Investigations on the thermoelectric and thermodynamic properties of quaternary coinage metal HgSBr

Abstract

We present the findings of a thorough first-principles analysis of physical parameters related to the ground state, elastic, electronic, optical, thermodynamic, and transport properties of the quaternary coinage metal-based compound CuHgSBr using the WIEN2k package. The computed equilibrium lattice parameters align well with their experimental equivalents, providing strong support for the validity of the findings. We performed numerical and computational calculations to estimate the elastic constants for the orthorhombic structure with space group Pbam. The band structure analysis of CuHgSBr reveals an indirect band gap semiconductor of 0.76 eV, classifying it as a p-type semiconductor. We also calculated the optical properties within the energy range of 0–13.56 eV. Moreover, we investigated the effective mass, exciton binding energy, and exciton Bohr radius, which indicated that CuHgSBr exhibits a weak exciton binding energy and belongs to the Mott-Wannier type exciton category. Using the Boltzmann transport theory, along with the constant relaxation time and Slack equations, we determined the thermoelectric properties and lattice thermal conductivity of CuHgSBr. Notably, the figure of merit at 800 K is calculated to be 0.54, which is encouraging for potential thermoelectric applications. The comprehensive research study we conducted provides valuable insights for experimental research across multiple physical properties, as the material is being theoretically examined for the first time in this full prospectus.

1. Introduction

Around the world, active research is being conducted in the hunt for effective materials for use in photovoltaics, optoelectronics and thermoelectric devices. Waste heat causes a significant loss of useable energy, with this lost heat being of excellent quality but dispersed. As the need for electricity among humans continues to grow, sustainable energy-producing techniques like thermoelectric energy generation are becoming increasingly crucial [1]. The discovery made by Seebeck in 1821 revealed that thermoelectric (TE) materials have the capability to produce power by utilizing a difference in temperature, which is known as the Seebeck effect. Conversely, the Peltier effect generates a temperature difference using electricity. As a result, there is a need for thermoelectric materials with exceptional performance to efficiently transform heat into electric energy on a continuous basis [2,3].

In recent times, the exploration of high-performance thermoelectric substances has garnered increasing attention from the fields of energy and the environment, driven by the prospect of potential commercial applications. The complexity of thermoelectric materials at the structural, electrical, and compositional levels necessitates close collaboration among a broad scientific community consisting of chemists, physicists, and materials scientists [[4], [5], [6]]. Notably, spacecraft have been continuously utilizing this energy-generating technique since early 1960s [7,8]. For the past two decades, thermoelectric materials have been at the forefront of research aimed at utilizing waste heat in stationary applications and automobiles to reduce the burden on alternators [9].

Several organisations have led the way in global-scale thermoelectric development, with numerous significant advancements made possible by fundamental insights based on first-principles. The DFT framework enables the estimation of band structure projections, phase diagrams through cluster expansion techniques, and phase stability of unidentified compounds. Over the past 15 years, the thermoelectric figure of merit (ZT) of half-Heusler alloys, fascinating class of thermoelectric materials, has experienced significant improvement. Recently, there has been a notable interest in chalcogenide materials for narrow/wide bandgap semiconductors, which are frequently employed in optoelectronic devices. Photovoltaic technology relies on a variety of materials, including primarily organic, inorganic, or organic-inorganic mixtures, for efficient photon-to-electron conversion [10].

Recent experimental and theoretical reports have highlighted several ternary and quaternary chalcogenide-based materials [[11], [12], [13], [14], [15]] for optoelectronics [[16], [17], [18], [19], [20]] and thermoelectric [[21], [22], [23], [24], [25], [26]] applications. Consequently, we conducted DFT computation on the quaternary chalcogenide-based material CuHgSBr, which has demonstrated extraordinarily high efficiency and provides a fundamental overview of all digital tools. The synthesis involves a hydrothermal reaction within sealed glass ampoules at 670 K, yielding yellow-orange CuHgSBr crystals. The most prevalent crystal structure for our chemical is orthorhombic with space group Pbam. The structures comprise flattened HgS chains that are interconnected by a combination of distorted CuS₂Br₂ tetrahedra sharing an edge with Br atoms. Notably, the copper atoms are positioned off-centre, as corroborated by experimental investigations [27]. The prior structural method, reliant on powder diffraction data [28], was invalidated upon the determination of a single crystal structure. The structural archetype of CuHgSBr bears a striking resemblance to previously identified compounds like CuHgSeBr, AgHgSBr, AgHgSI (collectively known as MHgYX) [29].

