The electronic structure, optical, and thermoelectric properties of novel Bi₂PbCh₄ (Ch = Se, Te) materials: insights from first-principles study

Abstract

Ternary chalcogenides have attracted much interest because of their potential for use in sustainable energy applications due to their tunable electronic, optical, and transport characteristics. This work examined the structural, electronic, optoelectronic, and thermoelectric properties of novel Bi₂PbSe₄ and Bi₂PbTe₄ chalcogenides through density functional theory. The predicted energy gap values measured with the TB-mBJ and PBE-GGA are 1.12 and 0.71 eV for Bi₂PbSe₄ and 1.08 and 0.82 eV for Bi₂PbTe_4, respectively. Both materials behave as semiconductors and have direct energy gaps, which makes them attractive for solar energy applications. COHP study illustrates that strong Bi-chalcogen bonding characterizes the valence band, whereas antibonding states prevail above the Fermi level in both Bi₂PbSe₄ and Bi₂PbTe₄. Their promise as absorber materials in photovoltaic devices is highlighted by optical investigations that show considerable absorption in the visible and infrared ranges, high dielectric constants, and higher photoconversion performance. The Seebeck coefficient, lattice thermal conductivity, and electrical conductivity were employed to assess thermoelectric features. These ternary materials are suitable for integrated solar energy collecting and conversion systems because of their outstanding optical absorption and thermoelectric potential. The structure–property interactions of these materials are explained by this study, opening the door for testing and more optimization for improved energy devices.

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1. Introduction

Due to the rapid rise in carbon production and global warming caused by fossil fuel consumption, there has been a significant global focus across multiple industries on providing clean energy and reducing greenhouse gases.^1–4 Globally, solar (PV) technology is growing at the fastest rate among energy technologies. It is now more inexpensive than fossil fuels, with costs that are almost 90% lower.⁵ The worldwide energy supply from solar cells is 2%, but this might increase greatly if manufacturing challenges are successfully handled.⁶ The semiconductor compounds used in producing photovoltaic solar sheets have an important effect on solar cell production.^7–9 To overcome the barriers and limitations that prevent the widespread use of solar panels, extensive experimental and theoretical research has been recently conducted to identify compounds that are both highly effective and reasonably priced for use in solar panel design and manufacture.^10–13 The ability of semiconductor materials to absorb solar energy is heavily influenced by their energy conversion efficiency, which is directly proportional to their energy band gap. Because of their optoelectronic capabilities, semiconductor materials are vital for the progress of optoelectronic devices and manufacturing methods.^14–17 Unlike optically active direct band gap semiconductors, which perform well, optically inert indirect band gap semiconductors are inefficient because optical transitions need the involvement of phonons.^18,19 With the increasing necessity for environmentally approachable and renewable energy resources in recent times, thermoelectric compounds have received a lot of attention.^20–24 The figure of merit governs a thermoelectric material's energy conversion efficiency. Recently, the TE behavior of IV–V–VI materials has sparked attention in the thermoelectrical field due to their small thermal conductivity. Y. Gan et al. anticipate 56 exceptional semiconductors from the family IV–V–VI (IV = Ge, Sn, Si, Pb; V = Sb, Bi, As; VI = S, Se, Te) and show that the majority of these materials have thermal conductivity less than 1.0 W m⁻¹ K⁻¹ at normal temperature.²⁵ PbBi₂S₄ material exhibits very small lattice thermal conductivity of 0.46 W m⁻¹ K⁻¹ around the temperature range at 800 K, with a value of zT of 0.46.²⁶ Singh et al.²⁷ discovered Bi₂GeTe₄ to be an n-type material having a zT value of 0.10 and a small value of S²σ of about 1.54 μW cm⁻¹ K⁻² at temperature of 350 K. Bi₂GeTe₄ IV–V–VI ternary thermoelectric material was studied to have near-room temperatures of small thermal conductivity of 0.28 W m⁻¹ K⁻¹ at 350 K.²⁷ Schroeder et al.²⁸ first stated that p-type Bi₂GeTe₄ has a zT of 0.050 at ambient temperature. Konstantinov et al.²⁹ exposed that Bi₂GeTe₄ exhibits relatively adjacent p–n transition point with slight Ge content adjustments, implying that the E_F of Bi₂GeTe₄ would be in the middle of the energy gap. The electrical band structures of bulk Bi₂GeTe₄, both with and without SOC, exhibit a limited band gap. Spin orbit coupling is vital in forecasting proper dispersion when the energy band gap increases from 0.380 eV (LDA) to ∼0.1 eV (LDA + SOC), which is near the described bulk band gap of ∼0.18 eV.^30–34 Due to their potential uses in optoelectronics and thermoelectric devices, the electrical, optical, and thermoelectric properties of Bi₂PbSe₄ and Bi₂PbTe₄ have been extensively studied here using Density Functional Theory (DFT). The potential of Bi₂PbSe₄ and Bi₂PbTe₄ as cutting-edge materials for energy conversion technologies is emphasized by these results. Though additional experimental validation and investigation of doping methods are required to fully demonstrate their potential for practical use, yet DFT has proven to be a useful tool for comprehending and optimizing their properties.

