First-principles identification of PtTiGe and PtTiPb as high-efficiency thermoelectric half-Heuslers

Abstract

This study presents a comprehensive first-principles investigation of the structural, mechanical, electronic, optical, thermoelectric, and thermodynamic properties of half-Heusler PtTiZ (Z = Ge, Pb) compounds using the full-potential linearized augmented plane-wave (FP-LAPW) method combined with semiclassical Boltzmann transport theory. Exchange–correlation effects were treated within the LDA, PBE-GGA, and Tran–Blaha modified Becke–Johnson (TB-mBJ) schemes to achieve accurate electronic descriptions. Both alloys crystallize in a stable cubic F-43 m structure and exhibit indirect semiconducting behavior with band gaps of 0.66 eV (PtTiGe) and 0.387 eV (PtTiPb). The density-of-states analysis indicates that the valence region is dominated by Ti-3d and Z-p hybridized states, confirming strong p–d interactions. Mechanical stability criteria and positive elastic constants verify that both compounds are mechanically robust, with PtTiGe being stiffer and harder than PtTiPb. Optical results reveal pronounced absorption and high optical conductivity in the ultraviolet region, suggesting potential for optoelectronic applications. Thermoelectric analysis demonstrates p-type character with Seebeck coefficients of 229.21 µV K⁻¹ (PtTiGe) and 236.21 µV K⁻¹ (PtTiPb) at 300 K, and 235.05 µV K⁻¹ and 237.31 µV K⁻¹ at 1200 K, respectively. The corresponding lattice thermal conductivities decrease to 0.45 W m⁻¹ K⁻¹ and 0.32 W m⁻¹ K⁻¹, yielding maximum dimensionless figures of merit (ZT) of 0.68 and 0.70 at 1200 K. Thermodynamic results confirm that the Debye temperature increases with pressure while heat capacity decreases, ensuring stability at elevated conditions. Overall, the synergistic combination of electronic tunability, optical responsiveness, and favorable thermoelectric performance highlights PtTiZ (Z = Ge, Pb) as promising candidates for high-temperature thermoelectric and ultraviolet-optoelectronic applications.

Introduction

Safe and alternative energy sources that are not dependent on fossil fuels are urgently needed¹. For these choices to be regarded as alternative energy sources, they must be affordable and ecologically benign². All of these requirements can be met by thermoelectric energy production, which turns temperature variations between specific materials into electrical power^3,4,5. Compact and adaptable, thermoelectric generators (TEGs) have been shown to exploit waste heat as a source of temperature bias in residential, commercial, and automotive settings⁶. The temperature bias among the many n and p-type thermal electricity legs that make up TEGs causes the generation of mobile electrons and holes⁷. Due to the massive consumption of fossil fuels in various forms of energy, today’s world is rapidly advancing technologically, leaving fewer fossil fuels for future generations⁸. Alternative, environmentally acceptable ways to turn waste heat into useful electricity are therefore desperately needed⁹. In this regard, it is anticipated that thermoelectric materials which transform waste heat into electricity will be crucial in offering a substitute remedy for the world’s energy dilemma^10,11 Thermoelectric (TE) energy generation offers a sustainable solution to the global energy crisis by directly converting waste heat into useful electrical energy through the Seebeck effect, without the need for moving parts or greenhouse-gas emissions¹². A significant portion of industrial and automotive energy is lost as heat is estimated to exceed 60% of the total primary energy input and recovering even a fraction of this waste can substantially improve overall efficiency¹³. TE devices based on half-Heusler materials are particularly attractive because of their high thermal stability, mechanical strength, and environmentally benign composition. By enabling continuous, maintenance-free energy harvesting from residual heat sources, thermoelectric technology can reduce fossil-fuel dependence, lower carbon emissions, and contribute to the development of decentralized, clean energy systems crucial for a sustainable future¹⁴. The Seebeck effect is used by thermoelectric materials to transform heat into electrical power¹⁵. The figure of merit ZT can be used to determine the material’s efficiency in converting heat into electricity^16,17,18,19. The interplay between electronic and thermal properties significantly influences the performance of thermoelectric materials, making it challenging to identify optimal candidates. A high-quality thermoelectric material generally demonstrates semiconducting behavior with significant doping, exhibiting a Seebeck coefficient (S) between 150 µV/K and 250 µV/K. To comprehend the thermoelectric properties of various materials, such as PbTe compounds, chalcogenides, and skutterudites research has been done^20,21,22. Half-Heusler compounds, with an 18-valence electron configuration and a semiconducting band structure, are considered promising candidates for thermoelectric applications due to their strong electrical conductivity, high Seebeck coefficient, and exceptional temperature stability^23,24. According to recent research, ZT values of 0.46, 0.35, and 0.29 at 1200 K have been found for n-type Ni-based half-Heusler alloys²⁵. RhScTe was discovered to be an intriguing thermoelectric substance with p- type conductivity by Adebambo et al.²⁶. After that they examined the thermoelectric efficiency of the Rh-based Heusler alloy²⁷. Researchers are also becoming interested in Pd-based alloys because to its respectable thermoelectric performance²⁸. According to Ashani et al.‘s study²⁹, the thermoelectric efficiency of PdTiSn with p-type character is expected to be more effective than that of n-type character in this situation^30,31. In a similar way, Nagura et al.^32,33, examined PdHfSn’s thermoelectric performance and discovered that its Seebeck coefficient was 56.11 µV/K. Furthermore, p-type PdZrGe has been found to have a ZT value of 0.759 at 300 K and a Seebeck coefficient worth 237µV/K³⁴. Exploring the thermoelectric properties of Pd-based materials could offer valuable insights into identifying suitable candidates for thermoelectric energy devices. Using the chemical formula ABC, there is a particularly interesting class of ternary substances that crystallize in the C1_b structure³⁵. Materials from different parts of the periodic chart are included in this structure: transition alloys (A, B) and a main group metal (C)^36,37. Several structural models can be used to describe half-Heusler compounds, such as a rock salt structure where extra atoms (A) occupy particular interstitial positions or a filled tetrahedral arrangement derived from the zinc-blende lattice³⁸. When these substances have a certain number of valence electrons, they may behave semiconducting, according to theoretical predictions that utilize electron counting principles. The freedom to select components for A, B, and C enables a wide range of adjustable attributes, including both electrical and physical traits (such as lattice parameters and elastic moduli)³⁹. Heterostructures, which blend two different ternary substances (ABC and A’B’C’), further increase this adaptability. These combinations can show notable variances in some aspects (valence band position, lattice constant) but striking similarities in another^40,41. Binary substances rarely provide for this degree of control over their characteristics. These materials’ inherent flexibility makes material design easier, which allows scientists to forecast and create materials with particular desired qualities. The theoretical predicted CoTaSn serves as an illustration of this methodology⁴², Doping this ternary semiconductor results in both visible transparency and p-type electrical conductivity. Another application of the concept of materials design is the optimization of thermoelectric materials, a class of materials that have the ability to convert heat into electricity. Some recent research has examined methods for optimizing binary thermo-electrics through doping, while others have concentrated on increasing conductive properties in half-Heusler alloys^43,44. The choice of the major group element in Heusler alloys has a significant impact on the material’s characteristics, allowing for customization for specific uses. This manipulation gives rise to a wide range of electronic behaviors, like metallic, spin-semi-metallic, half-metallic, spin-gapless, semiconducting, and topological or superconducting characteristics⁴⁵. Interestingly, half Heusler materials with narrow bandgaps show an especially attractive set of characteristics. These materials are often characterized by high melting points, elevated Curie temperatures, significant thermopower, and either inherent or negligible magnetism⁴⁶. The majority of C1b ternary compounds still lack a thorough understanding of the material properties essential for semiconductor technology, despite the fact that high-throughput computing technologies have made it easier to explore the fundamental features of this vast field. This contains comprehensive data on the charge carrier’s efficient masses and deform potentials. By offering a comprehensive theoretical characterization of two prominent C1_b materials, PtZrSi and PtZrGe, this work seeks to close this gap. These materials’ prospective uses in thermoelectric systems are the main driving force behind research into them. Recent theoretical research has investigated NiZrSn’s thermoelectric characteristics⁴⁷. Half-Heusler materials like NiTiSn, Ni(Zr, Hf)Sn, and CoTiSb, with similar compositions, have yielded encouraging experimental results. Experimental observations indicate that ternary compounds with the general formula ABC, where C is a group IV element, commonly exhibit n-type semiconductor behavior when doped. On the other hand, group V substances containing element C are more commonly used for p-type doping. The remarkable thermoelectric (TE) capabilities of half-Heusler compounds have sparked a recent wave of interest in thermoelectric. By optimizing their composition, these ternary materials can deliver enhanced thermoelectric performance at high temperatures and can be engineered to exhibit either n-type or p-type conductivity. Notably, half-Heusler alloys based on p-type FeNbSb have attained an impressive figure of merit (ZT) of 1.5 at 1200 K. Thermoelectric materials such as n-type XNiSn and p-type XCoSb, with X representing Ti, Zr, or Hf, have shown zT values greater than 1.0 at high temperatures, indicating strong potential for energy conversion applications. The Slater-Pauling rule, which produces a semi-conducting electronic structure perfect for TE applications, is the reason half-Heusler materials are successful in thermoelectric devices. These materials stand out due to their combination of high power output, low cost, mechanical strength, eco-friendliness, and excellent long-term stability. As a result, half-Heusler metals are regarded as leaders in the creation of thermoelectric devices of the future.

