₆, K₂AuBiBr₆, and K₂AuBiI₆, respectively. These materials also show good optical absorption in the visible spectrum range, have refractive indices within 1 to 3 and are low-loss optical materials, which are suitable for optoelectronic devices. Also, at room temperature, ZT values of 0.76, 0.78, 0.77, and 0.80, despite recent advances made in the study of Heusler fluorides, thorough studies incorporating their structural stability, and other physical properties are still very limited. This paper aims to fill the research gap by systematically studying the structural properties and physical properties of K₂TiHgF₆ and Na₂TiHgF₆ through first-principles calculations to establish the relationship between the crystal structures and the multifunctional properties. The findings provide a useful guide to the influence of alkali metals in terms of physical properties and revealed that the materials are excellent ultraviolet optoelectronics and thermoelectrics.

2. Computational details

Electronic structure calculations employing the full-potential linearized augmented plane wave (FP-LAPW) method that is available in WIEN2k code³⁴ are carried out. PBE-GGA scheme has frequently been adopted for structural optimizations and the properties of the ground state, whereas the TB-mBJ potential has been employed for the fine-tuning of electronic and optical properties. The PBE-GGA functional improves on the local density approximation (LDA) by including both electron density and spatial gradient into the exchange–correlation potential, resulting in a more accurate description of heterogeneous electron systems.³⁵ As a result, PBE-GGA offers consistent lattice parameters, equilibrium volumes, bulk moduli, and total energies, which makes it a good choice for optimization and stability studies. Although PBE-GGA shows good results for describing the properties of the ground state, it still provides underestimated values of semiconductor and insulator band gaps. In consequence, the conduction bands of materials are shifted to lower values of energy, leading to narrower band gaps and improper estimation of the optical transitions. To avoid such an obstacle, the Tran–Blaha modified Becke–Johnson potential (TB-mBJ) is applied.³⁶ TB-mBJ model is a semilocal exchange potential developed specifically for a better description of excitation energies and band structure of materials without imposing huge computational resources typical for hybrid approaches and many-body GW calculations. In contrast to PBE-GGA, the TB-mBJ potential is dependent on electron density and kinetic energy density at the same time, which gives a more realistic picture of the exchange interactions between electrons. This approach enables the movement of the conduction-band states upwards while the valence-band states remain largely untouched, thus producing better predictions of the band gap, which often agree with experimental values. In practical applications, the crystal structure optimized by using PBE-GGA is taken as the starting geometry in the TB-mBJ calculations to ensure that the structural information remains physically valid while the electronic properties are altered. The dispersions of the valence bands and conduction bands in both PBE-GGA and TB-mBJ are alike, and this means that the underlying orbital compositions and orbital hybridizations have remained the same. However, TB-mBJ normally predicts wider and more realistic band gaps because of the better description of the exchange effects. In the present study, the calculations were performed using a scalar-relativistic approach with an RMT × K_max = 9, where RMT denotes the smallest muffin-tin sphere radius, and K_max represents the maximum reciprocal-lattice wave vector. Brillouin-zone sampling was performed using a 12 × 12 × 12 Monkhorst–Pack k-point mesh. To ensure numerical accuracy and convergence, energy and charge convergence criteria of 10⁻⁵ Ry and 10⁻³e, respectively, were employed. SOC can modify the energies and dispersion of electronic states, particularly those having significant contributions from heavier elements, and may consequently produce moderate changes in the calculated band gaps, band-edge positions, and optical transition energies. In the electronic, optical and transport properties of the materials under study were calculated using the scalar-relativistic FP-LAPW approach. Nevertheless, quantitative SOC effects can be verified through future fully relativistic calculations. Furthermore, the precise determination of the electronic band gap leads to more reliable predictions of the carrier transport behaviour and thermoelectric performance, as the BoltzTraP package computes transport coefficients from the calculated band structure based on the framework of Boltzmann transport theory in the constant relaxation time approximation (CRTA).³⁷

3. Results and discussion

3.1. Structural properties

The crystallographic structure of K₂TiHgF₆ and Na₂TiHgF₆ were studied, and both compounds have been identified to possess the cubic Heusler-type structure in the space group Fm3̄m (see Fig. 1). Both compounds have a zero-dimensional structural network that comprises Hg, alkali-metal (K/Na), and TiF₆ structural motifs. K₂TiHgF₆ has a unit cell comprising four Hg clusters, eight K clusters, and four TiF₆ octahedral clusters. The Hg atoms exist in an environment with eight-fold coordination, and each of the K atoms is surrounded by twelve other atoms. This gives a very highly coordinated alkali-metal sublattice. The titanium atoms are positioned at the centre of the regular TiF₆ octahedra. Each of the titanium atoms is surrounded by six F atoms with equal bond lengths between them. Each of the fluorine atoms bonds to a titanium atom and thus exists in regular TiF₆ octahedra. Similarly, Na₂TiHgF₆ crystallizes in a cubic Fm3̄m Heusler structure and possesses a zero-dimensional structure comprising four Hg clusters, eight Na clusters, and four TiF₆ octahedral clusters per unit cell. The Hg atoms exist in an environment with eight-fold coordination, and each of the Na atoms is surrounded by twelve other atoms. The TiF₆ octahedra still do not change, because each Ti atom is in the octahedral coordinated environment of six equivalent fluorine atoms having a fixed Ti–F bond distance. Each F atom makes a single bond with Ti.

Fig. 1. The calculated (a) polyhedra and (b) crystallographic structure for X₂TiHgF₆ (X = K, Na) halide double perovskites.

Parabolic E–V optimization curves (Fig. 2(a and b)) of K₂TiHgF₆ and Na₂TiHgF₆ exhibit the characteristic trend for structurally stable crystals, where the minimum point on the graph represents the equilibrium state. The E–V curve obtained from calculations performed with the obligatory cubic symmetry condition gives only the minimum energy of the corresponding structure and cannot be used alone as proof of the unrestricted ground state. The aim of the present optimization was to study the cubic Fm3̄m double-perovskite structure and the E–V approach inquires the energy–volume dependence within this framework. The minimum value of total energy (E₀) for K₂TiHgF₆ is at an equilibrium volume (V₀) of about 1490–1500 a.u.³, with the total energy of roughly −44639.616 Ry, whereas that of Na₂TiHgF₆ is at a lower equilibrium volume of roughly 1410–1420 a.u.³ with a total energy of about −42880.696 Ry. The well-defined minima of both graphs suggest that the optimized structural configurations represent energetically favourable and stable states. This is because of the higher equilibrium volume of K₂TiHgF_6, which has to do with the larger ionic radius of K⁺ compared to Na⁺. Potassium being much bigger in size than sodium means that substituting Na with K increases the size of the crystal lattice and increases interatomic distances, hence increasing the equilibrium volume. The observed deviations are in accordance with the structural properties of alkali metal-based fluorides and hence confirm that the ion size is the predominant factor influencing the optimal volume of the lattice. The shape of the E–V curve near the point of equilibrium provides valuable information concerning the mechanical stiffness, bulk modulus and compressibility. It can be seen that the curve for Na₂TiHgF₆ is much sharper compared to the minimum in the case of K₂TiHgF₆. Na₂TiHgF₆ is predicted to possess a high bulk modulus and low compressibility, while K₂TiHgF₆ has a low bulk modulus and high compressibility. This occurs due to the compact structure of Na₂TiHgF₆ and strong atomic interactions. The shortening of bond lengths enhances electrostatic interaction and orbital overlap. From the electronic point of view, the response of the material to mechanical stimulus is directly dependent on valence electron arrangement and orbital hybridization. It has been proposed earlier by DOS analysis and band structure that the valence bands consist of the hybridized F 2p and Hg 5d orbitals, and conduction bands come primarily from the Ti 3d orbitals. The extensive Hg 5d/F 2p hybridization produces bonding orbitals, which help in stabilizing the crystal and lowering energy. The reduced lattice volume of Na₂TiHgF₆ makes it easier for orbitals to overlap with Ti, Hg, and F, increasing chemical bonding and energy stiffness. The greater overlap leads to an increase in the contribution of covalent bonds superimposed on the ionic Ti–F, Hg–F, and alkali–F interactions. An increased lattice volume of K₂TiHgF₆ limits orbital overlap and bonding, thus leading to broadening of the energy well and energy stiffness. This suggests thermodynamic stability and no presence of structural instability in the studied range of volume. The optimal structural relaxation of the system towards the equilibrium shape minimizes the attraction between bonding contacts and short-range repulsions, thus reducing energy. These optimized structural parameters directly influence the electrical properties because the distances between bonds and their overlaps determine the position of bands and their edges. Strong interactions between Ti 3d, F 2p and Hg 5d ions at equilibrium lead to the formation of well-defined valence and conduction bands. This is consistent with the semiconducting nature of the calculated electronic structures. As can be seen from the E–V dependence, both K₂TiHgF₆ and Na₂TiHgF₆ are mechanically and thermodynamically stable semiconductors. The sodium compound possesses somewhat stronger bonds, more rigidity and less compressibility than the potassium compound, having a smaller equilibrium volume.

Fig. 2. Energy–volume optimization curves for (a) K₂TiHgF₆ and (b) Na₂TiHgF₆ halide double perovskites.