2. Computational method

The structural parameters of Bi₂PbCh₄ (Ch = Se and Te) materialss as well as their electronic properties, were computed using the (FP-LAPW) approach employed in the WIEN2k code.³⁵ The PBE-GGA was used to analyze the effects of electronic interchange and correlation on structural characteristics. It is well accepted that energy gaps anticipated using typical approximations are smaller than observed.³⁶ Tran and Blaha formed a novel, useful potential known as the Tran–Blaha modified Beck–Johnson (TB-mBJ) potential, which gives a more accurate depiction of electrical characteristics and band gap predictions.³⁷ The cut-off value for the plane wave basis set K_max was established as R_min × MT = 11 (where R_min × MT is the minimal radii of the muffin-tin sphere). We replaced the Brillouin zone integration with a total k-points of 14 × 14 × 8 Monkhorst–Pack. The self-consistent field repetitions continued till the crystal's whole energy was less than 10⁻⁵ Ry. The Crystal Orbital Hamilton Population (COHP) calculations were carried out using the electronic structures acquired from Quantum ESPRESSO,³⁸ which were then post-processed to give the projected COHP curves for vital atomic pairs. Together with the Boltzmann transport calculations, including the rigid group and continually decreasing time approximations utilized in the BoltzTraP package, the thermodynamic properties were calculated using first-principles methods.³⁹

3. Results and discussions

3.1 Structure properties

Bi₂PbSe₄ has a trigonal crystal structure with R3̄m space group (see Fig. 1). Pb²⁺ is connected to six identical Se²⁻ atoms, forming PbSe₆ octahedra, which have edges with six identical BiSe₆ octahedra, corners with six corresponding PBS₆ octahedra, and corners with six identical BiSe₆ octahedra. The length of every Pb–Se bond measures 3.06 Å. Bi³⁺ is linked with six Se²⁻, forming octahedra BiSe₆, which attach corners and edges to three identical PbSe₆ octahedra and edges with six extra BiSe₆ octahedra. Three of the Bi–Se bond lengths measure 2.75 Å, while three others measure 3.06 Å. Two non-equivalent Se²⁻ sites exist. In the first Se²⁻ site, three corresponding Bi³⁺ atoms form bonds with Se²⁻ in a three-coordinate arrangement. The second site, for Se²⁻ is connected to three equivalent Pb²⁺ atoms and three equivalent Bi³⁺ atoms, resulting in a grouping of edge- and corner-sharing SeBi₃Pb₃ octahedra. Bi₂PbSe₄ have a trigonal crystal structure with R3̄m space group. Six corresponding Te²⁻ atoms bond with Pb²⁺ to produce PbTe₆ octahedra, sharing corners with six similar BiTe₆ octahedra and corners with six corresponding PbTe₆ and BiTe₆ octahedra. The length of each Pb–Te bond measures 3.17 Å. Six Te²⁻ atoms bond with Bi³⁺ to produce BiTe₆ octahedra, which combine corners and edges with three equivalent PbTe₆ octahedra and six equivalent BiTe₆ octahedra. Two inequivalent Te²⁻ sites exist. In the first Te²⁻ site, three Bi³⁺ atoms form bonds with Te²⁻ atoms. Also Te²⁻ atoms connects to three Pb²⁺ atoms and three identical Bi³⁺ atoms, resulting in an association of corner- and edge-sharing Bi₃Pb₃Te octahedra. The energy and volume optimization plots for Bi₂PbSe₄ and Bi₂PbTe₄ show the link between the system's total energy and unit cell volume. Table 1 presents the atomic coordinates and lattice constants of two chalcogenides: Bi₂PbSe₄ and Bi₂PbTe₄. Likewise Bi₂PbSe₄ possesses lattice constants of a = b = 4.78 Å and c = 32.27 Å, while Bi₂PbTe₄ has slightly greater values of a = b = 4.95 Å and c = 32.89 Å. The increased lattice parameters for Bi₂PbTe₄ can be related to the greater atomic radius of tellurium (Te) compared to selenium (Se). This results in a general expansion of the crystal lattice when Te replaces Se. Bi₂PbSe₄ possesses larger cell dimensions, implying a more relaxed structure because of weaker bonding and increased polarizability of Te atoms. The atomic locations indicate that the Bi, Pb, and chalcogen atoms possess separate fractional coordinates along the x, y, and z dimensions. In Bi₂PbSe₄, Bi is at (0.376, 0.697, 0.263), Pb at (0.387, 0.635, 0.698), and Se at (0.686, 0.359, 0.058). In Bi₂PbTe₄ the Bi, Pb, and Te atoms have positions at (0.381, 0.737, 0.371), (0.394, 0.672, 0.719), and (0.745, 0.473, 0.069), respectively. The shift in atomic coordinates for Bi and Pb between the two materials indicates that chalcogen substitution caused small shifts in bonding environments and interatomic distances. In particular, the chalcogen atoms (Se and Te) have different spatial arrangements, especially in the x and y dimensions, which could influence electronic distribution and local symmetry. In general, these structural differences, though modest, are vital for understanding the materials' electronic, optical, and thermoelectric properties, as these come directly from the interaction of atomic size and lattice geometry. Fig. 2(a) and (b) can be fitted with models based on the Birch–Murnaghan equation of state to get parameters such as bulk modulus, equilibrium volume, and pressure derivatives. The lowest point on the curve, 1460 ų for Bi₂PbSe₄ and 1725.4 ų for Bi₂PbTe_4, represents the balance volume (V₀), where the compound is more stable. At this volume, the system's energy is minimized, indicating the most energetically favorable structure. The two graphs show the optimization of energy vs. volume for the materials Bi₂PbSe₄ and Bi₂PbTe₄. The equilibrium volume is often calculated using (DFT) computations, and these curves show the system's whole energy as a function of its unit cell volume. The parabolic form of both graphs displays that the energy spreads its least value at a specific volume, which is the material's equilibrium configuration. The curvature of the E–V curve for Bi₂PbSe₄ near equilibrium volume is steeper compared to Bi₂PbTe₄, implying a relatively higher bulk modulus. The wider curve for Bi₂PbTe₄, on the other hand, represents greater compressibility and less stiffness. The equilibrium volume difference between the two materials is highlighted by the minima's position along the volume axis, which reproduces their different structural characteristics. The greater equilibrium volume (V₀) for Bi₂PbTe₄ (see Table 1) illustrates Te's larger ionic radius compared to Se. Bi₂PbSe₄ possesses greater bulk modulus, implying more incompressibility than Bi₂PbTe₄.