In the current work, we employed the comprehensive full potential linearized augmented plane wave (FP-LAPW) technique using the WIEN2k software to analyze the structural, elastic, electronic, and optical aspects of CuHgSBr. We are not aware of any literature describing the structural, transport, electronic, optical, thermodynamic, and elastic properties of these materials.

2. Computational details

The DFT computation were performed for the quaternary compound CuHgSBr to estimate its ground state, electronic behavior, elasticity, optical and transport characteristics. We used the WIEN2k [30] package to calculate all these properties. To achieve this, we employed the WIEN2k software [30], utilizing the Generalized Gradient Approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) method [31] to account for electron exchange-correlation effects. For CuHgSBr, we set an energy separation of −6.0 Ry between the valence and core states. The parameters R_MT x K_max were chosen as 8.0, with a k value of 4 × 4 × 4 using Monkhorst-Pack method. Notably, 2000 k points were designated solely for optical property calculations. Muffin tin radii, which prevent core charge leakage, were set at 2.19, 2.44, 1.79, and 2.14 for Cu, Hg, S, and Br, respectively. The calculations for CuHgSBr's structure were based on lattice constants reported experimentally [27]. To analyze structural parameters and their dependence on energy against the c/a ratio, Murnaghan's state equation was employed. The determination of elastic constants was facilitated by the IR-Elast package [32], integrated within the WIEN2k code, leading to the assessment of various elastic parameters. Optical properties were explored through the complex dielectric function, while transport properties were studied using the BoltzTraP code [33].

3. Results and discussion

3.1. Geometry optimization

The quaternary compound CuHgSBr exhibits an orthorhombic crystal structure with the Pbam space group and Brillouin zone plot, as shown in Fig. 1(a) and (b). Structural optimization was executed using the reported experimental lattice parameters [27]. The total energy was fitted into the Murnaghan equation of state as a function of c/a, and this relationship was visualized in Fig. 2. Based on the total energy vs c/a ratio, the optimized lattice parameters, total energy and equilibrium volume for different functionals were computed and listed in Table 2. Among the exchange-correlation functionals considered, GGA-PBE exhibited the lowest ground state energy, as indicated by the obtained total energy values. Consequently, GGA-PBE was selected as the functional for computing all the physical properties discussed in this paper. Our computed outcomes demonstrated favorable consistency with experimental data. Furthermore, we determined the bond lengths for CuHgSBr, and these values are detailed in Table 1.

Fig. 1.

a) Crystal structure of CuHgSBr, and b) First Brillouin zone representing high symmetry k points for CuHgSBr compound.

Fig. 2.

Optimized c/a variations using GGA-PBE.

Table 2.

Calculated optimized lattice constants, equilibrium volume, band gap (eV) and total energy (Ry) using different exchange-correlation functional of CuHgSBr.

CuHgSBr	a (Å)	b (Å)	c (Å)	V (Å3)	Eg (eV)	E0 (Ry)
Experimental [27]	10.037	18.336	4.124	758.97	–	–
Theoretical (GGA-PBE)	10.137	18.517	4.042	758.71	0.767	−389164.053
GGA-WC	10.138	18.515	4.042	758.70	1.329	−389124.524
LDA	10.133	18.507	4.045	758.56	1.309	−388879.370
mBJ	–	–	–	–	1.243	−388874.601

Table 1.

Elucidated atomic bond lengths within the crystal structure of CuHgSBr.

Compound	Bond	Bond distance (Å)
CuHgSBr	Cu–Br	2.4553
	Hg–S	2.3799
	Cu–S	2.2817

3.2. Electronic properties

The energy dispersion within CuHgSBr defines its electronic characteristics, a feature elucidated by the calculated band structure presented in Fig. 3. This band structure assessment is conducted across significant directions within the Brillouin zone, highlighting high-symmetry paths. The valence band maximum (VBM) and conduction band minimum (CBM) predominantly consist of Cu-d and S-p states, respectively. Notably, the material possesses an indirect narrow band gap of 0.76 eV, determined through the application of the PBE-GGA method. The CBM is situated at the high-symmetry gamma point, while the VBM resides between the R and Γ points, indicative of an asymmetrical location. Fig. 1(b), portrays the first Brillouin zone alongside high-symmetry k paths. The presence of non-symmetrical points within the band structure holds promise for thermoelectric materials, a trait previously acknowledges in high-performance material studies [[34], [35], [36], [37]].

