Introduction

To get an essential growth in the energy production, new technological devices are required specifically, in the field of photovoltaics and thermoelectric technology [1,2,3]. The most common materials studied for this purpose include cadmium telluride, gallium arsenide, copper indium diselenide, nitrides, oxides and organic materials. Such materials were studied practically for the solar energy conversion [4,5,6,7]. Several high-figure-of-merit (ZT) thermoelectric materials were reviewed by Wei et al. [8]; many copper chalcogenide materials possessed attractive performance such Cu2−2xAg2xSe1−xSx; a ZT value of 1.6 was obtained in the Cu1.8Ag0.2Se0.9S0.1 sample at 900 K [9]. The importance of the thermoelectric materials with low thermal conductivity is their use in water evaporation systems which are energy effective, with efficiencies ranging from 60 to over 90% [10]. There is a variety of solar water evaporation systems based on biomass and other natural products such wood and plants which have been investigated in different forms; their original shape allows them to be used as photothermal surfaces for the leaves [10]. Perovskite solar cells have also emerged as a promising and highly efficient solar technology. A recent study by Doolin et al. [11] presents a methodology for green solvent selection; it shows demonstrate the application of green chemistry principles to solvent selection for perovskite photovoltaic manufacturing. Building integrated photovoltaics and smart windows have gained considerable research attention, due to their promising prospects in terms of energy production from renewables. The use of cobalt redox electrolytes in partly covered photoelectrochromic devices was investigated experimentally by Dokouzis et al. [12] The result devices exhibited rather good transparency in the bleach. Furthermore, sustainable solar energy harvesting technologies able to provide enhanced performance under low irradiation are highly suited as complementary renewable sources to Si-based photovoltaics from a smart-grid energy perspective. Among them, dye-sensitized solar cells (DSSCs) are one of the most interesting choices [13].

Previously, some binary-type chalcogenides were of interest for the thermoelectric applications, such as Bi2Te3 lead telluride, inorganic clathrates, oxides and perovskite solar cells [14,15,16,17,18,19]. Synthesis of such ternary and quaternary chalcogenide materials with an increasingly complex composition becomes a principal direction in modern science of materials [20]. These ternary chalcogenides are mostly semiconductor materials with energy band gaps ranging from 0.6 to 4.0 eV [21]. The basic applications of these materials include light-emitting diodes (LED), infrared materials, the fabrication of solar energy converters, nonlinear optical tools and field-effect transistors [22,23,24]. Nonlinear optical devices provide a means of extending the frequency range of available laser sources. LiInSe2 and LiIn(S0.5Se0.5)2 crystals among the most effective crystals for the creation of middle IR optical parametric oscillators and difference frequency generators under pumping by all-solid-state and common near-IR lasers [25]. Numerous new effective ternary sulfides and binary selenides highly nonlinear optical (NLO) have been compiled and classified for optoelectronic device applications by Atuchin et al. [26] as HgGa2S4 which investigated experimentally with X-ray photoelectron spectroscopy [27]. The silver chalcogenides are promising superionic semiconductors with low heat conductivity of the lattice as the Ag2Se–Ho2Se3 compound [28]. The high potential of LiGaTe2 for the optical frequency conversion in the mid-IR spectral range has been demonstrated in several studies [29]. These results indicate that LiGaTe2, besides its well-known pronounced linear and nonlinear optical properties in IR spectral range, possesses the specific structural effects. A recent investigation on the electronic, optical and thermoelectric properties of nine ternary chalcogenides, NaAX2, where A = As, Sb, Bi and X = S, Se, Te was carried out using DFT calculations. Their band gaps ranged from 0.51 to 1.95 eV. The estimation of thermoelectric parameters showed high Seebeck coefficients of 500 and 300 μV/K for NaSbS2 and NaSbSe2 at 600 K, respectively. These results suggest that these two compounds are potential candidates for photovoltaic structures, especially in tandem solar cells, and thermoelectric applications [30].

Among the family of the copper-based, ternary and quaternary semiconductor compounds are the most widely used as absorbers materials in thin film solar cells, due to their unique structural and photoelectric properties [31, 32]. Moreover, Cu2ZnSn(S,Se)4 (CZTSSe) is also an interesting absorber material for thin film solar cells, which could replace the current absorber layers like Cu(In, Ga)(S,Se)2 (SIGSSe) and CdTe. In this context, a detailed study of band alignment parameters at the interface of n-type MZO and p-type CZTSSe system was investigated by Sengar et al. [33], a Cd-free n-type buffer layers with two different Mg-doped ZnO layers (Mg0.26Zn0.74O, Mg0.30Zn0.70O) have been examined using ultraviolet photoelectron spectroscopy. In addition, a very recent study was carried about improving the Cu2ZnSn(S,Se)4-based photovoltaic conversion efficiency by back-contact modification having a low toxicity, natural abundance, outstanding light absorption, and higher theoretical efficiency [33, 34].