Fig. 1 The crystal structure for the Bi₂PbCh₄ (Ch = Se, Te) chalcogenides.

Table 1 The coordinates of atomic sites, lattice constants bulk modulus, ground state energy, and equilibrium volume for Bi₂PbCh₄ (Ch = Se, Te) chalcogenides

Materials	PBE-GGA				a (Å)	b (Å)	c (Å)	B0 (GPa)	E0 (Ry)	V0 (Å)
Bi2PbSe4(R3̄m)	Atoms	x	y	z	4.78	4.78	32.27	74.78	−148.32	289.09
	Bi	0.376	0.697	0.263
	Pb	0.387	0.635	0.698
	Se	0.686	0.359	0.058
Bi2PbTe4(R3̄m)	Atoms	x	y	z	4.95	4.95	32.89	62.35	−174.65	323.16
	Bi	0.381	0.737	0.371
	Pb	0.394	0.672	0.719
	Te	0.745	0.473	0.069

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Fig. 2 The optimization plots for the (a) Bi₂PbSe₄ and (b) Bi₂PbTe₄ chalcogenides.

3.2 Electronic properties

The Crystal Orbital Hamilton Population (COHP) plots (see Fig. 3(a) and (b)) demonstrate bonding and antibonding interactions between key atomic pairs such as Bi–Pb, Bi–Se (Bi–Te), and Pb–Se (Pb–Te), as well as the total COHP across the valence and the conduction band. In both materials, negative COHP values suggest bonding interactions, whereas positive values imply antibonding states. Strong bonding dominates Bi₂PbSe₄ in the energy range from roughly −6 eV to just below E_F, with the most intense bonding peaks resulting from Bi–Se and Pb–Se interactions, especially at −5 eV and −2 eV, consistent with the strong covalent character between Bi/Se and Pb/Se. The Bi–Pb interaction additionally shows bonding behavior in this location, though to a lesser extent, reflecting weak covalency. Additionally, as the energy approaches E_F from below, the COHP curves for Bi–Se and Pb–Se fall quickly to zero before crossing into minor positive values, showing the onset of antibonding states. Above E_F, particularly in the conduction band region (0 to +3 eV), the total COHP is mostly positive, dominated by antibonding states from Bi–Se and Pb–Se, indicating that more electrons might destabilize the structure by populating antibonding orbitals. The heavier and more polarizable Te atom produces a little shift in bonding in Bi₂PbTe₄. Bi–Te interactions display strong bonding features deeper in the valence band (−5.5 to −2 eV), but with larger peaks than Bi–Se, signifying more delocalized bonding. Pb–Te contributions are significantly lower over the energy range, illustrating that Pb and Te have low covalency in comparison to Pb–Se. Bi–Pb interactions in Bi₂PbTe₄ are stronger than in Bi₂PbSe₄, particularly at −4 eV and −1 eV. This indicates that Te substitution promotes Bi–Pb overlap, potentially because of structural modifications that decrease Bi–Pb distances or change orbital orientation. As we approach E_F, Bi–Te bonding weakens and transitions into antibonding states close to the Fermi level, with a sharp positive peak between +0.5 and +2 eV, indicating that electron doping could swiftly destabilize Bi–Te bonds. The total COHP for Bi₂PbTe₄ shows an important antibonding region directly above E_F, however, with slightly smaller intensity than Bi₂PbSe₄. This implies that Bi₂PbTe₄ may tolerate modest electron doping compared to Bi₂PbSe₄ before structural destabilization occurs. Bi₂PbTe₄ exhibits higher Pb–anion bonding (Pb–Se) and a somewhat more symmetric bonding/antibonding distribution, whereas Bi₂PbTe₄ focuses on Bi–Pb and Bi–Te pairs, with Pb–Te bonds having little impact on overall stability. This is in line with the decreased electronegativity difference between Pb and Te, which reduces bond polarity and overlap strength. The deeper and sharper bonding peaks in Bi₂PbSe₄ show a more localized covalent framework, while the broader peaks in Bi₂PbTe₄ imply higher orbital delocalization as well as a more metallic nature. The COHP study reveals that both compounds exhibit strong Bi–chalcogen bonds in the valence band, but differ in their secondary bonding routes. Pb–Se plays a role in Bi₂PbSe₄, while Bi–Pb becomes more relevant in Bi₂PbTe₄. Electron doping can affect bonding integrity, as antibonding states dominate the conduction bands in both cases. However, Bi₂PbTe₄ has a smaller total antibonding peak at E_F, suggesting slightly more tolerance to these effects. Bi₂PbSe₄'s stronger covalency could favor lower carrier mobility but higher lattice stability, while Bi₂PbTe₄'s more delocalized bonding may enhance carrier transport at the expense of less effective anion–cation binding.