Fig. 3.

Electronic band structure of CuHgSBr.

We also estimated the band gap values using various exchange-correlation methods within the WIEN2k software, and these outcomes are listed in Table 2. Near the Fermi level E_F, the band structure shows very flat energy dispersion in the X-Γ direction, indicating a high hole effective mass. Flat bands, characterized by minimal energy dispersion, have significant effects on electronic properties and can lead to interesting physical phenomena, such as a high density of states, which enhances various electronic and transport properties, including thermoelectric efficiency.

The valence band region exhibits a collection of more dispersive flat bands and less dispersive parabolic bands. Flat bands find applications in 2D and 3D materials, such as ferromagnetism, superconductivity, Mott insulators, thermoelectric devices, quantum Hall effect, strongly correlated systems, Kagome lattice, and several topological states [[38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49]]. In our case, the presence of flat bands in the VB enhances the performance of thermoelectric devices, leading to an elevation in the Seebeck coefficient. All of these aspects are discussed in the transport properties section.

Regarding the conduction band (CB), there is only one band at the bottom, with an average effective mass of roughly 0.1 electrons, signifying the prevalence of electrons as the main charge carriers. The collective density of states (DOS) and partial density of states (PDOS) are shown in Fig. 4, Fig. 5. The calculated S-p peak is at 1–3 eV below the CBM, and the Cu-d peak is at 0 to −3 eV. The lower VB is composed of Hg-d orbitals, while the higher VB closest to the Fermi level is made up of Cu-d, S-p and Br-p orbitals. S-p orbitals dominate the bottom of the CB, accounting to the primary significant peak in the CB DOS at 2 eV. The s states in the PDOS plots were neglected due to their deep energy level. It is important to note that there is currently no experimental research available for comparison regarding the electronic properties of CuHgSBr.

Fig. 4.

TDOS of CuHgSBr.

Fig. 5.

PDOS for a) Copper (Cu), b) Mercury (Hg), c) Sulphur (S) and d) Bromine (Br).

3.3. Elastic and mechanical properties

The ability of materials to regain their original size and shape after the stress within them is removed can be explained using elastic attributes, as long as the stress doesn't exceed the elastic limit. The mechanical traits of materials (represented by C_ij) are established by the elastic constants of single crystals. These constants are linked to the Debye temperature and Grüneisen parameter. The mechanical robustness of materials that are sought after for technological purposes, including factors like melting point, fracture toughness, load deflection, sound velocity, and thermoelastic stress, plays a significant role in their performance. These parameters also provide crucial insights into stiffness, material bonding, and the interatomic forces present within solids.

The CuHgSBr quaternary compound has an orthorhombic crystal structure [50], which is characterized by nine distinct elastic constants: C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, and C₆₆, C₁₂, C₁₃, and C₂₃. These elastic constants must adhere to specific conditions (expressed in equations (1), (2), (3), (4), (5), (6), (7))) [51] to ensure the elastic stability of the orthorhombic crystal system, as defined by the Born-Huang criteria. The estimated values for the elastic constants of CuHgSBr are provided in Table 3, and they satisfy the prerequisites for maintaining elastic stability within the orthorhombic framework.

$$C_{11}>0$$ (1)

$$C_{11}{C}_{22}>C_{12}^2$$ (2)

$$C_{11}{C}_{22}{C}_{33}+2C_{12}{C}_{13}{C}_{23}$$ (3)

$$-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-C_{33}{C}_{12}^2>0$$ (4)

$$C_{44}>0$$ (5)

$$C_{55}>0$$ (6)

$$C_{66}>0$$ (7)

Materials with higher values of elastic constants (C_ij) are more incompressible and require greater external force to deform them, while materials with lower values of C_ij are more easily deformed. Table 4 displayed additional mechanical properties. To understand the internal strains described by equation (8) [52], the Kleinman parameter ($Ϛ)$ was calculated.

$$Ϛ=\frac{C_{11}+8C_{12}}{7C_{11}+2C_{12}}$$ (8)

Table 3.