The thermodynamic stability of K₂TiHgF₆ and Na₂TiHgF₆ was evaluated based on their theoretical formation energy and cohesive energy. The values of the formation energy are −3.64 (eV f.u.⁻¹) for K₂TiHgF₆ and −3.75 eV f.u.⁻¹ for Na₂TiHgF₆, while the cohesive energies of these materials amount to −4.43 eV f.u.⁻¹ and −4.52 eV f.u.⁻¹, respectively (Table 1). The negative values of the above quantities prove that the synthesis of these compounds is thermodynamically favored. Moreover, the formation and cohesive energies of Na₂TiHgF₆, being slightly lower, indicate stronger bonding. In both materials, the Ti–F bonds present in the separate TiF₆ octahedra represent the main contribution to the stability of the structure due to their strongly ionic–covalent nature. In this case, the alkali-metal ions (K or Na) serve as charge-compensating entities, ensuring the stability of the three-dimensional crystal via electrostatic stabilization in relation to the fluorine skeleton. As a result of having a smaller ionic radius, Na is responsible for slightly better lattice stability compared to K. Additionally, Hg contributes to the framework via structural bonding and metal–fluorine interactions affecting crystal stability. Such energetically favourable properties show the chemical stability of both Heusler fluorides. Phonon dispersion spectra have been calculated to confirm the dynamic stability of K₂TiHgF₆ and Na₂TiHgF₆ in the cubic phase (see Fig. 3). In the case of K₂TiHgF₆, all branches along the high-symmetry W–L–Γ–X–W–K line are at non-negative frequencies, where the acoustic branches approach zero at the Γ point, as expected for translational acoustic branches. There are no imaginary modes, i.e., negative frequencies, in the entire Brillouin zone, implying the dynamic stability of the optimized structure against small perturbations in the crystal structure. Na₂TiHgF₆ shows a similar phonon spectrum, where all branches are positive except for the convergence to zero at the Γ point, as expected for acoustic modes. The dynamic stability of the optimized structure of Na₂TiHgF₆ is again confirmed by the absence of imaginary modes. Both materials show some separated acoustic and optical branches up to 9–10 THz due to their stable lattice framework and various atomic masses. The high-frequency optical branches may be largely attributed to vibrational motions of the relatively lighter F atoms while the low-frequency branches are likely to have considerable contribution from the relatively heavier K/Na, Ti, and Hg ions. In comparison with K₂TiHgF₆, Na₂TiHgF₆ has slightly different acoustic branch dispersion and achieves similar maximum phonon frequencies due to the effect of A-site substitution on the lattice dynamics of these materials. More importantly, no soft modes are observed away from Γ for either material, which constitutes more conclusive evidence for the dynamical stability of the two compounds than the analyses based only on energy–volume and elastic calculations. Hence, the phonon computations validate the stability of K₂TiHgF₆ and Na₂TiHgF₆ at 0 K based on the harmonic approximation. Such findings are consistent with the negative formation/cohesive energies and mechanical stability obtained previously through energy–volume and elastic calculations.

Table 1. The Wyckoff positions, atomic coordinates, lattice constants, formation energy and cohesive energy values for X₂TiHgF₆ (X = K, Na) halide double perovskites.

Materials	Atoms	Wyckoff	PBE-GGA			a (Å)	b (Å)	c (Å)	E form (eV f.u.−1)	E coh (eV f.u.−1)
			x	y	z
K2TiHgF6 (Fm3̄m)						18.16	18.16	18.16	−3.64	−4.43
	K	4a	0.750	0.250	0.250
	T	4b	0.000	0.000	0.000
	Hg	8c	0.500	0.000	0.000
	F	24e	0.500	0.000	0.303
Na2TiHgF6 (Fm3̄m)						17.87	17.87	17.87	−3.75	−4.52
	Na	4a	0.750	0.250	0.250
	T	4b	0.000	0.000	0.000
	Hg	8c	0.500	0.000	0.000
	F	24e	0.500	0.000	0.301

Fig. 3. Phonon dispersion plots for (a) K₂TiHgF₆ and (b) Na₂TiHgF₆ halide double perovskites.

3.2. Mechanical properties

The mechanical response of K₂TiHgF₆ and Na₂TiHgF₆ was studied with respect to elastic constants and the polycrystalline mechanical properties calculated from Voigt–Reuss–Hill approximations. As seen from the calculations, the compounds meet the criteria of Born's mechanical stability for a cubic structure, which means that their structures are stable and can resist any stress without distortion of their structure. According to the calculated bulk modulus (B), the value is equal to 60.33 GPa for K₂TiHgF₆ and 64.87 GPa for Na₂TiHgF₆, which means that both materials have moderate strength to resist uniform compression. It is worth mentioning that the larger bulk modulus value of Na₂TiHgF₆ compared to K₂TiHgF₆ indicates the increased lattice density due to the replacement of potassium with sodium ions and increased average interaction in the lattice. As we know, this leads to better resistance to changes in volume due to hydrostatic pressure. While there is not much variation between these values, there is evidence that the crystal structure of Na₂TiHgF₆ is relatively less compressible. There is a similar pattern with respect to the shear modulus (G). While K₂TiHgF₆ has a shear modulus of 38.89 GPa, that of Na₂TiHgF₆ is 42.20 GPa (see Table 2). The increased value indicates greater resistance to shear deformation. As shear deformation implies deformation with respect to volume constant, it also suggests the greater rigidity of the Na-based compound with respect to distortions. In other words, Na₂TiHgF₆ has stronger interatomic interactions in its crystal structure since shear deformation does not involve volume changes but only relative interatomic distances. As such, this variation comes mainly from the greater effectiveness of the packing with Na, which results in stronger bonding between atoms in Na₂TiHgF₆. Young's modulus represents the material stiffness under uniaxial stresses and is one of the key properties associated with the elastic rigidity. The computed values range from 96.04 GPa for K₂TiHgF₆ up to 104.04 GPa for Na₂TiHgF₆. This means that the latter salt is more elastic–rigid than its potassium counterpart. A higher Young's modulus means that a greater stress value is needed to achieve the same elastic strain in Na₂TiHgF₆ due to better resistance to elastic deformation. It can be explained by the smaller size of the Na cation leading to denser crystal packing and higher strength of all bonds within the lattice. However, both materials have rather low Young's modulus values compared to high-stiffness oxide ceramic materials. This means that they have enough elastic rigidity but still preserve some degree of plasticity. Shear constant (C′) equal to (C₁₁ − C₁₂)/2 gives additional information about the resistance to tetragonal deformations. Positive and high values of 36.20 GPa for K₂TiHgF₆ and 38.20 GPa for Na₂TiHgF₆ (see Table 2) prove that the cubic structure is stable against tetragonal shear deformation. Positive values of C′ are an indispensable requirement for mechanical stability in cubic systems because otherwise it would mean an instability that leads to spontaneous structure modification. The slightly higher value of C′ for Na₂TiHgF₆ once again implies the presence of higher lattice rigidity and increased ability to resist symmetry-lowering deformation. This further supports our inference that sodium incorporation increases the mechanical stability of the crystalline system. The Cauchy pressure (C″ = C₁₂ − C₄₄) is a useful parameter giving insights into the type of chemical bonding. Both compounds have negative Cauchy pressures, having the values of −4.60 GPa and −5.70 GPa, respectively, for K₂TiHgF₆ and Na₂TiHgF₆. The negative value of Cauchy pressure means that there is considerable directionality in the interatomic chemical bonding. Thus, the bonding is not totally metallic in nature, but rather consists of ionic–covalent bonding. The Ti–F bonds in the TiF₆ octahedra of the fluorides have ionic–covalent character dominating the bonding pattern, while the contributions from the alkali metal ions are only electrostatic in nature.

Table 2. The computed elastic coefficients C₁₁, C₁₂, and C_44, elastic moduli; bulk modulus, shear modulus, Poisson's-ratio, Young's modulus, Cauchy pressure, shear constant, Pugh ratio, and anisotropy factor of K₂TiHgF₆ and Na₂TiHgF₆ systems.

Elastic parameters	K2TiHgF6	Na2TiHgF6
C 11	108.60	115.80
C 12	36.20	39.40
C 44	40.80	45.10
Bulk modulus (B)	60.33	64.87
Shear modulus (G)	38.89	42.20
Poisson's ratio (ν)	0.235	0.233
Young's modulus (Y)	96.04	104.04
Cauchy pressure C″	−4.60	−5.70
Shear constant (C′)	36.20	38.20
Pugh ratio (B/G)	1.551	1.537
Elastic anisotropy factor (A)	1.127	1.181