Fig. 3 The Crystal Orbital Hamilton Population (COHP) plots for (a) Bi₂PbSe₄ and (b) Bi₂PbTe₄ chalcogenides presenting the total and pairwise orbital interactions between Bi–Pb, Bi–Se/Te, and Pb–Se/Te. Negative COHP values correspond to the bonding states, whereas the positive values correspond to antibonding states.

To recognize the distribution of electronic states, the density of states in energy ranging from −8.0 to 8.0 eV was determined. By calculating the partial density of states for Bi₂PbSe₄ and Bi₂PbTe₄, we studied the electron distribution in the valence band (VB) and conduction band (CB). Fig. 4(a) and (b) depicts the partial density of states of both Bi₂PbSe₄ and Bi₂PbTe₄ in the VB and CB regions. Bi-p orbitals strongly hybridize with p states of Se and Te, ranging from −4.8 eV to 0 eV. The bonding performance of Bi and Se/Te is largely covalent, with the Bi-p orbitals contributing to the expansion of bonding states. The Bi-d and Bi-s orbitals are less complicated in bonding because they are more confined. As a result, their contributions to the VB are low and limited to a higher energy range −2.0 eV to 0 eV. Because they are involved in making antibonding states with Se/Te-p orbitals, Bi-p states make a main impact in the CB from 1.0 eV–6.0 eV. These antibonding states arise at higher energy levels as a result of atomic orbital repulsion. At higher energy levels, 6.0 eV to 6.5 eV, the contribution of Bi-d states is small because of the maximum energy and less important function in generating the conduction band. For Bi₂PbSe₄ and Bi₂PbTe₄, the experimental contributions of s and p states of Pb in the VB and CB, as well as the substantial existence of Pb-s states from −7.8 eV to −6.3 eV, show that these orbitals are deeply bonded and belong to the lower-energy valence band. This is characteristic of s-orbitals, which are more limited and have lower energy due to their round symmetry and close overlap with the nearby atomic potentials. The Pb-s states also contribute, albeit less significantly, between −1.0 eV and the Fermi level. This shows hybridization with other orbitals, such as the Bi-p and Se/Te-p states, which results in bonding and antibonding states nearer 0 eV. Pb-p states lead the conduction band between 1.8 and 7.0 eV. This is because Pb-p states have with larger energy than Pb-s states and can efficiently overlap with the antibonding states formed by hybridization with the surrounding Bi-p and Se/Te-p states. The observed dominance of p states of Se and Te in the valence band in the energy range −4.5 eV to the Fermi level and CB from 1.5 eV to 6 eV for Bi₂PbSe₄ and Bi₂PbTe₄ compounds. This happens because p states of Se and Te are energetically well-positioned to establish strong covalent bonding and antibonding states with the lattice adjacent atoms, especially Bi and Pb. The CB is formed by antibonding combinations of the p states of chalcogens, along with some additional influence from other orbitals (e.g., s\d-states) of Se, Te, and nearby Bi and Pb atoms. The s-orbitals of Se and Te contribute less to bonding because they are more localized and lower in energy than the p-orbitals. Their contributions to that conduction band in that energy range of 1.8 eV to 6.3 eV are attributable to hybridization at higher energy levels.

Fig. 4 The projected density of states for (a) Bi₂PbSe₄ and (b) Bi₂PbTe₄ chalcogenides.