Elucidated elastic constants and Kleinman parameter ($Ϛ)$ for CuHgSBr using GGA-PBE functional.

XC	${\mathit{C}}_{\mathbf{11}}$	${\mathit{C}}_{\mathbf{22}}$	${\mathit{C}}_{\mathbf{33}}$	${\mathit{C}}_{\mathbf{44}}$	${\mathit{C}}_{\mathbf{55}}$	${\mathit{C}}_{\mathbf{66}}$	${\mathit{C}}_{\mathbf{12}}$	${\mathit{C}}_{\mathbf{13}}$	${\mathit{C}}_{\mathbf{23}}$	$\mathit{Ϛ}$
GGA-PBE	104.80	102.50	138.66	3.34	4.06	32.93	33.40	44.06	20.82	1.15

Table 4.

Elucidated Bulk modulus (GPa), Young's modulus (GPa), Shear modulus (GPa), Poisson's ratio, and Pugh's ratio for CuHgSBr.

XC	${\mathit{B}}_{\mathit{v}}$	${\mathit{B}}_{\mathit{R}}$	$\mathit{B}$	${\mathit{E}}_{\mathit{v}}$	${\mathit{E}}_{\mathit{R}}$	$\mathit{E}$	${\mathit{G}}_{\mathit{v}}$	${\mathit{G}}_{\mathit{R}}$	$\mathit{G}$	${\mathit{n}}_{\mathit{v}}$	${\mathit{n}}_{\mathit{R}}$	$\mathit{n}$	$\raisebox{1ex}{$\mathit{B}$}\!\left/ \!\raisebox{-1ex}{$\mathit{G}$}\right.$
GGA-PBE	60.28	59.04	59.66	64.91	22.89	44.75	24.57	7.97	16.27	0.320	0.435	0.374	3.666

The $Ϛ$ provides information about a compound's ability to resist bending and stretching forces. The capacity to withstand alterations in volume when subjected to external pressure is quantified through the bulk modulus. Equations (9), (10), (11), (12) [53] can be employed to calculate the bulk modulus, shear modulus, Young's modulus, and Poisson's ratio. The bulk modulus signifies how well a material resists changes in volume, while the shear modulus gauges its resilience to reversible deformations. Young's modulus denotes the material's rigidity against alterations in length. Based on the computed bulk modulus values, CuHgSBr is not considered a hard material. Additionally, the size of the bulk modulus is related to bond distance in the crystal structure, with smaller bond lengths resulting in larger bulk modulus values. In the case of CuHgSBr, the obtained bond lengths are shorter.

$$B=\frac{B_v+B_R}{2}$$ (9)

$$G=\frac{G_v+G_R}{2}$$ (10)

$$E=\frac{9GB}{3B+G}$$ (11)

$$n=\frac{3B-2G}{6B+2G}$$ (12)

Through the utilization of Pugh's ratio B/G and Poisson's ratio, n, it becomes possible to discern whether a material possesses ductile (ionic) or brittle (covalent) characteristics. A B/G value exceeding 1.75 indicates brittleness, while values below 1.75 suggest ductility. For pure covalent and ionic materials, Poisson's ratio (n) is typically around 0.1–0.25 and 0.25–0.5 [[54], [55], [56]], respectively. Therefore, CuHgSBr can be considered ductile and strongly ionic character. Young's modulus serves as a gauge for determining a material's rigidity. The investigation also covered elastic anisotropy, contributing to the emergence of microcracks. The shear anisotropy ratio concerning (100) shear planes between the (011) and (010) orientations was calculated using equations (13), (14), (15) [57]. The calculated shear anisotropy factor values (A₁₀₀, A₀₁₀, A₀₀₁) using the GGA-PBE functional are 0.0860 GPa, 0.0813 GPa, and 0.9375 GPa, respectively. The shear anisotropy ratios A₁₀₀ and A₀₀₁ are both less than 1, indicating the anisotropy nature of CuHgSBr structure. Additionally, A₀₁₀ also have values less than 1.

$$A_{100}=\frac{{4C}_{44}}{C_{11}+C_{33}-2C_{13}}$$ (13)

$$A_{010}=\frac{{4C}_{55}}{C_{22}+C_{33}-2C_{23}}$$ (14)

$$A_{001}=\frac{{4C}_{66}}{C_{11}+C_{22}-2C_{12}}$$ (15)

The orthorhombic structure exhibits elastic anisotropy due to both shear anisotropy and the anisotropy of the linear bulk modulus.