The somewhat negative value of the Cauchy pressure of Na₂TiHgF₆ indicates marginally greater bond directionality and covalent character than that of K₂TiHgF₆, which is in line with greater lattice strength estimated through higher bulk and shear moduli. Poisson's ratio (ν) calculations give 0.235 for K₂TiHgF₆ and 0.233 for Na₂TiHgF₆ (see Table 2). Such ν values lie in the expected range for ionic and mixed ionic–covalent solids and reflect rather moderate lateral strain upon longitudinal stress. Very little difference between these values shows that replacement of K by Na has rather minor effects on the elastic deformation process. Usually, Poisson's ratio of materials equal to 0.25 reflects a balance between ionic and covalent bonding, while much greater ratios refer to ductile metallic materials. Thus, obtained ν values confirm that the bonding in the two materials contains rather substantial directionality due to the Ti–F framework. The calculated values of B/G are equal to 1.551 for K₂TiHgF₆ and 1.537 for Na₂TiHgF₆ (see Table 2). These numbers are less than the critical value of 1.75 suggested by Pugh. Therefore, both materials have brittle mechanical properties. Nothing is surprising about this result since fluoride inorganic compounds with rigid TiF₆ octahedra are characterized by directional bonds and low capability of plastic deformation. The smaller B/G ratio for Na₂TiHgF₆ means that this compound is slightly more brittle than K₂TiHgF₆ despite having larger stiffness and less elastic deformation. This contradiction appears because stronger bonds make the crystal lattice less ductile. This brittleness is tolerable for many optoelectronic and dielectric materials since they experience low mechanical loads. Another feature of the directional dependence of elasticity is the elastic anisotropy factor (A). Isotropic cubic crystals have the anisotropy factor of exactly one. Values of 1.127 for K₂TiHgF₆ and 1.181 for Na₂TiHgF₆ (see Table 2) prove a small degree of elastic anisotropy, which indicates that both studied materials have nearly isotropic mechanical properties. Such values mean that elastic properties are not highly dependent on the direction in the crystal, which contributes to the lack of internal stresses and cracks arising during processing or heat treatment. A slightly greater anisotropy factor of Na₂TiHgF₆ proves that replacement of K by Na results in a slight increase in the directional dependence of the elastic properties; nevertheless, the deviations are still small enough to classify both materials as nearly elastically isotropic. All of the mentioned above mechanical characteristics clearly prove that both K₂TiHgF₆ and Na₂TiHgF₆ have good mechanical stability, stiffness, resistance to compressive and shear deformation, positive resistance to the tetragonal deformation and very small elastic anisotropy. In comparing the two compounds, Na₂TiHgF₆ shows significantly larger bulk, shear, Young's, and shear modulus values compared to K₂TiHgF₆ because of its stronger lattice rigidity and increased strength of interatomic bonds due to replacing the large K ion with the smaller Na ion. On the other hand, K₂TiHgF₆ displays marginally lower rigidity, but a higher B/G ratio at the same time, which suggests its increased brittleness resistance to some extent compared to Na₂TiHgF₆. Both materials have always exhibited negative Cauchy pressures and intermediate Poisson's ratios, which show their dominance of mixed ionic–covalent bonds with predominance of strong Ti–F octahedra, whereas alkali metal ions are mainly present as ionic bonds. Thus, in considering all elastic constants together, one can state that both Heusler fluorides have sufficient mechanical stability, stiffness, predictability of elastic properties and reliability of structure, which makes them promising materials for further investigation in relation to the development of multifunctional fluoride-based devices.

3.3. Electronic properties

From the band structures of K₂TiHgF₆ and Na₂TiHgF₆ determined from both PBE-GGA and TB-mBJ exchange–correlation potentials (Fig. 4(a–d)). The projected energy gaps of K₂TiHgF₆ and Na₂TiHgF₆ from the PBE-GGA are about 1.56 eV and 1.36 eV, respectively. The more precise TB-mBJ functional boosts the gaps to about 2.14 eV and 1.88 eV, respectively, without affecting the fundamental character of the electronic structure. The retention of the VBM and CBM at the L point suggests that the cubic fluoroperovskite lattice's symmetry promotes both the highest occupied and lowest unoccupied electronic states at the same k-vector, allowing for electronic transitions without the use of phonons. This energy gap nature is caused by the orbital symmetry and energy matching of the constituent atomic states that compose the band edges. The VBM corresponds to F(2p)/Hg(5d) bonding states, whereas the CBM relates to Ti(3d)-based antibonding states. The observed band gap is the result of the states near energy alignment at the L point. Valence bands ranging from −6.5 eV to the Fermi level show low dispersion, indicating limited orbital overlap between fluorine and mercury. The large contribution of F 2p states near the VBM reveals that fluorine governs the hole-producing states, whereas the accompanying Hg 5d contribution implies strong p–d hybridization within the [TiHgF₆] framework. This combination causes an increase in the width of the valence bands and gives the crystal some covalent properties. In comparison, the contribution from the alkali metal cations K and Na is minimal at the edges of the bands, where they act only as sources of electrons to stabilize the lattice. The conduction bands immediately above the Fermi level are dominated by Ti 3d states and exhibit relatively low dispersion, particularly in TB-mBJ computations. This pattern is consistent with the strong Ti d peaks detected in the DOS, indicating that the conduction states are confined. The confined Ti 3d orbitals form relatively flat conduction bands around the CBM, assuming greater electron effective masses. The valence-band edge, on the other hand, has a little higher curvature because F 2p orbitals are more spatially stretched and hybridized with Hg 5d states, resulting in lighter hole effective masses. As a result, holes are projected to have greater mobility than electrons, albeit both carrier types are constrained by the materials' overall band gap properties. When comparing K₂TiHgF₆ and Na₂TiHgF_6, the band topology is essentially comparable, reflecting the common TiHgF₆ octahedral framework. The Na-containing molecule has narrower band dispersion and a smaller band gap in PBE-GGA. This is due to the reduced ionic radius of Na⁺, which affects bond lengths and crystal-field effects. Despite these minor changes, these compounds have comparable orbital compositions and direct gap properties. The heavy Hg atom is especially significant in determining the low-energy electronic structure. Its occupied 5d states make a major contribution to the upper valence-band region and strongly hybridize with F 2p orbitals, resulting in the prominent valence-band features. Due to the large radial size of the Hg 5d orbitals, such hybridization results in increased width of the valence band and changes in band curvature near the VBM. In contrast, the Ti atoms determine the electrical properties of the conduction region through the empty 3d states forming the CBM. Therefore, the elementary electronic transition is a charge-transfer from the F 2p/Hg 5d bonding states to the Ti 3d antibonding states. The strong overlap of the F 2p and Hg 5d states in the valence band is good evidence for the presence of strong covalent interaction, while the weak participation of K or Na states in the energy region close to the Fermi level and predominance of the fluorine states suggest the existence of strong ionic interaction. The absence of any band that crosses the Fermi level confirms that both materials do not show any metallic or semi-metallic behaviour. The fact that both of these materials show a definite forbidden gap and DOS = 0 at E_F qualifies both of these materials as semiconductors. The absence of free carriers means that there is no electrical conductivity under equilibrium conditions. To conduct charges, either thermal excitation or the photoelectric effect needs to take place. The character of the energy gap makes these semiconductors particularly attractive for use in optoelectronics since there is no transfer of momentum. Both of these semiconductors are classified as semiconductors and can be used in UV photodetectors, transparent optoelectronic devices, dielectric coatings, and radiation-hardened electronics. From a thermoelectric standpoint, the presence of localized conduction states and modest valence-band dispersion could increase the density of states along the band edges. The strong concordance between band structures and DOS results reinforces the electrical explanation. The DOS reveals that the VBM is derived from F 2p and Hg 5d states, precisely matching the valence bands around the 0 eV, whereas the Ti 3d-dominated CBM corresponds to the first conduction bands above the gap. The relatively flat conduction bands reflect the acute DOS features associated with Ti d states, whereas the larger F p/Hg d DOS distribution corresponds to the more dispersive valence band. The electronic structure near the Fermi level influences optical and transport properties, resulting in energy gap semiconducting behaviour, low equilibrium carrier density, strong ultraviolet optical transitions, mixed ionic–covalent bonding, and promising band gap fluoroperovskites for optoelectronic and dielectric applications.

Fig. 4. Electronic band profiles for (a and b) K₂TiHgF₆ and (c and d) Na₂TiHgF₆ halide double perovskites computed using the Tran–Blaha modified Beck–Johnson (TB-mBJ) and GGA-PBE potential.