Fig. 5(a)–(d) depicts the projected EB structures for these Bi₂PbSe₄ and Bi₂PbTe₄ compounds at their balanced structural parameters, comparing the PBE-GGA and TB-mBJ methods. Bi₂PbSe₄ and Bi₂PbTe₄ have similar energy band characteristics. The energy gap of both Bi₂PbSe₄ and Bi₂PbTe₄ is noticed as direct (Γ–Γ). The energy gaps with the TB-mBJ and PBE-GGA are 1.12 and 0.71 eV for Bi₂PbSe₄ and 1.08 and 0.82 eV for Bi₂PbTe₄. Where Bi₂PbSe₄ has a slightly larger band gap than Bi₂PbTe₄ under both the TB-mBJ and PBE-GGA methods, since selenium (Se) is smaller and less electronegative than tellurium. Stronger bonding in Bi₂PbSe₄ causes a wider energy difference, resulting in larger band gap. Bi₂PbTe₄ has weaker bonding due to the greater size and higher polarizability of Te, resulting in a smaller band gap. The electronic band structure and orbital contributions reported for Bi₂PbSe₄ and Bi₂PbTe₄ are determined by their atomic composition, chemical bonding type, and orbital hybridization. Because of their large atomic number, bismuth (Bi) atoms provide a significant contribution to the valence band, resulting in relativistic effects. These effects induce a considerable splitting of energy levels, resulting in the stabilization of Bi-p states from −4.8 eV to the Fermi level maximum. Lead (Pb) atoms contribute via their s-orbitals at −7.8 to −6.3 eV. This is due to the lower energy of Pb-s states, which are predominantly involved in core-like bonding interactions. Their contribution to bonding is less important than that of Bi-p or chalcogen-p states; hence, they are further down the valence band. The overlap of Bi-p and Se/Te-p states gives bonding and antibonding states in the VB. Bi-p states lead in the CB between 1.0 and 6.0 eV due to the antibonding character of the Bi-p and chalcogen-p connections. Pb-p states are more energetic than Pb-s states and contribute importantly to the CB between 1.8 and 7.0 eV. This is because Pb-p states overlap weakly with Bi-p states and donate antibonding interactions, causing them to scatter across the CB. In the 1.5 to 6.01 eV range, these Se-p and Te-p states contribute through hybridization with the p states of Bi and Pb. The chalcogen-p states mainly control the (DOS) around the CBM.

Fig. 5 The energy band profiles for (a and b) Bi₂PbSe₄ and (c and d) Bi₂PbTe₄ chalcogenides.

3.3 Optical properties

The real part of the frequency-dependent dielectric constant ε₁(ω) delivers key information regarding the optical nature of materials such as Bi₂PbSe₄ and Bi₂PbTe₄. Fig. 6(a) illustrates the ε₁(ω) in Bi₂PbSe₄ and Bi₂PbTe₄. Bi₂PbSe₄ and Bi₂PbTe₄ have ε₁(0) = 15.0 and 20.0, respectively. A higher value for the Bi₂PbTe₄ suggests more polarizability and lower interband transition energy than Bi₂PbSe₄. Moreover, Bi₂PbTe₄ has a lower band gap at 2.0 eV compared to Bi₂PbSe₄ at 2.5 eV. The peaks in ε₁(ω) indicate resonances caused by electrical transitions between bands. The lower peak energy of Bi₂PbTe₄ indicates that its band structure has smaller energy gaps for some optical transitions than Bi₂PbSe₄. Following the peaks, ε₁(ω) drops, and Bi₂PbSe₄ and Bi₂PbTe₄ reach negative ε₁(ω) values at 3.0 and 2.5 eV, respectively. When ε₁(ω) goes negative, the material exhibits metallic optical behavior (plasmonic behavior). This phenomenon results from a strong interaction between free charge carriers and incident light; Bi₂PbTe₄ achieves this state quickly because of its larger carrier density and more delocalized electrons. The imaginary part of the dielectric function, ε₂(ω), represents absorption of electromagnetic radiation caused by interband electronic transitions. Fig. 6(b) indicates that the threshold values of the ε₂(ω) are 1.5 eV and 1.0 eV for Bi₂PbSe₄ and Bi₂PbTe₄, respectively. The threshold value is the negligible energy required for interband electronic transitions. The highest peaks were seen at 3.0 and 2.5 eV for Bi₂PbSe₄ and Bi₂PbTe₄, respectively. Bi₂PbTe₄ drop to lower energy indicates a denser and more accessible conduction band structure than in Bi₂PbSe₄. After the peak, the imaginary part declines because fewer electron states exist for high-energy transitions. This drop is normal as photon energy rises over the threshold where interband transitions dominate, leaving only weaker transitions or higher-order effects.

Fig. 6 Calculated optical properties of Bi₂PbCh₄ (Ch = Se, Te): (a) and (b) dielectric function components, (c) refractive index, (d) absorption coefficient, (e) reflectivity spectra, and (f) energy loss function.