3.4. Optical properties

The evaluation of a material's potential for solar cell and optoelectronic applications necessitates an examination of how the optical properties respond to incident photon radiation [58]. The electronic characteristics of a material are encapsulated in a complex dielectric function, represented by the equation ϵ(ω) = ϵ₁(ω) + iϵ₂(ω), wherein ϵ₁(ω) and ϵ₂(ω) stand for the real and imaginary components of the complex dielectric wave function, respectively. Equations (16), (17), (18), (19), (20) [59] are employed to compute additional optical properties, including absorption coefficients (α), refractive index n(ω), extinction coefficient κ(ω), reflectivity R(ω), and energy loss function L(ω) (as illustrated in Fig. 6, Fig. 7):

$$\alpha \left(\omega \right)=\sqrt{2}\omega \sqrt{\sqrt{{ϵ}_1^2\left(\omega \right)+{ϵ}_2^2\left(\omega \right)}-{ϵ}_1\left(\omega \right)}$$ (16)

$$n\left(\omega \right)=\left(\frac{\sqrt{\sqrt{{ϵ}_1^2\left(\omega \right)+{ϵ}_2^2\left(\omega \right)}+{ϵ}_1\left(\omega \right)}}{\sqrt{2}}\right)$$ (17)

$$\kappa \left(\omega \right)=\left(\frac{\sqrt{\sqrt{{ϵ}_1^2\left(\omega \right)+{ϵ}_2^2\left(\omega \right)}{-ϵ}_1\left(\omega \right)}}{\sqrt{2}}\right)$$ (18)

$$L\left(\omega \right)=\frac{{ϵ}_2\left(\omega \right)}{{ϵ}_1^2\left(\omega \right)+{ϵ}_2^2\left(\omega \right)}$$ (19)

$$R\left(\omega \right)=\frac{{\left(n-1\right)}^2+{\kappa }^2}{{\left(n+1\right)}^2+{\kappa }^2}$$ (20)

In crystalline materials, photon absorption is described by the ${ϵ}_2\left(\omega \right)$ component of the dielectric function. Peaks in ϵ₂(ω) are attributed to electronic transitions from the VB to CB. The initiation of absorption in ϵ₂(ω) defines the material's bandgap, situated within the visible spectrum, specifically (<3.1 eV for the CuHgSBr compound). This points to robust interband transitions within the material. As a result, CuHgSBr is an attractive material for solar cell applications. Additionally, compared to wide bandgaps, materials with narrow bandgaps enable faster electron transitions.

Fig. 6.

Estimated a) Absorption coefficient, b) Reflectivity, and c) Energy loss for CuHgSBr.

Fig. 7.

Calculated a) Real part, b) Imaginary component, c) Refractive index, and d) Extinction coefficient.

The CuHgSBr compound displays persistent light absorption across the energy range of 0.8–6 eV, encompassing infrared, visible, and ultraviolet light. This behavior is primarily attributed to the elevated value of the imaginary components ${ϵ}_2\left(\omega \right)$ in its dielectric function within this energy span along the x, y, and z directions, as illustrated in Fig. 7(b). The static value, commonly known as ϵ₁ (Energy = 0), is the primary characteristic of the ϵ₁(ω) curve. In the area of optical and photonic devices, the refractive index n(ω) is a crucial optical parameter to investigate a material's possible uses. The refractive index of the material is correlated to this static value as $n=\sqrt{\left({ϵ}_1\left(0\right)\right).}$ According to Fig. 7(a), the computed values for the static dielectric constant at the zero-frequency limit ϵ₁(0) are 9.84, 7.62, and 6.67 along x, y and z directions, respectively. At Energy = 0, the ϵ₁(ω) graph reaches substantial peak at low energy along x, y, and z regions, which are 0.88 eV, 2.21 eV, and 2.27 eV for CuHgSBr, respectively. Notably, the value exceeding 1 for the y and z directions indicates that the interaction between photons and the material's electrons has caused a reduction in the speed of photons. Photon transmission persists until energy ranges of 5.72–6.38 eV and 8.53–13.56 eV, where the ϵ₁(ω) value turned negative. At this energy range, incident photon radiations are believed to be completely absorbed, leading to metallic behavior in the compounds. Similarly, the real component ${ϵ}_1\left(\omega \right)$ of the dielectric function of CuHgSBr has highest values within the energy range of 0–3 eV, signifying significant electromagnetic wave interaction within this span. The refractive index n(ω) (refer to Fig. 7(c)) achieves its peak values in the same energy range, indicating that electromagnetic waves are largely bent when travelling through the CuHgSBr crystal. The calculated values of static refractive index at the zero-frequency limit n(0) for CuHgSBr along x, y, and z directions are 3.14, 2.76 and 2.58, respectively.