In Fig. 5(a and b), the electronic structures of K₂TiHgF₆ and Na₂TiHgF₆ are similar, indicating that both materials are band gap fluoroperovskite semiconductors with highly localized electronic states and negligible electronic density at the (0 eV). In both materials, the Fermi level is located in a clearly prohibited energy region where the entire DOS vanishes completely, revealing the absence of partially occupied bands and verifying the nonmetallic character anticipated by the band structures. The lack of states at E_F shows that there are no free charge carriers available at equilibrium, implying limited electrical conductivity and transport dominated by thermally excited electrons and holes. The valence band in K₂TiHgF₆ ranges from −7.0 to −4.5 eV, whereas the conduction band starts around +2.8–3.0 eV and grows to higher energies. Na₂TiHgF₆ has a valence-band area of −7.0 to −4.4 eV and a conduction band onset of +2.7–2.9 eV, indicating a significant band gap in the electronic band structure. The upper valence-band area between −6.8 and −4.5 eV is dominated by F 2p states in both materials, as demonstrated by the influential PDOS peaks of fluorine, which coincide with the greatest features in the total DOS. These F 2p states form the valence-band maximum (VBM), showing that hole formation and transport are largely connected with fluorine-derived orbitals. Hg 5d orbitals contribute significantly to the valence band, particularly in the energy range of −6.5 to −5.0 eV. A considerable number of similarities between Hg d and F p states in the discussed energy range certainly proves that the Hg(5d)/F(2p) hybridization is strong enough. This p–d mixing results in widening of the valence band, meaning that besides the ionic interaction, there is some covalency as well, which is caused by orbital overlap. The similarity of the Hg d and F p states at nearly equal energies proves the creation of bonding states, which ensure stability of the crystal structure owing to reduction of its electronic energy. The deep valence bands (below −6.5 eV) are mainly composed of F 2p states with Hg d localized orbitals. These orbitals are strongly bonded, and therefore they affect only the structural stability and insignificantly the electrical properties of the material. In the vicinity of the CBM, there is an absolutely new electrical behaviour. Ti 3d states are mostly represented in the CBM of both K₂TiHgF₆ and Na₂TiHgF₆ with very strong peaks between +2.8 and +3.4 eV. The sharp and narrow character of these Ti d peaks suggests highly concentrated unoccupied states and relatively flat conduction bands, which is consistent with a large electron effective mass. Small contributions from F p states are seen toward the bottom of the conduction band, indicating weak Ti(3d)/F(2p) antibonding hybridization; however, the Ti d orbitals obviously remain the primary member of the CBM. As a result, the fundamental band gap transition in both materials may be explained as a charge-transfer excitation from F 2p/Hg 5d hybridized valence states to Ti 3d conduction states. Such transitions are common among fluoride-based perovskites and have a significant impact on their optical sensitivity. As a result, both materials are projected to have substantial UV absorption once photon energy exceeds the band gap. Higher-energy conduction states above +5.5 eV are dominated by K s/K p orbitals in K₂TiHgF₆ and Na s/Na p orbitals in Na₂TiHgF₆, with minimal Hg p contributions. These states are located well above the CBM and hence do not directly engage in low-energy electronic transport, but instead contribute to higher-energy optical excitations and dielectric polarization processes. However, there are slight changes in DOS profiles because of the substitution of K with Na. Though the electronic structure of the material is nearly identical, the Na-based compound is a little bit more contracted and localized in terms of the DOS profile because of a smaller ionic radius and a stronger crystal field effect of Na⁺ than K⁺. Nevertheless, the electronic states near the band edges are virtually determined by the Ti–F–Hg system, which is responsible for the similarity in electronic properties of the materials. Interaction between Hg d and F p orbitals in valence bands confirms the partial covalent character of the material that consists mainly of ionic bonds due to interaction of K⁺/Na⁺ ions with [TiHgF₆]²⁻ units. Correlation with band structures suggests that the VBM is mostly formed from dispersive F p/Hg d hybridized bands, whereas the CBM is derived from confined Ti d bands. Because the DOS near the VBM is quite broad and evenly distributed, holes are expected to have lower effective masses than electrons. In contrast, the abrupt and powerful Ti d DOS peak at the CBM indicates flatter conduction bands and, as a result, higher electron effective masses. The imbalance between the valence and conduction band edges may result in better hole mobility than electron mobility.

Fig. 5. The projected density of states for (a) K₂TiHgF₆ and (b) Na₂TiHgF₆ halide double perovskites.

3.4. Optical properties

Fig. 6(a) shows the real part of the dielectric function, ε₁(ω), which provides immediate information on dielectric screening, electronic polarizability, and the material's ability to retain electromagnetic energy. The ε₁(ω) spectra of K₂TiHgF₆ and Na₂TiHgF₆ show virtually comparable behaviour over the energy range, demonstrating that the substitution of K by Na has only a modest impact on the fundamental electronic and optical properties. This resemblance stems from the fact that the optical response is predominantly regulated by the Ti–Hg–F octahedral framework, whereas the alkali-metal cations serve mostly as charge-balancing ions. At zero photon energy, the static dielectric constant ε₁(0) is around 2.1–2.3, which is lower than in narrow band gap semiconductors. Such low values are typical of band-gap materials because the huge energy difference between occupied valence states and unoccupied conduction states restricts the ease with which electrons can be polarized. The electronic band structure and DOS results show that the valence-band area is dominated by hybridized F 2p and Hg 5d orbitals, whereas the conduction-band edge is primarily formed of Ti 3d states. The huge spacing between these states lessens the likelihood of low-energy electronic excitation, which lowers the static dielectric response. In the low-energy region below 3.0 eV, ε₁(ω) slowly grows with photon energy while remaining positive. Because this energy range sits below the optical band gap, photons do not have enough energy to move electrons from the valence band to the conduction band. As a result, only virtual electronic excitations occur, resulting in electronic polarization rather than true optical absorption. This pattern accounts for the high transparency expected in the visible and low-energy UV areas. As the photon energy comes close to 3.0–3.2 eV for optical absorption, ε₁(ω) begins increasing quickly as permitted interband transitions take place. The dielectric function attains its maximum value between 6 and 6.7 eV, where ε₁(ω) exceeds 5. This is due to strong charge transfer excitations from the filled F 2p/Hg 5d bonding states to the vacant Ti 3d antibonding states. High DOS in this energy range coupled with high p–d hybridizations facilitates transitions resulting in higher dielectric polarizations. The periodic structures observed between 5 and 10 eV are caused by several interband transitions involving deep valence levels and higher conduction bands. Within the energy band, electron transitions may occur not only from the F 2p and Hg 5d bands, but also from deeper Hg-based bands into Ti 3d, Hg 6p, and s/p alkali metal conduction bands. ε₁(ω) has a complicated valley-peak structure due to multiple permitted transitions. Hg atoms are very important due to the pronounced hybridization of the 5d orbitals with the F 2p state, which leads to enhanced electronic polarization and oscillator strength. Negative ε₁(ω) in the spectra is observed in the energy range of 7.3–8.5 eV, which is an important peculiarity of the process. It means that the collective oscillations of the electrons become powerful enough to take control of the dielectric behaviour, and plasma-like properties appear. In this case, the electromagnetic radiation gets greatly deflected and does not penetrate into the medium any more. This kind of effect is usually associated with reflectance. Starting from about 10 eV, ε₁(ω) starts decreasing slowly and reaches smaller positive values. It is due to the absence of low-energy interband transitions.

Fig. 6. Calculated optical properties for X₂TiHgF₆ (X = K, Na) halide double perovskites, including (a) real part of the dielectric function, (b) imaginary part, (c) refractive index, and (d) extinction coefficient.

The imaginary part of the dielectric function, ε₂(ω), presented in Fig. 6(b) reflects the absorptive behaviour of the medium and provides direct insights into the electronic transitions stimulated by photons between filled and empty bands. Different from ε₁(ω), which accounts for the polarizability, ε₂(ω) characterizes the efficiency of the material in absorbing incoming photons. The shape and magnitude of ε₂(ω) are highly dependent on the electronic band structure, density of states (DOS) and the transition probability. For K₂TiHgF₆ and Na₂TiHgF₆, ε₂(ω) is almost zero between 0 eV and 3.0 eV. Such behaviour indicates that the energy of photons within this range is insufficient to excite the electrons across the fundamental band gap. Therefore, there is no appreciable optical absorption, which confirms the semiconductivity and transparency of the materials in the low-energy spectral range. The beginning of ε₂(ω) occurs at 3.0–3.2 eV and corresponds to TB-mBJ direct energy gap values of 2.14 eV for K₂TiHgF₆ and 1.88 eV for Na₂TiHgF₆. Such agreement provides a good correspondence between optical and electronic structure calculations. The first visible absorption peak occurs at 3.5 eV, which corresponds to the onset of direct interband transitions. As the valence-band maximum is dominated by F p/Hg d orbitals and the conduction-band minimum is primarily Ti d, the first optical excitation can be classed as a charge-transfer transition from the fluorine–mercury framework to titanium-derived conduction states. As photon energy rises, ε₂(ω) reaches its peak intensity about 6.5–7.0 eV. This prominent peak represents the highest transition probability in the entire spectrum, resulting from transitions from deeper F 2p and Hg 5d states to Ti 3d and Hg 6p conduction levels. This feature's high intensity is exactly proportional to the DOS associated with these orbitals. A high concentration of occupied states below the Fermi level and available vacant states above the conduction-band edge boosts optical transition probability and consequently absorption. Additional peaks between 7 and 9 eV are caused by more complex transitions involving deeper Hg 5d levels and higher-energy Ti d, Hg p, and alkali-metal (K/Na) s and p conduction states. Although K and Na make very little contribution near the band boundaries, their higher-energy states participate in optical excitations at high photon energies. The heavy Hg atom influences the optical response significantly. Its localized 5d orbitals strongly hybridize with F 2p states, boosting the density of accessible electronic states and improving the optical matrix elements that absorb photons. This substantial p–d hybridization increases the intensity of ε₂(ω) peaks and contributes to the strong UV absorption reported in both materials. Beyond 10 eV, ε₂(ω) steadily declines due to a decrease in the density of accessible transitions and a lower probability of further interband excitation. In general, the spectra of ε₂(ω) exhibit weak absorption in the visible domain and strong absorption in the UV domain. These materials have great potential as ultraviolet filters, UV-protective coatings, photonic structures, and transparent optoelectronics applications due to their high transmission capability below the band gap energy and UV-absorption properties.