Fig. 6(c) depicts how the refractive index n(ω) varies with photon energy for materials Bi₂PbSe₄ and Bi₂PbTe₄. The static n(0) values for Bi₂PbSe₄ and Bi₂PbTe₄ are 3.8 and 4.6, respectively. The n(ω) first rises due to the intense resonance from interband electronic transitions. The highest peaks are 2.8 eV for Bi₂PbSe₄ and 2.3 eV for Bi₂PbTe₄. The transitions resonate with the input photon energy, resulting in higher polarizability and a high refractive index. Bi₂PbSe₄ has a greater band gap than Bi₂PbTe₄, resulting in a peak at somewhat higher photon energy. The heavier Te atom enhances spin–orbit coupling and polarizability in Bi₂PbTe₄, resulting in a higher static refractive index. At higher energies, materials show plasma oscillations of free carriers or interband transitions, which minimize the contribution of bound electrons to the refractive index. The n(ω) declines for both materials when photon energy increases from 2.8 eV to 24.0 eV in Bi₂PbSe₄ and from 2.3 eV to 24.0 eV in Bi₂PbTe₄. Fig. 6(d) shows the observed trend in the absorption coefficient I(ω) for Bi₂PbSe₄ and Bi₂PbTe₄ materials. The threshold absorption coefficient represents the minimal photon energy essential to stimulate an electron from the VB to the CB. Bi₂PbSe₄ at 1.6 eV and Bi₂PbTe₄ at 1.3 eV have values near their band gap energies. This is when interband transitions begin, resulting in considerable absorption. In the photon energy range of 3.5 eV to 12.5 eV for Bi₂PbSe₄ and 2.7 eV to 12.0 eV for Bi₂PbTe₄, substantial absorption occurs due to transitions between deeper valence bands and higher conduction bands. These transitions entail a denser electronic state distribution, resulting in a larger density of optical transitions and, thus, higher absorption coefficients. Due to Bi₂PbTe₄ lower band gap, the range begins significantly earlier at 2.7 eV than Bi₂PbSe₄ at 3.5 eV. The specific electronic structure of Bi₂PbSe₄ and Bi₂PbTe₄ dictates where major transitions terminate. Beyond 12.5 eV (Bi₂PbSe₄) and 12.0 eV (Bi₂PbTe₄), the states no longer line well with the incoming photon energy, resulting in decreased absorption.

Fig. 6(e) shows that Bi₂PbSe₄ and Bi₂PbTe₄ have R(0) of 0.35 and 0.40, respectively, indicating the material's inherent capacity to reflect light in the low-energy regime. Differences in R(0) result from changes in the electronic structure, particularly the DOS at or around the Fermi energy. Bi₂PbSe₄ and Bi₂PbTe₄ have the largest R(ω) peaks at 4.3 eV and 3.8 eV, respectively. The discrepancy in peak positions (4.3 eV vs. 3.8 eV) indicates differences in the band structure of Bi₂PbSe₄ and Bi₂PbTe₄. Bi₂PbTe₄ has a smaller energy gap between the electronic states involved in this optical transition. The band structures of Bi₂PbSe₄ and Bi₂PbTe₄ differ due to the replacement of selenium with tellurium. Tellurium, being heavier, causes higher spin–orbit coupling and potentially narrower band gaps. At higher photon energies, the materials absorb light due to the start of various transitions or the excitation of electrons to states deep in the conduction band, diminishing total reflectivity. The L(ω), including its peaks and subsequent reduction, is intimately related to optical characteristics and collective excitations in the material. In Fig. 6(f), the threshold energy of L(ω) is 4.2 eV for Bi₂PbSe₄ and 4.0 eV for Bi₂PbTe₄, indicating the start of considerable energy loss. This threshold frequently coincides with interband transitions or excitations, in which an electron jumps between energy bands and bridges the band gap. The largest peaks at 18.50 eV for Bi₂PbSe₄ and 16.7 eV for Bi₂PbTe₄ parallel to the plasmon resonance frequency, which occurs when the conduction electrons' collective oscillations match the incident electromagnetic wave. The energy loss function L(ω) for Bi₂PbSe₄ and Bi₂PbTe₄ declines after reaching 18.5 eV and 16.7 eV, respectively. The dielectric function ε(ω) is less sensitive at higher frequencies due to electrons' inability to follow the quickly fluctuating field, resulting in a decreased L(ω).