The ternary compounds such as CuInSe2, CuAlSe2 and CuGaSe2 [35,36,37,38] have also attracted the attention of researchers. The scarcity of both indium (In) and gallium (Ga) increases their cost, and, thus, using these elements as absorbers materials increases the cost of solar cells and limits their wide application in copper indium gallium selenide solar cell (CIGSS) [39]. Instead of In and Ga, which are group IIIB members, the group IIIA elements like Y and Sc can be employed. The chalcogenides containing Y and Sc have a unique electronic structure, a high quantum absorption at shorter wavelengths and subsequently emit light at longer wavelengths [40,41,42]. The structure and electronic properties of CuScS2 semiconductor were reported by Scanlon et al. [43]using the hybrid density functional theory. The optoelectronic and elastic properties of CuYS2 were studied by Brik et al. [44], using the first-principles methods. They found band gaps of 1.34 and 1.38 eV by the potential generalized gradient approximation (GGA) and local-density approximation (LDA), respectively. The synthesis and single-crystal structure determination of LaCuSe2 were reported by Julien-Pouzol [45].

A promising series of ternary chalcogenide materials is ACuSe2, where A = Y, Sc, La. Among these materials, yttrium copper diselenide (YCuSe2) is a candidate for solar cell applications. Its lifetime is smaller than that in materials made with silicon. It is a cheap compound with low toxicity [46]. Regarding the compounds interest, Ruixin et al. [46] prepared YCuSe2 using self-propagating high-temperature synthesis method at 1016.2 °C. They found that it was a promising photovoltaic material with an experimental optical gap of 1.53 eV. Similarly, Rugut et al. [47] analyzed the thermoelectric and transport proprieties of YCuSe2. The study showed that it has potential applications as thermoelectric devices. Although a few groups of researchers have explored thermoelectric properties of YCuSe2, there exists scarce information concerning ScCuSe2 and LaCuSe2 materials. Trigonal phase of ScCuSe2 was briefly explored experimentally by Pouzol, Guiltard et al. [48] reporting the structural stability, lattice constants and density.

To the best of our knowledge, rare theoretical or experimental investigations exist about the electronic structures, optical and thermoelectric proprieties of ScCuSe2, YCuSe2 and LaCuSe2 (Fig. 1). In this work, we focus on the electronic band structure, density of states, optical and thermoelectric properties of these three compounds using the density functional theory (DFT) methods. We calculated the electronic band structure, density of states, optical and thermoelectric properties of these three compounds using the density functional theory (DFT). The manuscript is organized as follows. The details of our calculations are reported in "Computational details" section. In "Result and discussion" section, the structural, electronic, optical as well as the thermoelectric properties of examined materials are discussed. In "Conclusion" section, we summarize the conclusions of the study and perspectives.

Figure 1
figure 1

Calculated unit cell structures for a ScCuSe2, b LaCuSe2, c YCuSe2, respectively

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Computational details

We used here the density functional theory (DFT), which is one of the most accurate theories to calculate the electronic structure of solid materials. This technique is increasingly used as an exploratory tool for material discovery and in computational experiments [49]. DFT simulation codes can calculate a vast range of structural, chemical, optical, spectroscopic, elastic, vibrational, and thermodynamic phenomena [50,51,52,53]. The ability to predict structure–property relationships has revolutionized experimental fields, such as vibrational and solid-state NMR spectroscopy. In semiconductor physics, great progress has been made in the electronic structure of bulk and defect states despite the severe challenges presented by the description of excited states. DFT is increasingly used as an exploratory tool for materials discovery and computational experiments, culminating in ex nihilo crystal structure prediction [54, 55].

All the calculations achieved in this work were performed using the full potential linearized augmented plane wave (FP-LAPW) in addition to local orbitals method as developed in the WIEN2k code [56]. The LDA along with PBE-GGA [57] approximations is used to described the exchange correlation potentials that calculate the total energy and are actually based on optimization of the exchange correlation energy. Furthermore, to obtain accurate electronic band structures, we used the recently modified Becke–Johnson potential (m-BJ) [58]. The space was divided into two parts: the spherical muffin-tin region around the nucleus where the radial solutions of the Schrödinger wave equation and its energy derivative were used as basic functions, and the interstitial muffin-tin region, where the base set consisted of plane waves. We selected RMT reduction by 5% so lattice parameters could optimize finely. The atomic sphere radii were selected to be 2.5 a.u., 2.34 a.u. and 2.23 a.u. for A = (Sc, Y, La), Cu, and Se, respectively, for each element of the compounds, i.e., ScCuSe2, YCuSe2 and LaCuSe2. The Brillouin zone integration is performed using a mesh of 500 k-points with RMT Kmax = 8 (a.u)−1, where RMT indicates the smallest muffin-tin radius and Kmax is the maximum size of reciprocal lattice vector, in addition to the Gaussian parameter Gmax = 12. We further fixed the Gaussian smearing value of 0.1 eV to get converged and much accurate electronic density of states, in addition to the convergence energy of 0.0001 Ry to get stability of the system within self-consistent calculations.