In Fig. 6(c), the refractive index, n(ω), is one of the most important optical parameters, which indicates the propagation of light in a certain medium and the decrease in the speed of light compared with that in vacuum. The virtually identical n(ω) spectra for K₂TiHgF₆ and Na₂TiHgF₆ show that substituting K with Na has only a minimal impact on the optical response. The Ti–Hg–F framework dominates the electronic states responsible for optical transitions, whereas alkali-metal ions contribute very little near the band boundaries, resulting in this resemblance. For both materials, the computed static refractive index, n(0), ranges from 1.50–1.51. These comparatively low values are typical of band-gap fluorides and much lower than those reported for several oxide perovskites. The low refractive index results from a energy gap between the occupied F 2p/Hg 5d valence states and the empty Ti 3d conduction states. Because electrons are more difficult to polarize across a band gap, optical polarizability is modest, resulting in a reduced refractive index. In the low-energy region below 3.0 eV, where photon energies remain below the fundamental band gap, both materials display normal dispersion, defined by a steady increase in n(ω) with increasing photon energy. In this region, incoming photons cannot promote electrons into the conduction band; hence, no meaningful absorption takes place. Instead, the optical response results from virtual electronic excitations between occupied F 2p/Hg 5d states and unoccupied Ti 3d levels. These virtual transitions lead to electronic polarization and raise the refractive index while retaining excellent transparency. As photon energy approaches the strong absorption area of 6–6.5 eV, the refractive index rapidly increases and reaches a maximum of roughly 2.2–2.3. This enhancement is closely related to the sharp peaks found in ε₂(ω) due to the interband electronic transitions. From the analysis conducted via the DOS and PDOS studies, these electronic transitions are attributed to excitations of electrons from the F 2p and Hg 5d valence states to the Ti 3d and Hg p conduction bands. The large number of available electronic states and the p–d hybridization result in enhanced probabilities for the transitions, hence the optical polarization and the rise in n(ω). Outside the main resonance range, the refractive index continuously decreases with increasing photon energy, exhibiting abnormal dispersion around the high absorption peaks. In this case, rapid changes in absorption affect the phase velocity of light, thus leading to a reduction in the refractive index contrary to the expected increase. Such a scenario is a typical resonance effect associated with severe interband transitions. The close connection between n(ω), the band structure, and the DOS indicates that the refractive properties are mainly governed by the transitions between valence and conduction bands. The relatively heavy Hg element plays an indirect role in enhancing the refractive effect through increasing the hybridization between Hg 5d and F 2p states, thus increasing the electronic polarizability. K₂TiHgF₆ and Na₂TiHgF₆ are characterized by low static refractive index, large optical transmission window, and moderate maximum refractive effect and are therefore promising materials for application in transparent optical coatings, dielectric layers, UV photonic devices, etc.

The extinction coefficient, k(ω), is shown in Fig. 6(d). It is an optical parameter characterizing the attenuation of electromagnetic radiation when passing through a material. From a physical point of view, it describes the imaginary refractive index. It is also directly related to optical absorption. Thus, the larger value of the parameter under consideration implies larger optical absorption and losses. On the other hand, close to zero values indicate high transparency. Consequently, the extinction coefficient allows getting information about the energy dependence of optical absorption as well as about electronic transitions leading to photon attenuation. The extinction coefficient k(ω) of K₂TiHgF₆ and Na₂TiHgF₆ is practically zero in the energy range between 0 eV and 3.0 eV. That is why photons with such energy are not able to excite electrons from the valence band to the conduction band. These results are supported by the estimation of the band gaps and the spectra of ε₂(ω). The onset of k(ω) at approximately 3.0–3.2 eV corresponds to the fundamental optical absorption edge and represents the starting point for allowed transitions of electrons. Based on band structure and DOS analysis, these transitions involve transitions of electrons from occupied F 2p and Hg 5d hybridized states of the valence band to unoccupied Ti 3d states of the conduction band. Since the valence band maxima are predominantly formed of F p and Hg d orbitals and conduction band minima contain mainly Ti d orbitals, the first optical transition has a strong charge transfer character from the F–Hg framework to Ti-derived conduction states. With increasing photon energy above the absorption edge, the value of k(ω) increases due to the increased probability of absorption. The highest intensity peaks can be observed in the region of 6.4–6.8 eV, corresponding to the pronounced peaks of ε₂(ω). Strong absorption peaks are related to the probability of interband transitions between occupied F 2p and Hg 5d states and available Ti 3d and Hg 6p conduction states. Both high densities of states in the region above and below the band gap enhance the optical transition matrix elements that yield strong UV attenuation. Further absorption structures observed between 7 and 9 eV are due to the transitions between deeper Hg 5d states and high-energy Ti d, Hg p, and K/Na s and p conduction bands. Even though there is little contribution by K and Na near the edges of the band structure, they contribute more to optical transitions because of their higher energy levels. Consequently, secondary peaks appear, making the UV absorption spectra broader. The heavy Hg atom plays an important role in the value of k(ω). The presence of localized Hg 5d orbitals helps in increasing the density of states and enhances the Hg(5d)/F(2p) hybridization. Moreover, due to the large spatial extent of Hg orbitals, electronic polarization is improved, and there is better interaction between them and incoming electromagnetic waves. From the value of 10 eV, the extinction coefficient consistently decreases with the reduction in density of electronic transitions available. Most valence-to-conduction excitations dominating at that point have occurred by then, and hence fewer states satisfy the conditions for optical transitions. Overall, the k(ω) spectra demonstrate that K₂TiHgF₆ and Na₂TiHgF₆ possess excellent transparency throughout the visible region due to their negligible absorption below the band-gap energy. However, they exhibit strong absorption in the ultraviolet region due to intense F 2p/Hg 5d → Ti 3d/Hg 6p interband transitions. Because of their low visible optical losses and excellent UV attenuation, these materials are ideal for UV filters, UV-protective coatings, transparent optical windows, dielectric layers, and ultraviolet photonic devices. The behaviour of k(ω), together with ε₁(ω), ε₂(ω), and n(ω), confirms that the optical response is principally dictated by the band gap electronic structure and the strong hybridization between F 2p, Hg 5d, and Ti 3d orbitals.

Fig. 7(a) depicts the real part of optical conductivity, σ(ω), which is a fundamental optical characteristic that provides direct insight into charge-carrier production, interband transport mechanisms, and the overall electronic structure of a material. For both K₂TiHgF₆ and Na₂TiHgF_6,σ(ω) remains practically zero up to 3.0 eV. It implies that incident photons do not have enough energy to promote the electrons through the fundamental band gap. In this case, no photo-excited charge carriers are produced in this region, thus showing the band gap behaviour of the semiconductors in question. The absence of low-energy conductivity is fully explained by the DOS and band structure data, showing an evident energy gap near the Fermi level, without any possibility of exciting any electronic states. It also accounts for the high transparency of the materials under visible light conditions. Upon reaching the critical value of band-gap energy, σ(ω) starts to increase, indicating the onset of allowed transitions between the bands. Such an initial increase in conductivity is basically due to the excitation of electrons from the mixed F 2p and Hg 5d valence-band states to Ti 3d conduction-band states. Since these orbitals play the major role in the low-energy optical spectrum and determine the starting point of the conductivity. Above the absorption edge, there is a sudden growth of σ(ω), with the conductivity peaks being at 6 and 8.5 eV. In this region, the conductivity is greater than, demonstrating that the excitation of carriers through photons is very effective. The peaks agree with the main properties that were found in the absorption coefficient, extinction coefficient, and imaginary part of the dielectric function spectra, implying that they are caused by the same interband transition of electrons. The large density of occupied and empty states in these energy regions increases the optical transition probability and intensifies the conductivity response. Additional conductivity features arise between 8 and 12 eV due to transitions into higher-energy Hg 6p and alkali-metal (K/Na) s–p conduction states. Although K and Na contribute very little to electronic states near the band boundaries, their higher-energy orbitals participate in optical excitations at high photon energies, resulting in secondary conductivity peaks. The DOS analysis demonstrates that the valence-band area comprises a high concentration of F p and Hg d states, whereas the conduction-band edge is dominated by Ti d. This advantageous configuration allows for a huge number of potential electronic transitions and generates high optical conductivity. Furthermore, the relatively flat Ti 3d conduction bands in the band structure indicate greater electron effective masses, whereas the slightly more dispersed F 2p/Hg 5d valence bands indicate lighter holes. As a result, interband excitation processes take precedence over free-carrier conduction in the optical response. Beyond 12 eV, σ(ω) steadily declines as the number of possible transitions reduces and the majority of the dominating valence-to-conduction excitations have already happened. The optical conductivity spectra show that K₂TiHgF₆ and Na₂TiHgF₆ are band gap semiconductors with minimal low-energy conductivity, high ultraviolet photoexcitation, and efficient interband charge-carrier production. These properties make them promising materials for ultraviolet photodetectors, UV optoelectronic devices, transparent electronic coatings, and other photonic applications that necessitate high transparency in the visible region as well as strong electronic activity under ultraviolet radiation.

Fig. 7. Computed optical response functions for X₂TiHgF₆ (X = K, Na) halide double perovskites, including (a) real part of optical conductivity, (b) absorption coefficient, (c) reflectivity, and (d) energy loss function.