The absorption coefficient serves as a significant optical indicator, providing valuable insights into the absorption of optical energy per unit length. The absorption process is influenced by the crystal's molecular structure and its band gap. Optical absorption takes place when the frequency of an incoming photon resonates with the transition frequency of an atom. This phenomenon is limited to specific materials due to the frequency-dependent nature of the absorption coefficient. The mechanism behind optical absorption involves electrons transitioning from occupied states in the upper valence band to available unoccupied states in the lower conduction band. As depicted in Fig. 6(a), the absorption rate α(ω) is minimal within the infrared range for energies below 2 eV, indicating the transparency of the CuHgSBr crystal to such light. However, the absorption rate α(ω) experiences a rapid rise from 0.77 eV upwards, reaching its peak values. Notably, incident light parallel to the y-z plane exhibits greater absorption compared to light parallel to the x-axis. The CuHgSBr compound displays higher absorption of ultraviolet light compared to infrared and visible light when energies surpass 9 eV, with diminished directional dependency. The absorption coefficient and the imaginary component of the dielectric function are closely related. The imaginary part of the dielectric function is responsible for describing the absorption of light or other electromagnetic waves in a material. It is represented by a complex number, where the real part accounts for the material's refractive index, and the imaginary part represents the absorption coefficient. In general, the absorption coefficient is directly proportional to the imaginary component of the dielectric function. If there is inconsistency between the absorption coefficient and the imaginary dielectric function, it would imply that the material's absorption behavior is not accurately represented by the dielectric function. The estimated absorption in Fig. 6(a) is not consistent with the imaginary components in the x plane, but it is consistent with y and z plane as shown in Fig. 7(b).

The interaction of fast-moving electrons as they enter substance is governed by two factors: the material's surface behavior, which is managed by reflectivity, and the extent of energy loss, regulated by the energy loss function. Notably, the key reflectivity peaks are consistently located at 13.56 eV in all three spatial directions: x, y, and z are illustrated in Fig. 6(b). According to the loss spectrum in Fig. 6(c), there was no discernible absorption in the visible wavelengths. The largest absorption peak is found in higher energy regions (>10 eV). The decline of electromagnetic wave intensity within the material is explained by the extinction coefficient (κ), as illustrated in Fig. 7(d). The maximum absorption occurs at 5.89 eV, 6.19 eV, and 8.28 eV along x, y, z directions for CuHgSBr, respectively. Based on these optical features, CuHgSBr is a strong contender for photovoltaic and solar cell materials.

3.5. Transport and thermodynamic properties

One direct approach to understanding a material's electronic properties is by employing the concept of effective mass. This measure is determined by employing a parabolic band approximation, where a polynomial fit of the relationship between wave vector (K) and energy (E) is applied to either the CBM or VBM. Maximum VBM, as depicted in Fig. 8. However, to calculate effective masses precisely, more computationally demanding approaches are needed. The effective mass can be obtained using equation (21), and Table 5 provides the elucidated effective masses expressed with respect to free electron mass.

$$\frac{1}{m_{eff}}=\frac{1}{{\mathrm{ħ}}^2}\frac{{\partial }^2\mathrm{E}}{\partial {\mathrm{k}}^2}$$ (21)