Therefore, using density functional theory (DFT), we have attempted to examine the thermoelectric characteristics of PtTiZ (Z = Ge, Pb) half Heusler compounds in both p as well as n-type nature in this study. Our goal is to determine the sort of temperature, chemical potential, carrier concentration at which compounds exhibit the highest level of thermoelectric performance. Therefore, in order to fully investigate the structural, electric, mechanical, as well as thermoelectric properties of PtTiZ (Z = Ge, Pb), we used a hybrid technique that combines first-principles calculations with Boltzmann transportation theory. The motivation behind investigating the PtTiZ (Z = Ge, Pb) half-Heusler alloys stems from the urgent need to develop advanced multifunctional materials capable of addressing the twin challenges of energy conversion efficiency and environmental sustainability. Half-Heusler systems have emerged as promising candidates for thermoelectric and optoelectronic applications owing to their inherent mechanical robustness, high thermal stability, and tunable electronic structures. Despite extensive studies on Ni-, Co-, and Fe-based Heuslers, the Pt–Ti-based family remains largely unexplored, particularly for combinations involving group-IV and group-V elements such as Ge and Pb. Their distinct electronic configurations, heavy-atom effects, and variable p–d hybridization strength offer new opportunities to tailor carrier transport and phonon scattering mechanisms. Therefore, the present study aims to systematically explore how compositional variation from Ge to Pb influences the structural, mechanical, optical, and thermoelectric responses of PtTiZ alloys, thereby contributing valuable insights toward the rational design of next-generation energy-efficient materials. The current investigation will be divided into four sections: (1) structural and computational details; (2) compound stabilization; (3) electronic, optical thermodynamic, and thermoelectric characteristics; and (4) Conclusion.

Computational methodology

The DFT⁴⁸ as applied to the WIEN2k program⁴⁹ was used to analyze the sample compounds’ structural and electrical characteristics. The generalized gradient approximate (GGA) with an implementation of Perdew-Burke-Ernzerhof (PBE) potential and the TB- mBJ potential⁵⁰ were used to treat the exchange-correlation interactions between the electrons. This approach improved the calculated bandgaps of the material. The basis function is expanded to R_MT×K_MAX = 7.0 to achieve the charge and energy convergence. The cut-off power which denotes the difference between the core and valence states, has been set at -6.0 Ry. In the unstable wedge of the Brillouin zone (BZ), a dense grid of 10,000 k-points was taken into consideration to carry out the consistent with itself cycle consistent field (SCF) for a given energy density. The muffin-tin (MT) radii corresponding to PtTiGe were determined to be 2.49, 2.25, and 2.25 a. u. for Pt, Ti, and Ge, respectively; in the case of PtTiPb, the values were 2.50, 2.40, and 2.50 a. u. The utilized electronic combinations are [Xe] 4f1⁵d16s² for Pt, [Ar] 3d²4s² for Ti, [Ar] 3d³4s² for Ge, [Ar] 3d⁵4s¹ for Ge, and [He] 2s²2p⁴ for Pb. The MT spheres’ spherical harmonics were extended to l_max = 10, and G_max=12 (a. u.) was assumed to be the Fourier expanded charge density. A dense sampled k-point grid is vital for capturing smooth electronic band dispersions and accurately computing transport characteristics. The spin-orbit coupling (SOC), one of the relativistic effects, has a remarkable effect on the separation energy of compound molecules with heavy atoms. The interacting particles that rely on their values, mutual orientations, and orbital and spin angular momenta are responsible for the SOC effect. Here, we utilize the variational method built into the WIEN2k code to incorporate the GGA and SOC calculations⁴⁶. The resulting band-edge positions and total energies showed only marginal variations (< 0.05 eV) compared with the GGA and mBJ results, confirming that electron–electron repulsion has an insignificant impact on the electronic structure of these weakly correlated 3d/5d systems. Therefore, the standard PBE-GGA and TB-mBJ exchange–correlation schemes were considered adequate for accurately describing the electronic behavior of PtTiZ (Z = Ge, Pb).