Fig. 7(b) shows the absorption coefficient, I(ω), which measures how well a material absorbs incident photons as light passes through it. It directly reflects the probability of electronic transitions from occupied states in the valence band to unoccupied states in the conduction band, providing important information about a material's optoelectronic behaviour. The absorption coefficient is close to zero at energies less than 3.0–3.2 eV in K₂TiHgF₆ and Na₂TiHgF₆, suggesting that the photons with energies less than this value cannot cause electronic transitions through the band gap. Such a trend clearly indicates that both materials behave as semiconductors and is well supported by the band gaps obtained from the TB-mBJ calculation, where the values for K₂TiHgF₆ and Na₂TiHgF₆ are 2.14 eV and 1.88 eV, respectively. The absence of any electronic states between the valence-band maximum and the conduction-band minimum in DOS calculations also confirms the absence of absorption in the region. When photon energy exceeds the band-gap energy, I(ω) suddenly rise, signaling the start of authorized interband transitions. The initial optical excitations predominantly involve electron transfer from hybridized F 2p and Hg 5d valence-band states to Ti 3d conduction-band levels at the band boundaries. Because these states correspond to the valence-band maximum and conduction-band minimum, respectively, they influence the absorption edge and dictate the material's fundamental optical response. Absorption is most intense between 6.5 and 7.5 eV, with I(ω) greater than 5 × 10⁴ cm⁻¹. Such high numbers indicate extraordinarily strong UV light absorption, implying that only a very thin layer of material is necessary to significantly attenuate high-energy photons. These prominent absorption peaks are caused by intensive charge-transfer transitions between deeper F 2p and Hg 5d states, as well as higher-energy Ti 3d and Hg 6p conduction states. The high density of occupied and unoccupied states in these energy regions, along with considerable orbital overlap, results in large optical transition probabilities. Additional absorption patterns between 8 and 10 eV are caused by transitions between deeper Hg 5d states and higher conduction-band manifolds comprising Ti d, Hg p, and modest K/Na s–p contributions. Although alkali-metal ions contribute little to electronic states at the band boundaries, their higher-energy conduction states have a role in optical transitions at high photon energies. The presence of Hg is an important factor in the strong absorption behaviour. The Hg 5d orbitals strongly hybridize with F 2p states, resulting in broad valence bands with a high density of states. This p–d hybridization improves the optical matrix elements by increasing the likelihood of photon-induced electronic transitions. Furthermore, the confined character of Hg d states adds to strong UV absorption by giving a large number of electronic states for excitation. Beyond roughly 12 eV, the absorption coefficient steadily declines as the number of energetically favourable interband transitions decreases. The majority of the dominating valence-to-conduction excitations have already occurred, decreasing the overall absorption likelihood. The optical absorption spectra show that K₂TiHgF₆ and Na₂TiHgF₆ have outstanding transparency in the visible range and extraordinary absorption in the ultraviolet region. This distinct combination makes them ideal candidates for UV-shielding coatings, ultraviolet photodetectors, optical filters, transparent protective windows, and advanced UV-optoelectronic devices that require efficient blocking of harmful high-energy radiation while maintaining visible-light transparency.

Fig. 7(c) shows the reflectivity spectrum, R(ω), which depicts the fraction of incident electromagnetic radiation reflected from a material's surface and offers significant information about its interaction with light. Reflectivity is very close to the dielectric function, refractive index, and electronic polarizability and therefore serves as an appropriate measure of electronic structure. At low energy, reflectivity in K₂TiHgF₆ and Na₂TiHgF₆ is generally less than 0.15–0.20; this implies that not much of the light falling on the material gets reflected, and most of the radiation gets transmitted through the material. This type of reflectivity profile is normally observed in band gap semiconductors and agrees well with the moderate static dielectric constant and relatively low refractive index. Reflectivity stays essentially constant and low in the energy region below the optical band gap (about 3.0–3.2 eV) because photons lack sufficient energy to drive electrons from the valence band to the conduction band. As a result, there are no meaningful interband transitions, and only weak electronic polarization occurs. The low absorption and electrical excitation in this area account for both materials' exceptional optical transparency in the visible spectral range. As photon energy increases beyond the absorption edge, reflectivity progressively increases and becomes more prominent in the UV range. Reflectivity peaks between 6 and 8 eV, with R(ω) values ranging from 0.40–0.45. These peaks are consistent with the prominent features detected in the dielectric function, absorption coefficient, extinction coefficient, and optical conductivity spectra, indicating a shared electronic origin. These reflectivity maxima, according to the band structure and DOS investigations, are predominantly caused by intensive interband transitions involving occupied F 2p and Hg 5d hybridized valence-band states and unoccupied Ti 3d and Hg 6p conduction-band states. The high density of states associated with these orbitals results in substantial optical oscillator strengths, which enhance the interaction of light with the electronic system. Reflectivity increases in this energy range in direct proportion to increased dielectric polarization. With an increase in the number of electrons that are excited by photons, there will be an increase in polarization and hence reflectivity of the material due to an increase in its capacity to reflect incident radiation. There will be a good overlap of the F 2p and Hg 5d orbitals, and thus p–d hybridization will take place and will enhance the polarizability and optical response. The heavy metal atom Hg, therefore, plays a great role in enhancing the reflectivity of the metal because it increases the formation of localized d-states and enhances the density of electronic transitions. In the plasma-like region, there is a sharp change in R(ω) near energies at which the real part of the dielectric function ε₁(ω) goes to zero. At high energies, there will be oscillations of valence electrons and hence a plasma-like response. Therefore, under such circumstances, the propagation of electromagnetic waves will not be possible, and hence light is reflected from the surface. Above 12 eV, the value of the reflectivity constantly decreases and reaches a stable level. This occurs owing to the fact that all the main inter-band transitions for effective polarization of light have occurred, thus leaving fewer possibilities for the electron excitation. As a result, as the photon energy becomes higher, the role of electricity in the reflection phenomenon decreases. From the equal shapes of the reflectance spectra of K₂TiHgF₆ and Na₂TiHgF₆, one can conclude that replacing potassium with sodium affects the optical properties insignificantly.

In Fig. 7(d), there is the energy loss function, L(ω), which is used to describe the energy lost by high-energy electrons when they travel through a material. It contains a lot of useful information on collective electronic excitations. L(ω) has great significance when it comes to the analysis of the data from EELS measurements. Under such conditions, the valence electrons will oscillate collectively because of an external disturbance, hence causing plasmon resonance. The energy loss function of both K₂TiHgF₆ and Na₂TiHgF₆ is extremely weak in the low-energy range of less than 5 eV. This is not surprising since both materials have band gaps and no free charge carriers near the Fermi level. Such behaviour is perfectly consistent with the DOS and band-structure data revealing the presence of a definite energy gap and no electronic states crossing the Fermi level. A clear and well-defined plasmon peak appears within the energy range of 6.5–8.0 eV, which is the most distinct feature of the loss spectrum. Such a resonance relates to the oscillations of valence electrons mostly from the F 2p and Hg 5d orbitals that make up the major part of valence band states. In such an energy range, the loss function reaches its point of sign change (crossing zero), meaning plasma oscillations. The high intensity of the plasmon peak can be considered as proof of strong collective electronic polarization and effective coupling of energetic electrons with the electron cloud of the crystal. The position of the plasmon resonance depends on the electronic density and the chemical bonding in the Ti–Hg–F matrix. Electronic structure, DOS distribution, and band-edge composition in K₂TiHgF₆ and Na₂TiHgF₆ are nearly identical, leading to nearly identical plasmon energies and loss spectra. Due to the small influence of alkali-metal ions on the electronic states responsible for the optical properties, such substitution leads to minor differences. The effect of the massive Hg element also plays an important role in the enhancement of the loss structure. The existence of Hg 5d electrons increases the electron density and also enhances the hybridization of Hg(5d)/F(2p), thereby improving the collective behaviour of the electronic system. In addition, the rather localized nature of the Hg d state causes increased electronic polarization and the development of plasmon resonance. The minor losses above the major plasmon peak arise due to the interband transition of Ti 3d, Hg 6p, and alkali metal s/p states.

3.5. Thermoelectric properties

Fig. 8(a) displays the Seebeck coefficient, S, and is the most important characteristic of thermoelectrics, as this quantity defines the electric potential produced by applying a temperature gradient to a sample. The polarity of this parameter corresponds to the type of charge carrier dominating in a material, while the magnitude characterizes the degree of asymmetry of electron states around the Fermi level and their energy dependence on transport. For K₂TiHgF₆ and Na₂TiHgF₆ compounds, the Seebeck coefficient values are negative within a temperature range from 50 to 700 K, which means that electrons act as major carriers; thus, the studied samples have semiconductor behaviour. The absence of a sign change with an increase in temperature suggests the absence of any significant effect of thermal generation of holes. At 700 K, the amplitude of S grows constantly, reaching approximately −18.5 × 10⁻⁶ V K⁻¹ for K₂TiHgF₆ and −20 × 10⁻⁶ V K⁻¹ for Na₂TiHgF_6. The reason for this increment is that higher temperatures widen the energy spectrum of carriers, thus providing participation of electrons whose energies deviate substantially from the Fermi level. Thus, the average energy of carriers compared to the Fermi level increases, leading to higher thermoelectric voltage. Both materials exhibit nearly linear temperature dependence, which is typical for semiconductors with transport dominated by thermally active electronic states. Na₂TiHgF₆ shows a higher absolute value of Seebeck coefficient in comparison with K₂TiHgF₆, meaning that the former has a better electronic structure for thermoelectric voltage generation. One possible reason behind this difference may be the smaller TB-mBJ band gap (1.88 eV) of Na₂TiHgF₆ compared to that of K₂TiHgF₆ (2.14 eV). The smaller gap makes it possible for the thermal excitation of electrons, while still providing enough carrier filtering, thus leading to higher thermopower. Moreover, the higher absolute value of S means that the DOS asymmetry around the Fermi level is higher and the carrier concentration is lower. It helps understand the described trends by looking at the orbital-resolved electronic structure. From the DOS analysis, it can be seen that the conduction band minimum is formed by localized Ti 3d states, while the valence band maximum is mainly composed of hybridized F 2p and Hg 5d states. The localized nature of Ti 3d states causes strong peaks of DOS near the conduction band minimum, enhancing the energy dependence of carrier transport and thus increasing the Seebeck coefficient in accordance with Mott's relation for charge transport. It was noted that the larger Seebeck coefficient of Na₂TiHgF₆ was caused by better state localization of Ti and increased DOS asymmetry near the band edge. It should also be noted that the heavy Hg atom plays its own indirect but important role here. It will enhance carrier energy filtering and enhance entropy per charge carrier, hence enhancing the Seebeck coefficient. Besides, the existence of Hg in the material enhances the edge band complexity, which is usually an attribute of enhanced thermoelectric performance since it facilitates enhanced DOS near transport levels. Generally, the continuous rise in S with respect to temperature and the dominant role played by Ti 3d in the conduction band and Hg 5d/F 2p in the valence band imply that the two compounds have favourable electrical characteristics as thermoelectric materials. Na₂TiHgF₆ is a superior thermoelectric material at high temperatures compared to K₂TiHgF₆ owing to a lower band gap and enhanced DOS asymmetry.