3.4 Thermoelectric properties

The behavior of the Seebeck coefficient (S) can be explained using the fundamental physics of thermoelectric materials, specifically the link between carrier concentration, scattering mechanisms, and temperature. Fig. 7(a) depicts the S for Bi₂PbSe₄ and Bi₂PbTe₄ materials at temperatures ranging from 0 to 700 K. Fig. 7(a) shows that Bi₂PbSe₄ and Bi₂PbTe₄ have maximal Seebeck coefficient values of 1.5 × 10⁻⁶ V K⁻¹ and 3.01 × 10⁻⁶ V K⁻¹ at 50 K, respectively. Bi₂PbTe₄ has a larger initial Seebeck coefficient at 50 K than Bi₂PbSe₄ due to variations in their band structures, carrier effective masses, and intrinsic doping levels. At low temperatures, the carrier density is low, resulting in a sharper energy dependency of the density of states and superior thermopower. The carriers are less thermally restless, and the transport properties are mostly dictated by the compound's basic electronic structure. This allows for larger asymmetry in the carrier energy distribution, resulting in a higher Seebeck coefficient (S). Higher temperatures cause thermal excitation of electrons and holes around the band gap, enhancing bipolar conduction. The Seebeck coefficient, which is the weighted number of contributions from both types of carriers, reduces as they tend to counterbalance one another. At 650 K, Bi₂PbSe₄ and Bi₂PbTe₄ have minimal Seebeck coefficients (S) of −12.5 × 10⁻⁶ V K⁻¹ and −13.0 × 10⁻⁶ V K⁻¹, respectively. The negative value shows that electrons are the main charge carriers in each Bi₂PbSe₄ and Bi₂PbTe₄. Bi₂PbTe₄ has a much larger negative Seebeck coefficient (S) at 650 K, representing stronger n-type behavior, which could be owing to a lesser band gap or more thermal carrier excitation. The reduction in electrical conductivity (σ/τ) for both Bi₂PbSe₄ and Bi₂PbTe₄ can be clarified by the interaction of carrier concentration, mobility, and scattering mechanisms. At higher temperatures, thermal excitation causes a slight rise in intrinsic carriers. However, this is inadequate to compensate for the considerable drop in mobility caused by scattering effects. Fig. 7(b) displays a reduction in electrical conductivity (σ/τ) from 50 K to 650 K for both Bi₂PbSe₄ and Bi₂PbTe₄. Fig. 7(b) shows that the maximum electrical conduction (σ/τ) numbers at 50 K are 1.44 × 10¹⁸ and 1.42 × 10¹⁸ Ω ms⁻¹ for Bi₂PbSe₄ and Bi₂PbTe₄, respectively. At 650 K, Bi₂PbSe₄ and Bi₂PbTe₄ have minimum electrical conductivity (σ/τ) values of 1.36 × 10¹⁸ and 1.32 × 10¹⁸ Ω ms⁻¹, respectively.

Fig. 7 Calculated thermoelectric properties of Bi₂PbCh₄ (Ch = Se, Te): (a) Seebeck coefficient, (b) electrical conductivity, (c) thermal conductivity, (d) figure of merit, (e) lattice thermal conductivity, and (f) power factor.

Bi₂PbSe₄ and Bi₂PbTe₄ display a linear increase in electronic thermal conductivity (κ_e) with temperatures from 50 K to 650 K, which could be credited to that material's electronic characteristics. Fig. 7(c) indicates a linear rise in electronic heat conductivity (κ_e) through temperature from 50 K to 650 K for Bi₂PbSe₄ and Bi₂PbTe₄. The Wiedemann–Franz law describes that electronic thermal conductivity (κ_e) is determined by the mobility of charge carriers in a material and is proportional to its electrical conductivity (σ): κ_e = LσT. Since T is directly in the Wiedemann–Franz equation, κ_e increases with temperature as long as σ does not fall significantly. Bi₂PbSe₄ and Bi₂PbTe₄ materials have a thermal conductivity of 0.90 and 1.05 (×10¹⁴ W m⁻¹ K⁻¹ s⁻¹) at −300 K, respectively, and reach an extreme of 1.95 and 2.10 (×10¹⁴ W m⁻¹ K⁻¹ s⁻¹) at partial density of states 600 K. Bi₂PbTe₄ has a lower band gap than Bi₂PbSe₄, resulting in higher carrier concentration at a given temperature and, thus, higher κ_e. The figure of merit zT for both materials (see Fig. 7(d)) surges as temperature rises from 50 K to 650 K. Fig. 7(d) indicates that at 300 K, Bi₂PbSe₄ and Bi₂PbTe₄ had zT values of 0.25 and 0.20, respectively. At 650 K, the highest zT values are 0.65 and 0.53, respectively. Differences and trends in zT values occur as thermoelectric characteristics fluctuate with temperature. At low temperatures (300 K), both materials have a low zT due to limited carrier excitation and increased thermal conductivity. Bi₂PbSe₄ and Bi₂PbTe₄ differ in their fundamental material features, including bonding strength, atomic masses, and phonon scattering mechanisms. Bi₂PbSe₄ often performs better in thermoelectric applications because of reduced thermal conductivity and a more favorable combination of electrical characteristics.