The IRelast program⁵¹ that interacted with the WIEN2k program⁵² was used to evaluate the mechanical characteristics. A particular q-mesh of order was used to analyze the phonon dispersal curve and generate dynamical matrices. The Gibbs2 code’s implementation of the quasi-harmonic approximation (QHA) was utilized⁵³, the thermodynamic characteristics of the sample materials were investigated. The electric thermoelectric properties were calculated utilizing the BoltzTrap code as a function of carrier concentration and temperature⁵⁴. The code utilizes the semi-classical Boltzmann theorem. In BoltzTrap code, the Seebeck coefficient (S) is independent of the relaxation time (τ), but the electrical conductivity (σ) and thermal conductivity (κ) can only be determined with respect to constant relaxation time (10^− 14 s). For PtTiZ compounds (Z = Ge, Pb), thermoelectric properties were evaluated using a highly dense 46 × 46 × 46 k-point mesh, totaling 100,000 points within the first Brillouin zone. We have performed phonon dispersion calculations using the finite displacement method as implemented in Phonopy^55,56, based on DFT calculations carried out with QE⁵⁷. The crystal structures were optimized using the PBE-GGA functional, with a plane-wave cutoff energy of 500 eV and a Monkhorst–Pack k-point grid of 8 × 8 × 8. A 2 × 2 × 2 supercell was constructed to calculate force constants, and phonon frequencies were obtained by diagonalizing the dynamical matrix.

Results and discussion

Structural and mechanical properties

In the PtTiZ (Z = Ge, Pb) substances, the Z = Ge, Pd atoms correspond to the Wyckoff positions 4c (0.25, 0.25, 0.25), 4b (0.5, 0.5, 0.5), and 4a (0, 0, 0), respectively. Figure 1 depicts the crystal structure of these materials, which form in a cubic face-centered (FCC) lattice with a C1_b-type configuration and belong to the F-43m space group. While Pt and Ti occupy the tetrahedral as well as octahedral locations as cations, Z = Ge, Pd is an anion that lives in the octahedral regions. To determine the optimal lattice parameters of the studied materials, various computed total energies and structural data were fitted using the Birch-Murnaghan equation of state²³. To guarantee stability, a thorough structural optimization process using the Murnaghan process was conducted. The results are shown in Tables 1 and include the characteristics of the lattice (a₀), bulk modulus (B), derivation of bulk modulus (B’’), and ground energy (E₀). The energy–volume (E–V) relationship of the PtTiZ (Z = Ge, Pb) compound was analyzed within both the Local Density Approximation (LDA) and the Generalized Gradient Approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) functional (Fig. 2a, b). The results show that the total energy obtained from PBE is lower than that from LDA, indicating that the PBE functional provides a more stable and energetically favorable configuration for the PtTi(Ge/Pb) compounds. Analysis of the results shows that the non-magnetic (NM) phase is energetically more stable than the ferromagnetic (FM) phase. Thus, it may be concluded that its most stable situation in terms power is the NM phase. Figure 2 presents the volume-energy curves corresponding to various atomic arrangements of the PtTiZ (Z = Ge, Pb) half-Heusler compounds. The NM phase, which has the lowest energy level of both the metals shown, is surprisingly the most stable arrangement (Fig. 2c, d). Table 1 provides the estimated lattice parameters (a_o) for the materials identified as stable in the non-magnetic configuration. The PtTiPb atom was found to have a greater lattice constant than PtTiGe due to its bigger electronic shell.

Fig. 1

Constructed the crystal structure by XCrsyden of half Heusler PtTiZ(Z = Ge, Pb) alloys.

Table 1 Comparison of the lattice constants (a, b, and c), volume (V), bulk modulus (B) and its pressure derivative (B′), cohesive energy, formation energy and bader charges of cubic Hhs PtTiZ (Z = Ge, Pb) calculated using LDA and PBE-GGA with previously obtained experimental and theoretical Results.

Fig. 2

Considered optimizations (a,b) of half Heusler PtTiZ(Z = Ge, Pb) alloy, to check the material stability via attaining the ground state energy.

Dynamics involving all ions and electrons is used to carefully relax the structure while meeting strict converging requirements for force, overall energy, and kinetic energy^58,59. The formation energy of the solid is the difference between the energy of a crystal and its constituents’ solid phases at zero temperature, which is given by:

$$\:{E}_{f}={E}_{tot}^{bulk}-E\left(Pt\right){E}_{tot}^{bulk}-E\left(Ti\right){E}_{tot}^{bulk}-E\left(Ge/Pb\right){E}_{tot}^{bulk}$$

(1)

.

where $\:{E\left(PtTiZ\right)}_{tot}^{bulk}$ is the total energy of the alloy. $\:{E\left(Pt\right)}_{tot}^{bulk}$, $\:{E\left(Ti\right)}_{tot}^{bulk}$ and $\:{E(Ge/Pb)}_{tot}^{bulk}$ are the energies of the fundamental state per atom of each elemental bulk for Pt, Ti, Ge and Pb. We find that the formation energies of the stoichiometric PtTiZ (Z = Ge, Pb) in half-Heusler structure are given in Table 1, which confirms the stability of the alloy in its half-Heusler structure.

The cohesive energy represents the energy necessary to break a compound down into its isolated constituent atoms. The cohesive energy (E_c) of PtTiZ (Z = Ge, Pb) is calculated using formula:

$$\:{E}_{c}={E\left(PtTiZ\right)}_{tot}^{bulk}-{E}_{atom}^{Pt}-{E}_{atom}^{Ti}-{E}_{atom}^{Z}$$

(2)

.

Where $\:{E\left(PtTiZ\right)}_{tot}^{bulk}$ is the total energy of the compound at equilibrium and $\:{E}_{atom}^{Pt}$, $\:{E}_{atom}^{Ti}$, and $\:{E}_{atom}^{Z}$ are the total energies of the pure atomic components. The structural stability of PtTiGe and PtTiPb is investigated by means of cohesive energy. Thus, the larger the calculated value, the more stable the crystal structure. The obtained cohesive energies of PtTiZ (Z = Ge, Pb) are presented in Table 1, and as one can see they are positive which means that the half Heusler compounds PtTiZ (Z = Ge, Pb) are chemically stable.

To evaluate the dynamical stability and lattice vibrational properties of PtTiZ (Z = Ge, Pb) compounds, phonon dispersion calculations were performed using first-principles density functional theory (DFT) combined with the finite displacement method as implemented in Phonopy. The crystal structures were fully optimized using the PBE-GGA functional with Quantum espresso pseudopotentials. A 2 × 2 × 2 supercell of the optimized unit cell was constructed, and small atomic displacements (~ 0.01 Å) were applied to calculate the force constants. Phonon frequencies were obtained by diagonalizing the dynamical matrix, and dispersion curves were plotted along the high-symmetry directions of the cubic Brillouin zone. The phonon dispersion curves of both compounds are shown in Fig. 3. No imaginary frequencies were observed, confirming the dynamical stability of PtTiGe and PtTiPb. As expected, PtTiGe exhibits higher phonon frequencies due to the lighter Ge atom and stronger Pt–Ti–Ge bonding, resulting in steeper acoustic modes. In contrast, PtTiPb shows softer phonon modes owing to the heavier Pb atom and slightly weaker bonding, suggesting a lower lattice thermal conductivity, which is favorable for thermoelectric performance. The phonon density of states (DOS) further illustrates the contributions of Pt, Ti, and Z atoms to the acoustic and optical branches, providing insights into thermodynamic properties and phonon-mediated transport. These results confirm that PtTiZ compounds are dynamically stable and promising candidates for thermoelectric applications.