Fig. 8. Computed thermoelectric properties for X₂TiHgF₆ (X = K, Na) halide double perovskites as a function of temperature: (a) Seebeck coefficient S, (b) electronic thermal conductivity κ_e/τ, (c) electrical conductivity σ/τ, (d) dimensionless figure of merit ZT (e) lattice thermal conductivity κ_l/τ, (f) power factor S²σ/τ.

Thermal conductivity κ_e/τ, is an important parameter in the study of heat transfer in semiconductors and thermoelectrics. In an electronic system, the very same electrons that conduct electricity also conduct heat. A rise in conductivity or temperature leads to a rise in thermal conductivity by electrons. This trend is clearly evident in Fig. 8(b) both K₂TiHgF₆ and Na₂TiHgF₆, where κ_e/τ grows almost linearly from about 1 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹ at 50 K to almost 16 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹ for K₂TiHgF₆ and around 14 × 10¹⁴ W m⁻¹ K⁻¹. This trend being nearly linear suggests that with rising temperature, electrons are more efficient in transporting heat through thermal stimulation. The values of the κ_e/τ ratio in K₂TiHgF₆ are higher because of high electrical conductivity. The transfer of both charge and heat takes place through electrons; therefore, K₂TiHgF₆ has better carrier mobility. The electronic band structures reveal that the conduction-band minimum is dominated by somewhat dispersive Ti 3d states, which are in charge of both electrical conduction and electronic heat transport. K₂TiHgF₆ has larger Ti-derived conduction bands that increase carrier velocities and reduce effective masses. This allows electrons to transport energy more effectively through the crystal lattice. Consequently, both σ/τ and κ_e/τ remain greater than those of Na₂TiHgF₆ across the temperature range. This conclusion is supported even further by the DOS analysis. Electronic states near the conduction band edge are mostly Ti 3d, but hybridized F 2p and Hg 5d orbitals dominate the valence band region. Hg is a heavy element that contributes significantly to the overall electronic structure via Hg 5d–F 2p hybridization, although it does not dominate the transport-active states necessary for heat conduction. Instead, Hg mostly alters the electrical environment surrounding the Ti-centred conduction network, altering band curvature and carrier dynamics indirectly. Thus, the amplitude of κ_e is regulated mostly by the properties of the Ti 3d conduction states rather than direct Hg-derived transport channels. The smaller Na ion in Na₂TiHgF₆ creates a slightly more compact lattice, leading to increased orbital localization and flatter conduction bands. These flatter bands have greater electron effective masses and reduced carrier mobility, lowering both electrical conductivity and electronic heat conductivity. Thus, κ_e/τ is consistently smaller than in K₂TiHgF₆. This reduction benefits thermoelectrics because excessive electrical heat transmission reduces thermoelectric efficiency by reducing the temperature gradient required for energy conversion. Ideally, a good thermoelectric material requires a high value of Seebeck coefficient, high electrical conductivity and low thermal conductivity. A lower value of the κ_e/τ ratio in Na₂TiHgF₆ would make a positive impact in such an environment because lower loss of heat and acceptable electrical transport can be achieved through this material. Relatively high value of thermopower, moderate electrical conductivity and low value of electrical thermal conductivity can help maintain a larger temperature gradient between two ends of the material, resulting in greater energy conversion. Higher electrical conductivity and carrier mobility are observed in K₂TiHgF_6, whereas Na₂TiHgF₆ has a comparatively low value of electrical thermal conductivity.

Fig. 8(c) illustrates the separation of electrical conductivity σ/τ, which is an important transport property that represents the natural ability of charge carriers to move across the medium subjected to electric field strength. Hence, it reveals the transport properties of electrons regardless of any particular scattering process. At low temperatures, σ/τ is around 2.50 × 10¹⁸ Ω⁻¹ m⁻¹ s⁻¹, while at 700 K, it is about 2.20 × 10¹⁸ Ω⁻¹ m⁻¹ s⁻¹. These relatively low drops prove that both compounds have quite stable electronic conduction characteristics within the wide temperature range, which is useful for thermoelectric devices working under high temperatures. K₂TiHgF₆ has relatively larger σ/τ values than Na₂TiHgF₆ in all temperature intervals, which proves better electrical conduction features. Electrical structures may be the reason for this difference. The conduction band minimum of these compounds includes Ti 3d orbitals, which serve as transport-active states for electron conduction. Band structure calculations reveal that these conduction bands derived from Ti 3d states have significant band dispersion and not complete band flatness. Large band dispersion means high carrier group velocity and small effective mass of carriers, which results in the improvement of electrical conduction properties. The better electrical conduction in K₂TiHgF₆ may be connected with the effect of structural changes induced by bigger K⁺ ions. Since potassium has a larger ionic radius than sodium, K₂TiHgF₆ has a slightly larger structure in comparison with Na₂TiHgF₆, which changes the Ti–F bond lengths and Ti–F–Hg orbital interactions. As a consequence, effective masses of charge carriers become smaller and mobility increases. The smaller size of the Na⁺ ion results in a reduced lattice in Na₂TiHgF₆. Reduced lattice sizes lead to improved localization of orbitals, mainly at Ti-centred conduction states, thus leading to flatter bands and higher carrier effective masses. Thus, electron mobility decreases, leading to reduced σ/τ ratios. This conclusion is confirmed by the DOS and PDOS analysis. The conduction zone consists mainly of Ti 3d states, while the valence zone is mostly formed by F 2p/Hg 5d hybridized states. While Hg 5d/F 2p states themselves do not contribute significantly to electron transport, high hybridization affects overall band curvature, bandwidth, and anisotropy of electron transport. Interaction between Hg 5d/F 2p states changes the crystal potential for conduction electrons and indirectly affects electron mobility through changes in Ti-based conduction bands. Increased temperature promotes the carrier scattering process. In spite of the CRTA approach, thermal broadening of the carrier distribution decreases the average carrier velocity and transport efficiency. High temperature further stimulates phonon motion, which, in practical materials, restricts mobility. That is why σ/τ decreases gradually for both compounds. But the decrease is very small, which means that the electronic structure does not depend on thermal effects and there are no abrupt transitions for transport states. Such features as dispersive character of Ti 3d conduction bands, strong Hg 5d/F 2p hybridization and broad band gap semiconductor behaviour result in rather high and temperature-independent conductivity. Na₂TiHgF₆ has low conductivity due to localization of carriers, whereas K₂TiHgF₆ has broader conduction bands and higher mobility. From the point of view of electrical transport, K₂TiHgF₆ is more conducting than Na₂TiHgF₆. However, such properties should be compared with the high Seebeck coefficient and low thermal conductivity of Na₂TiHgF_6.