Furthermore, the breakdown of excitons into electrons (n-type) and holes (p-type) can also be predicted using effective masses and dielectric constants. The binding energy of excitons can be estimated using equation (22) [60], and the values for CuHgSBr are 12.1 meV, 20.2 meV, and 26.4 meV along x, y, and z directions of static dielectric constant. Similarly, the exciton Bohr radius can be estimated using equation (24) [61,62], and the calculated are 60.25 Å, 46.65 Å, and 40.84 Å along x, y, and z directions of static dielectric constant. It is observed that CuHgSBr has a low exciton binding energy in all the three planes, and the value of $a^{*}$ is greater than the lattice parameters in all the three planes, indicating that it is a weak exciton and falls under Mott-Wannier type.

$$E_b=13.6\frac{m_{\mu }}{m_0}\frac{1}{{ϵ}_1{\left(0\right)}^2}$$ (22)

where, m_μ is reduced effective mass and it is consider using equation (23) [60],

$$\frac{1}{m_{\mu }}=\frac{1}{m_e^{eff}}+\frac{1}{m_h^{eff}}$$ (23)

$$a^{*}={ϵ}_1\left(0\right)\frac{m_0}{m_{\mu }}{a}_0$$ (24)

Fig. 8.

Calculated quadric curve of VBM and CBM for CuHgSBr.

Table 5.

Calculated effective mass of electron and effective mass of hole for CuHgSBr.

Compound	Electron	Hole
CuHgSBr	0.1210 $m_e$	0.3025 $m_e$

The investigation into the electronic and optical attributes proposes that CuHgSBr holds promise for photovoltaic applications. Due to the global energy crisis, significant research has been dedicated to advanced materials, including smart materials like thermoelectric materials (TE). These possess the capability to convert waste heat into electrical energy and vice versa. A material's thermoelectric performance is heavily influenced by its electronic structure at the VBM. In this study, the transport characteristics of CuHgSBr were simulated using BoltzTraP, a code baser on semi-classical Boltzmann theory. The efficiency of a material in this context is gauged by its elevated electrical conductivity and Seebeck coefficient as well as reduced electronic thermal conductivity.

This study focused on investigating thermoelectric parameters across the temperature range of 50 K–800 K. Various parameters such as electrical conductivity (σ/τ), electronic thermal conductivity (κ_e/τ), lattice thermal conductivity (κ_L), Seebeck coefficient (S), Power factor (P.F), and figure of merit (ZT) for CuHgSBr were calculated. The computed electrical conductivity (σ/τ) is depicted in Fig. 9(a), illustrating a rise in σ/τ values as temperature increases. As materials display enhanced semiconducting properties, their electrical conductivity also increases, enabling electrons to surmount the fermi level barrier. For CuHgSBr, the σ/τ values are 4.24 × 10¹⁶ (1/Ω cm s) at 50 K and 5.73 × 10¹⁸ (1/Ω cm s) at 800 K.

Fig. 9.

Calculated a) electrical conductivity, b) Seebeck coefficient, c) and d) electronic and lattice thermal conductivities.

The thermopower (S) of CuHgSBr, depicted in Fig. 9(b), displays a consistently positive trend across the entire temperature range, indicating a p-type characteristic of the compound. The computed values of S at 50 K and 800 K are 317 (μV/K) and 198 (μV/K), respectively. The higher Seebeck coefficient (S) at 50 K results from the presence of flat bands at the VB. As temperature increases, S significantly decreases, as depicted in Fig. 9(b). The electronic component of thermal conductivity (κ_e/τ) is depicted in Fig. 9(c), demonstrating a gradual increase with rising temperature. The κ_e/τ values at 50 K and 800 K are 2.48 × 10¹¹ (W/mKs) and 3.20 × 10¹⁴ (W/mKs), respectively.

Power factor (P.F), which combines electrical conductivity and Seebeck coefficient, serves as an efficiency descriptor for thermoelectric materials. Fig. 10(a) depicts the fluctuation of power factor with temperature. The lattice thermal conductivity (κ_L) was estimated using the Slack [63] equation (25), and the values are presented in Fig. 9(d).

$${\kappa }_l=A\frac{M_a{\theta }^3\delta }{{\gamma }^{2}T{n}^{2/3}}$$ (25)

where, A, θ, and γ, are represented as constant, Debye temperature, and Grüneisen parameter. The constant A can be calculated using equation (26).

$$A=\frac{2.43\times {10}^{-8}}{1-\frac{0.514}{\gamma }+\frac{0.228}{{\gamma }^2}}$$ (26)

Fig. 10.