Fig. 3

Computed phonon dispersion of half Heusler PtTiZ(Z = Ge, Pb) alloy.

Understanding a material’s response to external stress requires knowledge of its elastic constants, which are key indicators of its mechanical behavior. The mechanical robustness and stability of the material are indicated by these constants. To extract them, stress values corresponding to minor applied strains were evaluated under zero-pressure conditions. Energy calculations were performed using lattice strains that conserved the overall volume of the system. Elastic constants were determined using the IRelast tool, which is optimized for cubic structures and integrated with the WIEN2k computational framework. The mechanical as well as dynamical characteristics of the compounds being studied were examined to provide the long-term stability of the substances because the materials being studied were hypothetically synthesized. Table 2 displays the computed elastic constants C₁₁, C₁₂, and C₄₄ that describe the cubic symmetry of the materials under investigation. The positive value of such constants implies the physical stability of the examined substances. In addition to elastic constants, Table 2 presents several mechanical parameters calculated from them, such as the bulk modulus (B), shear modulus (G), Young’s modulus (Y), Debye temperature (θ_D), anisotropy factor (A), Pugh’s B/G ratio, Poisson’s ratio (ν), and melting temperature (T_m). We considered the mechanical properties in the 3D and also computed the data in the tables which are given. The B for the both half Heusler alloys were computed and compared to that in values and 3D graphs, The computed outcomes of bulk modulus revealed that the B for PtTiGe was smaller than the PtTiPb due to having the larger atomic number of Pb atom as shown in Fig S1 (a). Like the Bulk modulus, the linear compressibility also considered, the result of linear compressibility advocates that PtTiGe has more power for the compressibility as compared to the PtTiPb alloy as demonstrated in Fig S1 (b). According to the values B and Y, PtTiGe is predicted to be tougher and stiffer than PtTiPb as shown in Fig S2 (c). The poisson ratio with comparison of each other is also given in Fig S2 (d). For further understanding, we added the other parameters for 3D graphs are also given in Tables 3 and 4. This effect may arise from the p-electrons in Ge being more tightly bound due to a stronger Coulomb interaction than those in Pb. Additionally, the compounds being studied are predicted to be less stiff and more pliable than similar compounds such as PdHfGe, NbPdSi, and NbPtSi, based on the calculated values of B and Y^60,61. The strong interaction force between all the atoms in PtTiGe compatible with B is predicted by the value of θ_D. Furthermore, the θ_D indicates a reduced lattice thermal conductivity in PtTiPb. The A and B/G ratio values for the compounds, along with a Poisson’s ratio (ν) greater than 0.25, suggest the presence of ionic bonding, elastic anisotropy, and ductile behavior. According to Greaves and his team’s estimation, the Poisson’s ratio (v) among the polymerization is roughly 33, indicating that the materials under study exhibit characteristics of polymers. The equation below is employed to estimate the melting temperature (T_m), a key indicator of a material’s thermal and elastic characteristics.

Table 2 Values of elastic constants (C_ij), bulk modulus (B), shear modulus (G), young’s modulus (E), poisson’s ratio (σ), Pugh ratio, Frantsevich ratio, shear anisotropy factor (A) cauchy pressure C^P, sound velocities (m/s), Debye temperature Θ_D (K) of PtTiZ (Z = Ge, Pb).

Table 3 Calculated energy bandgap (in eV) of hHs PtTiGe) by PBE approximation.

Table 4 Calculated energy bandgap (in eV) of hHs PtTiGe) by PBE approximation.

The materials under study exhibit high T_m values, which suggest that they possess thermodynamic stability throughout a broad temperature range. The impact of lattice vibrating to thermal conductivity of the lattice is provided by the analysis of dynamic characteristics.

Electronic properties

All material’s band structure provides information on the power and momentum of its electrons, describing its behaviour and state. The electronic state and behaviour of a substance determines its electrical and optical characteristics, or how it reacts to electromagnetic (EM) radiation. As such, the use of approximate values is still essential. The authors used the potential functionals described in the computation details to perform SCF calculations in order to ascertain the electronic band structure as well as band gap. LDA and GGA functionals are widely recognized for their tendency to underestimate the band gap values in semiconductor materials. To resolve this limitation and achieve more accurate computational results, the TB-mBJ method was employed. The electronic behavior of the Pt-based PdTiGe half-Heusler compounds was examined using the modified Becke–Johnson (mBJ) potential to determine their band structure. The electronic energy levels close to the Fermi energy value (E_F) have a substantial impact on the material’s electrical conduction and Seebeck coefficient. Thus, from a thermoelectric perspective, figuring out an energy structure of bands is essential, and values of bandgap are given in Table 5. The energy bands of the materials were determined along the high-symmetry directions (W-L-Γ-X-W-K) in the Brillouin zone (BZ) between − 5 and 5 eV, as shown in Fig. 4. The energy band profiles of the two sample compounds were identical at the valence band region, showing a two-fold orbital degeneracy at the Γ point that was mainly controlled by Ti-d orbital characteristics, as shown in Fig. 5. The occurrence of both light and heavy bands in the E_F’s proximity is advantageous for high thermoelectric performance. There are slightly different energy band characteristics in the conduction area.

Table 5 Calculated energy bandgap (in eV) of hHs PtTiZ(Z = Ge, Pb) by different exchange-correlation potentials LDA, PBE, mBJ and mBJ + SOC.

Fig. 4

Calculated the electronic band structure with diverse approximations such as LDA, GGA, mBJ, and mBJ + SOC of half Heusler PtTiZ (Z = Ge, Pb) alloy, to check the material nature via attaining the band gap values.

Fig. 5

Calculated the total density of staes with diverse approximations such as LDA, GGA, mBJ, and mBJ + SOC of half Heusler PtTiZ(Z = Ge, Pb) alloy, to check the material nature via attaining the density values.

As shown in Fig. 6, both compounds exhibit two separate valleys at the X symmetry point in the conduction band, which originate from the partial Pt-d and Z-p states as well as Ti-d electronic states. When a Pb atom replaces a Ge atom, one among the valleys is pushed towards the E_F, widening the gap between them and reducing the amount of energy difference in PdTiPb. The Γ and X symmetry points displayed the highest valence band and the lowest conduction band, respectively, resulting in energy gaps of 0.66 eV for PtTiGe and 0.387 eV for PtTiPb. The functionals mBJ-GGA and mBJ-LDA produce a large gap and no experimental value. This observation is in a good agreement with the band structures that observed by Roy et al.⁶². Moreover, replacing Ge with Pb in the conduction region at the Γ symmetry point of PtTiGe leads to a shift of the light band, associated with Ti-d and Z-p electronic states, toward lower energy values. In the conduction region of both compounds, dense bands are observed at the W point, primarily resulting from the participation of Ti-d electronic states. There is a single, distinct maximum in the valence phase and many peaks in the conducting zone, according to the entire density of all states (Fig. 6). In the conduction region, the Ti-d electrons produce multiple peaks, while the Ge/Pb atoms’ p electronic states produce a definite peak in the valence band. Pt-d states provide very little when compared to Ti-d as well as Ge/Pb-p electronic levels. According to a density of states study, both Ti-d as well as Z-p states contribute to the valence region, whereas Ti-d states predominate in the conducting region.