The lattice thermal conductivity (κ_l) and power factor (PF) of Bi₂PbSe₄ and Bi₂PbTe₄ show distinct thermoelectric performance. The Fig. 7(e) for lattice thermal conductivity demonstrates that both materials experience a significant increase in κ_l as temperature rises from 50 K to generally 300 K. Although κ_l begins to saturate, especially with Bi₂PbSe₄. This thermal saturation indicates that phonon–phonon Umklapp scattering takes priority at high temperatures. Between 300 K and 650 K, Bi₂PbSe₄ shows a higher and flatter κ_l profile (7.5–7.8 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹), showing steady phonon transport in that range. Bi₂PbTe₄ has a slightly greater value κ_l at 50 K (2.58 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹), but rises slowly and consistently below Bi₂PbSe₄ after 150 K, attaining 7.31 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹ at 650 K. Te's higher atomic mass and phonon scattering decrease lattice conductivity, which makes it ideal for thermoelectric materials with low κ_l and improved zT. Yet, the power factor (PF) curves (see Fig. 7(f)) favor Bi₂PbSe₄. Its PF rises fast and linearly with temperature, starting at 8.77 × 10¹¹ W m⁻¹ K⁻² s⁻¹ at 50 K and reaching 6.67 × 10¹¹ W m⁻¹ K⁻² s⁻¹ at 650 K. This pattern shows a substantial increase in both Seebeck coefficient and electrical conductivity as temperature rises, suggesting optimal thermoelectric performance. Bi₂PbTe₄, on the other hand, displays an increasing PF with temperature, though at a consistently lower magnitude, starting at 1.41 × 10¹¹ W m⁻¹ K⁻² s⁻¹ at 50 K and declining at 4.49 × 10¹¹ W m⁻¹ K⁻² s⁻¹ at 650 K. At low temperatures, it has a slightly greater PF due to superior electrical conductivity, but it rapidly loses position to Bi₂PbSe₄ at and above 100 K. Bi₂PbSe₄'s Seebeck coefficient increases with temperature without losing electrical conductivity, but Bi₂PbTe₄ loses this balance. Bi₂PbSe₄ possesses better electrical transport performance (PF) and greater κ_l. Bi₂PbTe₄ has lower lattice thermal conductivity, which is helpful for thermal performance. Yet its power factor performance is insufficient, limiting its efficiency. Bi₂PbTe₄ may benefit from lower phonon heat conduction, although Bi₂PbSe₄ is more attractive thermoelectrically because of its superior electronic performance.

4. Conclusions

This study examined the optoelectronic and thermoelectric properties of novel Bi₂PbSe₄ and Bi₂PbTe₄ with trigonal structure and space group R3̄m, employing density functional theory. The minima's position along the volume axis effectively replicates the two materials' varying structural features, highlighting the equilibrium volume difference between them. Bi-p orbitals were substantially hybridized with p states of Se and Te in the valence band. Bi₂PbSe₄ and Bi₂PbTe₄ exhibit direct band gaps at (Γ–Γ) points, with predicted energy gaps of 1.12 and 0.71 eV for Bi₂PbSe₄ and 1.08 and 0.82 eV for Bi₂PbTe₄ using the TB-mBJ and PBE-GGA, respectively. Bi₂PbTe₄ has weak bonding due to Te larger size and its greater polarizability, leading to a smaller band gap. Because of their massive atomic number, bismuth atoms contributed significantly to the valence band, resulting in relativistic effects that caused considerable splitting of energy levels, resulting in the stabilization of Bi-p states. Bi₂PbTe₄ possesses stronger Bi–Pb interactions and lower Pb–chalcogen covalency than Bi₂PbSe₄, which can be due to Te's larger size and polarizability. The peaks in ε₁(ω) signify resonances induced by electrical transitions between bands. Bi₂PbTe₄ had a higher ε₁(0) value, signifying stronger polarizability and lower interband transition energy compared to Bi₂PbSe₄. The peaks in ε₂(ω) corresponded to a substantial density of states, where interband transitions probably occurred. Bi₂PbTe₄ decreases to lower energy, revealing a denser and more accessible conduction band structure than Bi₂PbSe₄. The heavier Te atom improved spin–orbit coupling and polarizability in Bi₂PbTe₄, leading to an increased static refractive index. Bi₂PbSe₄ and Bi₂PbTe₄ exhibit R(0) values of 0.35 and 0.40, indicating the material's natural ability to reflect light in the low-energy region. The L(ω), notably its peaks and subsequent decline, was significantly connected to optical properties and collective excitations in these materials. The dielectric function ε(ω) becomes less sensitive at higher frequencies because of electrons' inability to adapt to the rapidly fluctuating field, leading to a fall in L(ω). Bi₂PbTe₄ has a higher primary Seebeck coefficient of around 50 K than Bi₂PbSe₄ due to differences in band topologies, carrier effective masses, and intrinsic doping levels. The negative values of the Seebeck coefficients demonstrated that electrons were the primary charge carriers in both Bi₂PbSe₄ and Bi₂PbTe₄. The reduction in electrical conductivity with higher temperatures in Bi₂PbSe₄ and Bi₂PbTe₄ can be explained by the combination of carrier concentration, mobility, and scattering. Variations and trends in zT values occurred when thermoelectric properties changed with temperature.

Conflicts of interest

There are no conflicts to declare.

Data availability

Data are available upon request from the corresponding author.

Acknowledgements

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2503).

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