Calculated a) Power factor and b) Figure of merit (ZT).

The value of θ and γ are 183.84 K and 2.35 calculated using IR-Elast code, which is incorporated in WIEN2k package. The tendency of κ_L shows that as the temperature rises, heat transfer of phonon decreases. The κ_L values at 50 K and 800 K are 1.77 (W/mK) and 0.11 (W/mK), respectively, while the computed lattice thermal conductivity at 300 K is 0.29 (W/mK) as shown in Fig. 9(d).

Figure of merit (ZT), a widely-used transport parameter, assesses material effectiveness. As shown in Fig. 10(b), a higher ZT indicates greater suitability for thermoelectric applications, necessitating elevated electrical conductivity, Seebeck coefficient, and diminished thermal conductivity. In this study, the calculated figure of merit (ZT) for CuHgSBr at 50 K and 800 K are 0.0012 and 0.54, respectively, indicating an increase with temperature. This behavior is attributed to the flat band structure at the upper VB.

In conclusion, the band gap significantly influences a material's electrical conductivity, with CuHgSBr's properties affected by its unique band gap chemistry. With an increasing temperature, the figure of merit (ZT) also increases, making CuHgSBr a suitable candidate for thermoelectric applications. Furthermore, the temperature-dependent thermodynamic characteristics such as free energy F, entropy S, and heat capacity C_v were estimated using the harmonic approximation in the phonopy [64], interfaced using WIEN2k, as shown in Fig. 11. As temperature climbs, free energy decreases, while entropy and heat capacity increase in accordance with the third law of thermodynamics. CuHgSBr's heat capacity adheres to the T³ equation of the Debye model [65] at low temperatures, rapidly increasing up to 300 K. at higher temperatures, fluctuation diminishes, consistent with the Dulong-Petit rule [66].

Fig. 11.

Estimated thermodynamic properties of CuHgSBr.

4. Conclusion

To sum up, we conducted a comprehensive analysis of various properties including structural, electronic, elastic, optical, thermodynamic, and transport properties of the CuHgSBr quaternary compound using the Density Functional Theory (DFT) within the WIEN2k software. The optimized lattice constants were determined as a = 10.137, b = 18.517, c = 4.042 Å for CuHgSBr. Additionally, the bandgap was calculated to be 0.76 eV. Based on the results of the elastic constants, CuHgSBr was determined to be ductile (ionic), and mechanically stable at relatively low pressure, indicating its potential for use in resilient materials. Moreover, our studied compound shows promise for photovoltaic applications due to its predicted bandgap, high refractive index, and high absorption coefficient, particularly in the Ultra-violet to visible regions of the electromagnetic spectrum. Furthermore, we employed the semi-classical BoltzTrap code to compute the thermoelectric features of CuHgSBr. By fitting the band structure as VBM and CBM, we obtained the effective mass and exciton binding energy. Our analysis reveals the presence of a Mott-Wannier type exciton based on the exciton Bohr radius. Remarkably, CuHgSBr exhibits strong potential as a thermoelectric material, supported by power factor and ZT outcomes. These results are promising, driving us to delve deeper into the intricate and captivating domain of chalcogenide crystals in this field. In the context of future directions, we anticipate exploring methods to reduce thermal conductivities to further enhance the thermoelectric performance of CuHgSBr. The investigation of CuHgSBr as two-dimensional (2D) and one-dimensional (1D) materials could offer substantial improvements in this aspect. Advancements in DFT computations, such as employing potentials like the Becke-Johnson or hybrid potentials, hold promise for upcoming research endeavours. Ultimately, we foresee an ongoing focus on optimizing CuHgSBr's properties and envision its potential applications in various dimensions.

Author contribution statement

M. Hariharan: Analysed and interpreted the data; performed the experiments; wrote the paper.

R. D. Eithiraj: Conceived and designed the experiments; contributed regents, material, analysis tools or data; wrote the paper.

Date availability statement

All data that support the findings of this study are available from the corresponding author upon reasonable request.

Additional information

No additional information is available for this paper.

Declaration of competing interest

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