Fig. 6

Computed the partial density of states with diverse orbital of half Heusler PtTiZ (Z = Ge, Pb) alloy, to check the material nature via attaining the density values.

The mechanism of band-gap formation in the half-Heusler PtTiZ (Z = Ge, Pb) compounds originates from the hybridization and crystal-field splitting of transition-metal d and main-group p states within the cubic F-43 m symmetry. The valence-band maximum at the Γ-point is dominated by strongly hybridized Ti-3d and Z-p orbitals, while the conduction-band minimum at the X-point mainly arises from Ti-d and Pt-d contributions. This p–d hybridization results in splitting of the Ti-d manifold into t₂g and e_g sub-levels, producing an indirect Γ–X gap for PtTiGe and PtTiPb as shown in Fig. 5. When Ge is replaced by the heavier Pb atom, the reduced p–d overlap and larger lattice constant narrow the separation between t₂g and e_g states, thereby decreasing the band gap. The weaker covalent interaction in PtTiPb, evident from charge-density analysis (Fig. 7), further supports this reduction. Overall, the electronic configuration follows the 18-valence-electron rule typical for semiconducting half-Heusler phases, where the balance between bonding (Ti-Z) and antibonding (Ti-Pt) states governs the observed band topology and ensures semiconducting behavior.

Fig. 7

Calculated the 2D electrons localized function of half Heusler PtTiZ(Z = Ge, Pb) alloy, to check the material bonding and localization of the electrons via attaining the n (r) values.

Charge density profile

The amount known as charge density indicates the type of chemical bond and the extent of atom-to-atom interaction. Figure 7a, b illustrates the charge density contours for PtTiZ (Z = Ge, Pb) compounds on the (110) plane. In both compounds, the charge density surrounding the Z atom is notably lower compared to that around the Ti atom. The compounds under investigation demonstrate that Pt-Z, Pt-Ti, and Ti-Z have partly covalent interactions. As the Pb atom has a larger radius than the Ge atom, PtTiGe materials demonstrate stronger covalent bonding compared to PtTiPb materials. To better understand the chemical bonding in these compounds, the 2D-dimensional charge density is shown in Fig. 7a, b. The charge density plot in three dimensions shows a similar profile. It is observed that compared to the PtTiPb compound, the PtTiGe has a stronger covalent nature. Therefore, it appears from the charge density dispersion that the compounds under study exhibit a combination of covalent and metallic bonding.

Optical properties

Assessing a substance’s appropriateness for a variety of technical applications, especially in the field of photovoltaics as well as optoelectronic devices, requires a thorough understanding of its optical characteristics⁶³. The foundation for understanding the complex interaction between a compound’s reaction to absorbed light and the photon energy of the light is the complicated dielectric tensor, ε(ω)⁶⁴. Ehrenreich and Cohen formulated a mathematical model that provides an in-depth explanation of a material’s interaction with electromagnetic radiation. The dielectric tensor is expressed as ε(ω) = ε₁(ω) + iε₂(ω), where ε₁(ω) represents the material’s polarization response, and ε₂(ω) reflects its light absorption characteristics. The incident electromagnetic radiation’s angular frequency is indicated by the symbol ω. Using the Kramers-Kronig equation makes it easier to determine the real element of the function of dielectric, ε₁ (ω). A comprehensive analysis of the equilibrium structure’s optical characteristics in connection to the intensity of incident electromagnetic radiation is shown in Fig. 8a–h. The variation of ε₁(ω) for PtTiZ (Z = Ge, Pb) with respect to the energy of the incident radiation is depicted in Fig. 8a. A significant response to electromagnetic radiation is shown by the calculated static dielectric constant of ε₁ (0), which stands at 18.00 and 23.99. A closer look at Fig. 8 (a) shows that ε₁ (ω) increases gradually with increasing energy until it peaks at 1.74 eV and 2 eV, then declines before reaching another high point at 2.1 eV and 2.73 eV. After 9 eV, additional measurements show a negative value of ε₁ (ω), with a modest recovery towards zero. Strong light absorption in the visible portion of the electromagnetic spectrum is indicated by the measured maxima in the imaginary component of the dielectric value ε₂(ω). This property makes PtTiZ (Z = Ge, Pb) a viable option for uses requiring interaction with light at these particular wavelengths. Furthermore, the dielectric constant’s negative value suggests that PtTiZ (Z = Ge, Pb) is conductive. Because of its conductivity, it may be used in a variety of optoelectronic devices, such as filters, optical fibres, super lenses, and electromagnetic shielding devices. Table 6 lists the optical properties’ static values.

Fig. 8

Computed the optical parameters such (a) ε₁ (w), (b) ε₂ (w), (c) n (w), (d) k (w), (e) σ( w), (f) α( w), (g) R(w), and (h) L(w) of half Heusler PtTiZ(Z = Ge, Pb) alloy, to check the material nature by applying the EM waves.

Table 6 Calculated optical properties of hHs PtTiZ (Z = Ge, Pb) by mBJ.

Figure 8b presents a critical analysis of the energy dependence exhibited by the imaginary component of the dielectric function, ε₂(ω). This plot essentially serves as a spectral representation of the light absorption profile within PtTiZ (Z = Ge, Pb) compounds. The prominent peaks observed in ε₂ (ω) correspond to well-defined electronic transitions occurring between the valence band maxima and the conduction band minima. These characteristic peaks provide invaluable insights into the intrinsic energy loss mechanisms governing the light-matter interaction within these materials. The threshold energy, identified as the point where ε₂ (ω) experiences a significant rise, signifies the optical bandgap of PtTiZ (Z = Ge, Pb). This spectroscopic analysis establishes the optical bandgap values to be 1.72 eV and 1.25 eV for PtTiZ (Z = Ge, Pb) compounds, respectively. Notably, these experimentally determined values exhibit excellent concordance with the electronic bandgaps predicted through theoretical calculations, thereby serving as a strong validation of the computational approach. This close correspondence underscores the precision and reliability of the current findings. In essence, the analysis of ε₂(ω) provides valuable insights into the absorption behavior and electronic structure of PtTiZ (Z = Ge, Pb), serving as a foundation for further exploration into its optical properties and potential applications in optoelectronic devices.