The optical properties of a material are described using the real ε1(ω) and imaginary ε2(ω) parts of the complex dielectric function given by the expression:

$$\varepsilon \left( \omega \right) = \varepsilon_{1} \left( \omega \right) + i\varepsilon_{2} \left( \omega \right)$$
(1)

where ε1(ω) corresponds to dispersion of photons and ε2(ω) is associated with the energy absorption of the material. The imaginary part of the complex dielectric function can be calculated using the following relation:

$$\varepsilon_{2} \left( {\omega )} \right. = \frac{{2\pi e^{2} }}{{\Omega \varepsilon_{o} }}\mathop \sum \limits_{c,v} \mathop \sum \limits_{k} \left| {\psi_{k}^{c} |u r|\psi_{k}^{v} } \right|^{2} \delta \left( {E_{k}^{c} - E_{k}^{v} - h\omega } \right)$$
(2)

where Ω is unit cell volume, the valence and conduction band are denoted by (v, c), respectively, the ω term is the incident light frequency, and k defines the direction of polarization of the E field with a full isotropic type average value in the polycrystalline materials. Based on the complex dielectric function, we calculated the significant optical parameters such as the reflectivity R(ω), absorption coefficient I(ω), refractive indices n(ω), energy loss spectrum L(ω), extinction coefficient k(ω) and real optical conductivity σreal(ω) by the given relations [59].

$$L\left( \omega \right) = \frac{{\varepsilon_{2} \left( \omega \right)}}{{\varepsilon_{1}^{2} \left( \omega \right) + \varepsilon_{2}^{2} \left( \omega \right)}}$$
(3)
$$I\left( \omega \right) = \frac{\sqrt 2 \omega }{c}\left( {\sqrt {\varepsilon_{1}^{2} \left( \omega \right) + \varepsilon_{2}^{2} \left( \omega \right)} - \varepsilon_{1} \left( \omega \right)} \right)^{\frac{1}{2}}$$
(4)
$$R\left( \omega \right) = \left| {\frac{{\sqrt {\varepsilon_{1} \left( \omega \right) + i\varepsilon_{2} \left( \omega \right)} - 1}}{{\sqrt {\varepsilon_{1} \left( \omega \right) + i\varepsilon_{2} \left( \omega \right)} + 1}}} \right|^{2}$$
(5)
$$n\left( \omega \right) = \left( {\frac{1}{2}\left[ {\sqrt {\varepsilon_{1}^{2} \left( \omega \right) + \varepsilon_{2}^{2} \left( \omega \right)} + \varepsilon_{1} \left( \omega \right)} \right]} \right)^{\frac{1}{2}}$$
(6)
$$k\left( \omega \right) = \left( {\frac{1}{2}\left[ {\sqrt {\varepsilon_{1}^{2} \left( \omega \right) + \varepsilon_{2}^{2} \left( \omega \right)} - \varepsilon_{1} \left( \omega \right)} \right]} \right)^{\frac{1}{2}}$$
(7)
$$\sigma ^{{{\text{real}}}} \left( \omega \right) = ~\varepsilon _{0} \left( \omega \right)\varepsilon _{1} \left( \omega \right)$$
(8)

We have calculated also the thermoelectric proprieties, including thermal conductivity, Seebeck coefficient, specific heat capacity, power factor, electrical susceptibility and figure of merit, with the use the BoltzTrap code [60].

Result and discussion

Structural proprieties

The unit cell structure of our studied copper-based ternary chalcogenides (ACuSe2) is presented in Fig. 1. We show in Table 1 the computed values of lattice constants and volumes alongside the available theoretical and experimental values where possible. For the case of ScCuSe2, its trigonal form has been explored experimentally by Pouzol and Guiltard where in both structural stability, lattice constants and densities are reported in [48]. While investigating the interactions between the components in the Y2Se3-Cu2Se-SnSe and Y2Se3-Cu2Se-PbSe system at 870 K determined using X-ray powder diffraction, Shemet et al. [61] confirmed the existence of the trigonal YCuSe2 in the Cu2ErS2 structure having lattice constants as provided in Table 1. For LaCuSe2, our calculated structural parameters agree reasonably with experimentally reported in [62]. Hence, all the calculated structural properties agree reasonably well with previous results and experimental data. Furthermore, employing the (PBE-GGA) generalized gradient approximation we optimized the atomic positions using the minimized forces which act on atoms. The atomic coordinates and isotropic temperature factors are given in Table 2.