ZT in Fig. 8(d) is the most all-inclusive figure of merit for the analysis of thermoelectric efficiency since it includes Seebeck coefficient (S), electrical conductivity (σ), thermal conductivity (κ), and temperature (T). The value of ZT is described by the equation ZT = S²σT/κ. High ZT requires a very precise combination of such interrelated factors as a high Seebeck coefficient, which allows obtaining high thermoelectric voltage; high electrical conductivity, which makes charge transportation more effective; and low thermal conductivity, which keeps the temperature gradient to convert energy. Therefore, ZT is the most trustworthy predictor of a material's ability to generate electricity from waste heat. ZT of K₂TiHgF₆ and Na₂TiHgF₆ increases linearly up to 700 K, which shows considerable improvement in thermoelectric performance at elevated temperatures. At lower temperatures, ZT is less than 0.1 because thermal energy cannot provide enough activation of charge carriers in a band gap. As a result, the Seebeck coefficient and carrier transport remain limited, limiting thermoelectric efficiency. However, as temperature rises, carrier excitation becomes more effective, resulting in continuing improvement of thermoelectric performance. The constant expansion in the size of the Seebeck coefficient is the primary driver of the increase in ZT. Although electrical conductivity falls slightly with temperature, this decline is minimal and is more than offset by the increase in thermopower and the temperature factor in the ZT equation. As a result, both materials have significantly improved thermoelectric efficiency at high temperatures. At 700 K, K₂TiHgF₆ has a ZT value of around 0.82, while Na₂TiHgF₆ has around 1.0, indicating higher thermoelectric performance. Na₂TiHgF₆ has a higher ZT due to its advantageous electrical structure. First, it has a higher absolute Seebeck coefficient across the whole temperature range, indicating more energy-dependent carrier transport. This increased thermopower is accompanied by a higher imbalance in the density of states around the Fermi level and a more pronounced localization of transport-active electronic states. DOS calculations show that the conduction-band edge is dominated by Ti 3d orbitals, while the valence-band edge is dominated by hybridized F 2p and Hg 5d states. This orbital structure causes a sharp fluctuation in DOS near the band borders, which is advantageous for increasing the Seebeck coefficient. Secondly, Na₂TiHgF₆ has a slightly narrower TB-mBJ band gap (1.88 eV) than K₂TiHgF₆ (2.14 eV). The narrowed band gap allows for excitation of electrons by heat energy without losing semiconducting properties. Na₂TiHgF₆ provides the most adequate combination of carrier density and thermopower. High band gaps usually suppress carrier creation, while narrow gaps reduce the Seebeck coefficient. It can be said that the electrical structure of Na₂TiHgF₆ is close to the optimal balance required for thermoelectric performance. Reduced electronic thermal conductivity of Na₂TiHgF₆ is another important parameter. Electrical conductivity of K₂TiHgF₆ is higher owing to a wider conduction Ti 3d band and greater mobility of carriers. This parameter, however, results in increased heat conduction, according to the Wiedemann–Franz relation. Na₂TiHgF₆, however, exhibits low values of κ_e, ensuring the retention of temperature gradients and converting more thermal energy into electrical form. The heavy element Hg plays an indirect role in contributing to the high values of ZT in both compounds due to the existence of a strong interaction between Hg 5d–F 2p, which brings about changes in the curvature of the bands and makes the DOS more complicated near the Fermi level. This leads to an increase in transport asymmetry and thermopower. In the meantime, the states derived from the Ti 3d are responsible for the electron transport and are very important in determining conductivity and thermoelectricity. These characteristics balance its somewhat lower electrical conductivity, resulting in a greater ZT approaching unity at 700 K.

The lattice thermal conductivity of the two compounds shows a remarkable reduction (see Fig. 8(e)) with the increase of temperature in the lower temperature region and then gradually becomes flat at higher temperatures, indicating that the phonon mediated heat transport is strongly suppressed with the increase of thermal excitation. For K₂TiHgF₆, the lattice thermal conductivity begins with a relatively higher value at low temperature and falls rapidly in the initial temperature interval, and then decreases more slowly and the curve tends to an almost constant low κ_l regime. This is the impression of increasing phonon–phonon scattering with temperature: as the population of thermally activated phonons increases, so does the strength of the phonon–phonon interactions, which in turn reduces the phonon mean free path, and thus reduces the lattice contribution to heat conduction. The same overall trend is observed in Na₂TiHgF₆, confirming that the temperature dependence is governed by similar phonon-scattering mechanisms in the two materials. However, the Na-containing material remains slightly above the K-containing material throughout the entire range plotted, indicating a comparatively larger lattice thermal conductivity and thus somewhat more efficient phonon heat transport. The difference is most noticeable in the region where both curves show their high temperature peaks, but the separation between the two materials is rather modest. The lower κ_l of K₂TiHgF₆ is of technological relevance, since a reduced lattice thermal conductivity is generally beneficial for thermoelectric applications, provided the electrical transport is also maintained at a sufficiently high level at the same time. The nearly converging trend at high temperature also indicates that the difference in phonon transport between two materials becomes less pronounced with the increase of temperature. It is vital to point out that the decreasing trend is not simply a significance of increasing thermal energy but rather a competition between phonon population, phonon group velocities and temperature dependent scattering processes. This comparison shows that the replacement of K by the smaller Na cation modifies the phonon transport properties sufficiently to result in a slightly enhanced lattice contribution to the thermal conductivity. Therefore, K₂TiHgF₆ of the two compounds is more advantageous in suppressing the lattice thermal conductivity, while Na₂TiHgF₆ has a relatively larger phonon-mediated thermal conductivity. The overall common decreasing trend of both compounds and the eventual low-temperature-independent behavior at higher temperature indicate the increasing dominance of phonon scattering, and the systematic separation between the curves highlights the role of chemical composition in controlling the lattice thermal transport. Therefore, the plotted results provide useful evidence that both materials can exhibit low lattice thermal conductivity at elevated temperature, and K₂TiHgF₆ shows the more favorable thermal profile for thermoelectric performance.

The temperature-dependent power factor of K₂TiHgF₆ and Na₂TiHgF₆ is shown in the Fig. 8(f), which exhibits a very different behavior from the lattice thermal conductivity. Unlike the decreasing κ_l, the power factor of both materials increases systematically with temperature in the studied range, indicating the electronic transport contribution becomes more favorable at higher temperatures. The power factor for K₂TiHgF₆ begins at a relatively low value at the low temperature end and then gradually increases, and the increase is particularly obvious toward the high temperature region. The continuous improvement suggests that temperature improves the electronic transport processes that are responsible for the calculated power factor. This can be a result of increased carrier activation and thermoelectric transport under the conditions used in the calculations. Na₂TiHgF₆ also shows the same general increasing trend, indicating that the electronic-transport response of both materials is comparable with temperature. However, the Na compound still has a slightly higher power factor than K₂TiHgF₆ in the temperature range plotted and the difference becomes more significant at higher temperatures. This means Na₂TiHgF₆ displays a comparatively better electrical transport response in the calculated conditions, which is a good advantage. A large power factor is one of the essential requirements to obtain a good thermoelectric performance. Thus, the comparison of the two materials shows an interesting tradeoff between thermal and electronic transport properties. K₂TiHgF₆ shows the lower lattice thermal conductivity benefiting to the parasitic heat conduction. Na₂TiHgF₆ expresses the higher power factor, which indicates more favorable electronic transport. In fact, it is more informative to consider these two quantities simultaneously rather than evaluating either property separately, since thermoelectric performance requires high electrical transport and low lattice-mediated thermal transport. The near parallel increase of the power-factor curves shows that both compounds benefit from an increase in temperature, but the always larger values for Na₂TiHgF₆ indicate that its electronic structure and its carrier-transport properties are advantageous in this respect. The larger separation of the curves at high temperature shows the difference in their electronic behavior more clearly. The two compounds display complementary features when combining these results with the lattice thermal conductivity behavior, i.e., the advantage of reducing the lattice heat transport for K₂TiHgF₆ and the advantage of power-factor improvement for Na₂TiHgF₆. Therefore, the overall thermoelectric performance of the material cannot be simply judged by the power factor alone, and the electrical conductivity, Seebeck coefficient, total thermal conductivity and finally the dimensionless figure of merit ZT must be considered with consistency. However, the plotted power-factor results clearly show that the electronic transport potential of both K₂TiHgF₆ and Na₂TiHgF₆ is enhanced by increasing temperature, with Na₂TiHgF₆ showing the relatively stronger power-factor response across the whole temperature range investigated.

4. Conclusion

In conclusion, the first-principles results indicate that K₂TiHgF₆ and Na₂TiHgF₆ are promising cubic Heusler fluorides with very stable structures and multi-functional properties. Both compounds crystallize in the stable Fm3̄m phase, exhibit well-defined energy–volume minima, and possess negative formation and cohesive energies, confirming their thermodynamic feasibility and strong chemical stability. The smaller Na ion produces a more compact lattice with stronger interatomic interactions than the K analogue. Electronic structure calculations reveal band gap semiconducting behaviour with TB-mBJ energy gap values of 2.14 and 1.88 eV, respectively, while the valence-band edge is dominated by hybridized Hg 5d/F 2p states and the conduction-band edge originates primarily from Ti 3d orbitals, enabling efficient electronic transitions. The optical analysis indicates very high transparency at low energy levels, high ultraviolet absorption, high optical conductivity beyond the absorption edge, and the presence of plasmon peaks in the range of 6.5 to 8.0 eV, proving their excellent performance in ultraviolet photonic and optoelectronic devices. The mechanical analysis proves that both materials have the Born criteria, positive elastic constants, medium stiffness, low elastic anisotropy, and good compressibility and shear resistance, but Na₂TiHgF₆ is found to have high bulk, shear, and Young's moduli due to its dense crystalline structure. Mechanical calculations prove that both compounds comply with Born criteria of stability, have positive values of elastic moduli, reasonable stiffness, weak anisotropy of elastic properties and good resistance to compressive and shear deformation; Na₂TiHgF₆ has greater bulk, Young's and shear moduli due to its denser crystal structure. The negative value of Cauchy pressure, positive value of shear modulus and Pugh ratio less than the critical one demonstrate the presence of ionic–covalent bondings and brittleness of these fluoride semiconductor materials. Thermoelectric properties predict that both compounds are semiconductors with negative Seebeck coefficients at all temperatures. K₂TiHgF₆ is characterized by comparatively higher electrical conductivity due to broader Ti 3d conduction bands, Na₂TiHgF₆ has a larger Seebeck coefficient, lower electronic thermal conductivity and a better ratio of carrier concentration to thermopower, providing much better thermoelectric properties with a maximum ZT of about 1.0 at 700 K. In summary, excellent thermodynamic stability, a band gap of electronic structure, strong ultraviolet light absorption properties, stability under mechanical stress and promising thermoelectric behaviour make these Heusler fluorides prospective materials for future application in UV optoelectronics and energy conversion application.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data are available upon request from the corresponding author.