The refractive index, n(ω), determines the substance’s transparency and capacity to refract light. This parameter provides important insights about semiconductors by determining how much of a light deflection or reflection. Figure 8c showed how it correlated with the power of incident radiation. The static values of n(0) for PtTiZ (Z = Ge, Pb) materials are 4.48 and 4.65, respectively. As the energy of electromagnetic radiation increases, the value rises, particularly in the infrared (IR) range, with a noticeable increase in the near-visible region at 7.0 and 7.2. The transparency of the material is indicated by the refractive index, which consistently stays positive despite slight peaks in the visible spectrum. Additionally, n(ω) decreases but stays positive as the incident radiation energy increases, confirming the material’s transparency across the energy range. This behavior demonstrates the materials’ isotropic nature and shows consistent optical properties independent of directionality. The coefficient of extinction k(ω) in Fig. 8d shows a reduction in the intensity of the electric vector oscillations caused by incident radiation at various energy levels. This parameter depicts the behavior of ε₁(ω), the actual component of the dielectric response, and is directly related to light absorption. The point at which ε₁(ω) becomes zero exactly corresponds to the peak value of k(ω). A significant shift in the material’s interaction with incoming radiation is suggested by this alignment, which represents a fundamental change that affects the material’s optical behavior.

The coefficient of absorption α(ω) measures what a material can do for absorbing radiation per unit its thickness, indicating its effectiveness in photovoltaic conversion and the depth of energy penetration. In Fig. 8e, the variation of α(ω) with incident radiation energy is shown, providing an understanding of the compound’s responsiveness to electromagnetic radiation. The maximum measured values of α(ω) are roughly 90.5 and 80.99 × 10⁴/cm. Additionally, α(ω) provides valuable insights into how EM radiation affects chemical bonds inside a molecule through photoelectron conduction.

In Fig. 8f, the optical conductivity spectra σ(ω) are shown for a range of incident radiation energies, highlighting four clearly defined peaks. For PtTiGe and PtTiPb, the highest optical conductivity values are observed at 1.25 eV (10,550 Ω cm⁻¹) and 1.74 eV (13,000 Ω cm⁻¹), respectively. There is a noticeable relationship between optical conductivity and absorption, with both displaying similar patterns of peaks and valleys across the energy spectrum. This consistency emphasises both the faithfulness of the theoretical structure and the precision of its computing predictions. Furthermore, the optical reflectivity R(ω) plays a key role in evaluating the proportion of energy reflected relative to the incident energy. This parameter offers valuable insight into how the compound responds optically and interacts with electromagnetic waves.

Figure 8g shows the frequency-dependent reflectivity, R(ω), in relation to the incident energy. The static reflectivity values, R(0), are measured as 0.389% for PtTiGe and 0.423% for PtTiPb. The reflectivity spectrum of the studied material reveals pronounced peaks within the visible range (approximately 1.8–3.1 eV), with maximum reflectivity values of 0.61 and 0.60 observed in the higher energy portion of this spectrum. This demonstrates the substance’s potential as shields versus far visible radiation, as well as their efficacy for solar power in both the infrared and nearby visible areas. The energy losses function L(ω) describes the absorption of energy by mobile electrons in a substance. Figure 8h depicts its reaction to increasing levels of radiation energy. The energy loss function displays peaks that correspond to the resonant oscillations of valence electrons, known as plasmons, occurring at characteristic plasma frequencies (ω_P). The energy loss function exhibits peaks with increasing amplitude at higher energy levels, particularly beyond 4 eV, indicating the material’s potential for applications in optical devices.

Transport properties

The thermoelectric properties were calculated using the semiclassical Boltzmann transportation theory described in⁶⁵. This theory investigates how heat and electrical gradients affect the behaviour of electron transport within a material. It includes methods for interaction and scattering among carriers of charge (electrons or holes) and other carriers, impurities, and the crystal lattice. Highly efficient thermoelectric substances can be designed and engineered with the help of this theoretical framework. The semiclassical Boltzmann transport theory is implemented by the well-known computing tool BoltzTrap software. It gives scientists the ability to create and improve thermoelectric substances for a range of uses. This software offers a reliable numerical approach, enabling the derivation of analytical expressions and overcoming limitations associated with relaxation time constant (τ). An extensive investigation of key transport parameters was conducted to meticulously evaluate the thermoelectric performance of PtTiZ (Z = Ge, Pb) compound. These parameters included the Seebeck coefficient (S), electrical conductivity(σ/τ), thermal conductivity(κ/τ), power factor (PF), and figure of merit (zT). The computed values against the temperature at the room temperature are given in the Table 7. The study involved a thorough investigation of how these parameters varied with chemical potential (µ) across four different temperatures: 300 K, 700 K, 900 K, and 1200 K.

Table 7 Calculated thermoelectric properties of hHs PtTiZ(Z = Ge, Pb)) by mBJ at ambient temperature.

The beck coefficient (S) can be defined, induced change in voltage against change in temperature. The S considered due to analyzing the how much the change in voltage induced when we vary the temperature in the material. The S for two half Heusler PtTiZ (Z = Ge, Pb) alloys were computed at different temperatures. The values of S observed are approximately 229µV/K and 236 µV/K at 300 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys. As shown in Fig. 9a. Similarly, we observed other temperatures such as 700 K (229µV/K and 236 µV/K), 900 K (235µV/K and 240 µV/K), and 1200 (235µV/K and 236 µV/K) for both half Heusler PtTiZ (Z = Ge, Pb) alloys. We observed the highest value of induced voltage at 700 K as compared to the other temperatures.

Fig. 9

Computed the thermal parameters such (a) S, (b) PF, (c) σ/t, (d) k/t for PtTiGe (e) k/t for PtTiPb (f) zT of half Heusler PtTiZ (Z = Ge, Pb) alloy, to check the thermal behavior of the material by applying the temperature.

Power Factor (PF):

The PF for two half Heusler PtTiZ (Z = Ge, Pb) alloys were computed at different temperature. The values of PF observed are approximately 1.1 × 10¹¹W/K²ms and 1.21 × 10¹¹W/K²ms at 300 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys. As shown in Fig. 9b, similarly, we observed with other temperatures such as 700 K (6.1 × 10¹¹W/K²ms and 5.9 × 10¹¹W/K²ms), 900 K (9.5 × 10¹¹W/K²ms and 8.1 × 10¹¹W/K²ms), and 1200 K (16 × 10¹¹W/K²ms and 13.5 × 10¹¹W/K²ms) for both half Heusler PtTiZ (Z = Ge, Pb) alloys. We observed the highest value of power factor at 1200 K as compared to the other temperatures.

Electrical Conductivity (σ/τ):

The σ/τ for two half Heusler PtTiZ (Z = Ge, Pb) alloys were computed at different temperatures. The values of σ/τ observed are approximately 0.004 × 10¹⁹(Ω ms)⁻¹ and 0.005 × 10¹⁹(Ω ms)⁻¹ at 300 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys. As shown in Fig. 9c, similarly, we observed with other temperatures such as 700 K (0.1 × 10¹⁹(Ω ms)⁻¹ and 0.310¹⁹(Ω ms)⁻¹), 900 K (0.155 × 10¹⁹(Ω ms)⁻¹ and 0.165 × 10¹⁹(Ω ms)⁻¹), and 1200 K (0.28 × 10¹⁹(Ω ms)⁻¹ and 0.25 × 10¹⁹(Ω ms)⁻¹) for both half Heusler PtTiZ (Z = Ge, Pb) alloys. We observed the highest value of σ/τ at the at 1200 K as compared to the other temperatures.

Thermal conductivity (κ/τ):

The κ/τ for two half Heusler PtTiZ (Z = Ge, Pb) alloys were computed at different temperatures. The values of κ/τ observed are approximately 2.3 × 10¹⁵W/Kms at 300 K for half Heusler PtTiGe alloys as shown in Fig. 9d. Similarly, we observed with other temperatures such as 700 K (1.9 × 10¹⁵W/Kms), 900 K (2.1 × 10¹⁵W/Kms), and 1200 K (2.49 × 10¹⁵W/Kms) for half Heusler PtTiGe alloys. We observed the highest value of κ/τ at 1200 K as compared to the other temperatures. The values of κ/τ observed are approximately 0.42 × 10¹⁵W/Kms at 300 K for half Heusler PtTiPb alloys as shown in Fig. 9e. Similarly, we observed with other temperatures such as 700 K (1.2510¹⁵W/Kms), 900 K (1.5 × 10¹⁵W/Kms), and 1200 K (2.25 × 10¹⁵W/Kms) for half Heusler PtTiPb alloys. We observed the highest value of κ/τ at the1200K as compared to the other temperatures.

Figure of merit (zT):

The zT for two half Heusler PtTiZ (Z = Ge, Pb) alloys were computed at different temperatures. The values of zT observed are approximately 0.02 and 0.03 at 300 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys. As shown in Fig. 9f. Similarly, we observed other temperatures such as 700 K (0.35 and 0.40), 900 K (0.50 and 0.59), and 1200 K (0.65 and 0.70) for both half Heusler PtTiZ (Z = Ge, Pb) alloys. We observed the highest value of power factor at 1200 K as compared to the other temperatures.

Thermal properties versus chemical potential:

We calculated the overall Seebeck coefficient S to identify its nature against chemical potential. As may be shown, the Heusler combination has a positive and negative Seebeck coefficient. The holes’ dominance as charge carriers is explained by its positive sign of S and vice versa. Our considered Heusler compounds are p-type substances as a result.

The S for two half Heusler PtTiZ (Z = Ge, Pb) alloys were computed at different temperatures. The values of S observed are approximately 1800µV/K and 1900 µV/K at 300 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys in positive side as shown in Fig. 10a. For negative side, the values of S observed are approximately 1750µV/K and 1950 µV/K at 300 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys. Similary, we observed other temperatures such as 600 K (900µV/K and 1000 µV/K), and 900 K (600µV/K and 500µV/K) for both half Heusler PtTiZ (Z = Ge, Pb) alloys. For negative side, the values of S observed are approximately 600 K (-800µV/K and − 7500 µV/K), and 900 K (-400µV/K and 500µV/K) for both half Heusler PtTiZ (Z = Ge, Pb) alloys. We observed the highest value of induced voltage is in positive side and at the 300 K as compared to the other temperatures.

Fig. 10

Computed the thermal parameters such (a) S, (b) PF, (c) zT, (d) σ/t (e) k/t of half Heusler PtTiZ (Z = Ge, Pb) alloy, to check the thermal behavior of the material by applying the temperature.

The calculated transport coefficients are now utilized to assess the thermoelectric efficiency using the figure of merit zT measurement. If a material’s zT is about or larger than unity, it is regarded as a good element for thermoelectric devices³⁶. The Fig. 10b displays the variance of zT, demonstrates that zT decreased linearly with temperature for PtTiZ (Z = Ge, Pb) have respective values of 9.99 and 1 at 300 K against chemical potential. For other temperatures such as 600 K(0.95/0.96) and 900 K(0.90/0.91) for PtTiZ (Z = Ge, Pb) respectively.

Additionally, power factor PF represents the material’s suitability for thermoelectric technology. High efficiency is suggested by the material’s high-power factor. The PF for two half Heusler PtTiZ (Z = Ge, Pb) alloys were computed at different temperatures. The values of the first peak of PF observed are approximately 2.5 × 10¹¹W/K²ms and 2.4 × 10¹¹W/K²ms from (-0.5 eV to 0 eV) at 300 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys as shown in Fig. 10c. For the second peak with same temperature the values of PF observed are approximately 2.6 × 10¹¹W/K²ms and 4.2 × 10¹¹W/K²ms at 1.4 eV. The values of first peak of PF observed are approximately 7.6 × 10¹¹W/K²ms and 5.3 × 10¹¹W/K²ms from (-0.5 eV to 0 eV) at 600 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys. For the second peak with the same temperature the values of PF observed are approximately 7.2 × 10¹¹W/K²ms and 6.2 × 10¹¹W/K²ms at 1.4 eV. The values of first peak of PF observed are approximately 13.2 × 10¹¹W/K²ms and 6.5 × 10¹¹W/K²ms from (-0.5 eV to 0 eV) at 900 K for both half Heusler PtTiZ (Z = Ge, Pb) alloys. For the second peak with the same temperature the values of PF observed are approximately 9.89 × 10¹¹W/K²ms and 9.8 × 10¹¹W/K²ms at 1.4 eV.

Figure 10d plots the electrical conductivity variance (σ/τ) as an indication of temperature. Up to 300 K, the electrical conductivity (σ/τ) increases gradually. The values of σ/τ with 300 K/600K/900K were observed approximately 2.5/2.501/2.51 × 10¹⁹(Ωms)⁻¹ at -2µ-E_f (eV) for half Heusler PtTiGe alloys and for + 2µ-E_f (eV), the value is 4 × 10¹⁹(Ωms)⁻¹ for all temperatures. The values of σ/τ with 300 K/600K/900K were observed approximately 1.9/1.9/2.0 × 10¹⁹(Ωms)⁻¹ at -2µ-E_f (eV) for half Heusler PtTiPb alloys and for + 2µ-E_f (eV), the value is 2.5 × 10¹⁹(Ωms)⁻¹ for all temperatures.

Figure 10e plots the thermal conductivity (σ/τ) as an indication of temperature. Up to 300 K, the electrical conductivity ((σ/τ) increases gradually. The values of σ/τ with 300 K/600K/900K were observed approximately 1.6/4.1/6 × 10¹⁵(W/Kms) at -2µ-E_f (eV) for half Heusler PtTiGe alloys and for + 2µ-E_f (eV), the value is 3.5/7.4/9.1 × 10¹⁵(W/Kms) for all temperatures. The values of σ/τ with 300 K/600K/900K were observed approximately 1.5/3/0/5.0 × 10¹⁵(W/Kms) at -2µ-E_f (eV) for half Heusler PtTiPb alloys and for + 2µ-E_f (eV), the value is 1.5/4.26.8 × 10¹⁵(W/Kms) for all temperatures.

Thermodynamic properties

Using the quasi-harmonic approximate (QHA) Debye model, the thermodynamic characteristics of the sample substances were investigated. According to temperature and pressure, the Debye model can be used to estimate several thermodynamic quantities, including volume (V), pressure (B), heat concentration at volume constant (C_V), efficiency (S), Grunession parameter (γ), Debye temperature (θ_D), and coefficients of thermal contraction (α).

We have examined these characteristics in relation to temperature for various pressure ranges from 0GPa to 90 GPa with 15GPa difference in our current work. At lower temperatures (50–1200 K), both compounds demonstrate a marked increase in C_V as temperature rises. Only at low temperatures, where the downward tendency (14 J/moleK) is evident, is the impact of pressure on C_V shown to be substantial pressures as shown in Fig. 11a. At elevated temperatures, approaching the Dulong-Petit limit, C_V stabilizes at approximately 74.0 J/mol·K for PtTiGe. The coefficient of thermal expansion (α) rises abruptly with a temp up to 300 K for each combination of pressures as shown in Fig. 11b. As we can see from Fig. 11c, the θ_D has little decline at the end but with increasing the pressure across PtTiGe alloy. This implies that as hydrostatic pressure increases, θ_D increases as well, while the interatomic binding strength decreases with rising temperature. After that, the curve rises gradually and varies steadily at higher temperatures. Compared to PtTiGe the amount of α is somewhat greater than PdTiPb alloy Moreover, given the relationship α = γCV³B²⁹, where γ denotes the Grüneisen parameter and B is the bulk modulus, we found that the changes in α closely resemble those in C_V. This change in α was seen in the sample components because all γ and B are slightly dependent on temperature pressures as shown in Figs. 11 and 12d.

Fig. 11

Calculated the thermodynamics parameters such (a) C_V (b) α (c) Ө_D (d) γ of half Heusler PtTiGe alloy, to check the thermal behavior of the material by applying the temperature and pressure (0GPA- 90GPa).

Fig. 12

Calculated the thermodynamics parameters such (a) C_V (b) α (c) Ө_D (d) γ of half Heusler PtTiPb alloy, to check the thermal behavior of the material by applying the temperature and pressure (0GPA- 90GPa).

At low temperatures, the influence of pressure on C_V is notable, showing a decrease of approximately 20 J/mol·K at 90 GPa. However, at higher temperatures, approaching the Dulong-Petit limit, CV tends to stabilize around 74.5 J/mol·K for PtTiPb, as depicted in Fig. 12a.

As we can see from Fig. 12b the α has little increase at the end but with increasing the pressure across PtTiGe alloy. This indicates that as hydrostatic pressure increases, α also increases, while the interatomic binding force weakens with higher temperature. As we can see from Fig. 12c. the θ_D has little decline at the end but with increasing the pressure across PtTiPb alloy. This indicates that as hydrostatic pressure increases, θ_D also rises, whereas the atomic bonding strength tends to weaken with increasing temperature. As we can see from Fig. 12d the γ has little increase at the end but with increasing the pressure across PtTiPb alloy. This implies that γ increases under higher hydrostatic pressure, whereas the interatomic bonding strength diminishes as temperature rises^{66,67,68,69,70,71,72}.

As a comparison between PtTiGe and PtTiPb reveals systematic trends directly linked to the heavier atomic mass and larger covalent radius of Pb. Structurally, PtTiPb exhibits a slightly expanded lattice constant and a lower bulk modulus, reflecting weaker interatomic bonding relative to PtTiGe. The reduced stiffness of PtTiPb correlates with its smaller band gap, arising from diminished p–d hybridization strength between Ti-3d and Pb-6p orbitals compared with the stronger Ti-3d–Ge-4p interaction. This weaker bonding not only lowers the Debye temperature but also enhances phonon scattering, contributing to its lower lattice thermal conductivity. Consequently, PtTiPb shows superior thermoelectric efficiency due to the trade-off between decreased mechanical rigidity and improved carrier transport. In contrast, PtTiGe’s stronger covalency leads to higher optical absorption in the ultraviolet region and slightly better mechanical stability. These consistent cross-property correlations confirm that atomic-mass substitution from Ge → Pb systematically tunes mechanical, electronic, and transport properties toward higher thermoelectric performance while preserving overall structural integrity.

Finally, the findings of this study extend beyond the fundamental understanding of PtTiZ (Z = Ge, Pb) half-Heusler compounds and are directly relevant to energy-efficient material design and device applications. The coexistence of mechanical robustness, semiconducting behavior, and strong UV optical absorption makes these materials promising for multifunctional technologies. In particular, their high Seebeck coefficients, moderate band gaps, and low lattice thermal conductivities suggest that PtTiZ compounds can serve as efficient thermoelectric generators for waste-heat recovery in high-temperature environments such as automotive or aerospace systems. Simultaneously, the wide optical absorption in the ultraviolet range positions them as candidates for solar-blind photodetectors and optoelectronic coatings. These combined features emphasize the dual functional potential thermoelectric and optoelectronic of PtTiZ alloys, which fits squarely within interdisciplinary materials research with real-world technological impact.

Conclusion

The structural, mechanical, optoelectronic, thermoelectric, and thermodynamic properties of PtTiZ (Z = Ge, Pb) compounds were comprehensively examined using the full-potential linearized augmented plane wave (FP-LAPW) method combined with semi-classical Boltzmann transport theory. The exchange–correlation interactions were treated within the LDA, PBE-GGA, and TB-mBJ approximations. The calculated electronic band structures indicate that both compounds are indirect semiconductors, exhibiting band gaps of 0.66 eV and 0.387 eV for PtTiGe and PtTiPb, respectively, within the mBJ framework. The density of states analysis reveals that Ti-d and Z-p orbitals dominate the electronic states near the Fermi level. Mechanical stability and positive elastic constants confirm the robustness of the cubic phases, with PtTiGe identified as stiffer and harder than PtTiPb. Optical results show strong absorption and optical conductivity in the ultraviolet region, coupled with transparency to low-energy photons, suggesting their suitability for optoelectronic applications. The thermoelectric investigation reveals p-type conductivity with promising Seebeck coefficients of 229.21 µV/K and 236.21 µV/K at 300 K for PtTiGe and PtTiPb, respectively, increasing slightly at elevated temperatures. At 1200 K, the lattice thermal conductivities were found to be remarkably low—0.45 W/mK for PtTiGe and 0.32 W/mK for PtTiPb—resulting in dimensionless figures of merit (ZT) of 0.68 and 0.70, respectively. Thermodynamic analysis further demonstrates that the Debye temperature increases with pressure, while the heat capacity decreases correspondingly, supporting their stability under compression. Overall, PtTiGe and PtTiPb emerge as mechanically stable semiconductors with favorable optical absorption and thermoelectric efficiency, making them potential candidates for future optoelectronic and energy conversion devices. Nevertheless, it is important to recognize the methodological limitations inherent to density functional theory. The PBE-GGA and TB-mBJ approximations neglect explicit many-body and anharmonic effects, which may influence the accuracy of the predicted band gaps and thermodynamic parameters at elevated temperatures. In addition, BoltzTraP calculations were performed under the constant relaxation-time approximation, assuming temperature-independent scattering rates. While these assumptions are standard in theoretical modeling, they may lead to minor quantitative deviations from experimental data. Therefore, future experimental synthesis, temperature-dependent transport measurements, and phonon calculations including anharmonic corrections are strongly encouraged to validate and refine the present theoretical findings⁷³.