Table 1 The main parameters of processing and refinement
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Table 2 Atomic coordinates and isotropic temperature factors for the ACuSe2 compounds
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Band structure and density of states

The study of material band structure is very essential as it gives information about optoelectronic behavior at different frequency excitations, i.e., electron and hole transitions, and permits to examine the electronic properties. In addition, for technological applications of semiconductors, the nature of band structure is a key parameter, and it can be measured experimentally or determined theoretically. We calculated the electronic band structures (EBS) of the three materials along the high-symmetry directions, as displayed in Fig. 2. For the ScCuSe2 and YCuSe2 compounds, the valence band maximum and conduction band minimum are located at different points (Γ and M) of the first Brillion zone (FZB) resulting in an indirect band gaps (Γ-M) of 0.7 and 1.2 eV, respectively, while LaCuSe2 exhibits a direct band transition with a band gap value (Γv − Γc) of 0.8 eV. So, the indirect band gap increases from A = Sc (0.7 eV) to Y (1.2 eV) and then decreases to direct band gap for La (0.8 eV). The maximum band gap is observed for A = Y (1.2 eV). The decrease of the band-gap value from LaCuSe2 or ScCuSe2 to YCuSe2 can be accredited due to the shift related to conduction bands toward Fermi level that most likely reflects the significance of La and Sc in band gap opening if the corresponding valence band is allowed to be unchanged.

Figure 2
figure 2

Calculated band structures for a ScCuSe2, b YCuSe2, c LaCuSe2, respectively

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The band gaps for the examined materials were calculated using the modified Becke–Johnson exchange potential (mBJ). The obtained values of the fundamental gap of ScCuSe2 using PBE and hybrid exchange functional (HSE06) are 0.65 and 1.62 eV, respectively [41]. It can be observed that the theoretical DFT-based calculation of PBE and mBJ characteristically underestimate corresponding band-gap than HSE06. Hence, according to this calculation and previously reported values of ScCuSe2 band gaps, the most optimal value can range from 0.65 to 0.7 eV. The band gap of YCuSe2 calculated in the present study is comparable to the experimental optical gap that was estimated to be 1.53 eV using self-propagating high-temperature synthesis method [46].

Other DFT calculation confirms that YCuS2 has indirect band gap with a value of 1.3 eV for theoretical bulk modulus of 70 GPa, as computed using the PBE functional [44]. The PBE-DFT gap was computed to be 1.08 eV [47]. The presence of the fundamental band gap energy between valence and conduction bands indicates the semiconductor behavior for the three examined materials.

So, as to explain further the contributions related to the various electronic states in both valence and conduction bands, we also computed the density of states. In Fig. 3a, b and c, the total density of states (TDOS) and atomic-projected density of states (PDOS, in states/eV) are presented for the studied compounds. As seen, TDOS for the three materials is nearly similar due to the similarity in chemical bonding. In the case of valence band, TDOS of ScCuSe2 is divided into two parts: the first one situated in the region from − 14.1 to − 12.2 eV and the large region ranges from − 6.2 to 0.01 eV; in this region, we see the dominance of Tot-Cu while DOS of other elements is negligible. As seen, for this material, the principal contribution is through the DOS components Se-p (2 states/eV), Cu-d and Sc-p which gives a strong support to the partial density of states, while DOS of other elements is negligible. In the conduction band, DOS shows a large contribution range from 0.6 to 15 eV which is supported, especially by Sc-d, Se-p and Cu-p. For YCuSe2, there is the large contribution in the case of the valence band ranging from − 5.7 to 0 eV, and it is due to the Se-p and Cu-d states, while the conduction band consists mostly of Y-p, Cu-p and Se-p states with a small contribution of all other elements. For the last material LaCuSe2, in the valence band, DOS, divided into three parts, consisting of La-p, Se-s and Se-p states, while in the case of conduction band, which lies between 2.2 and 15 eV, a major contribution is from La-d with 1.5 states/eV.

Figure 3
figure 3figure 3

Calculated total and partial density of states for a ScCuSe2, b YCuSe2, and c LaCuSe2

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Optical proprieties

In this section, the optical parameters are calculated in the energy range of 0–14 eV. We calculated the dielectric functions ε1 and ε2 of the investigated compounds as a function of the photon energy, as shown in Fig. 4a, b, respectively. As seen in Fig. 4a, the real part of dielectric function ε1(ω) increases from ε1(0) to reach a maximum value at 1.3, 1.9 and 1.8 eV for ScCuSe2, YCuSe2 and LaCuSe2, respectively, and, then, it begins decreasing to negative values that show the reflection of light falling on the material surface and materials become metallic [63]. The zero limits of frequency for ScCuSe2, YCuSe2 and LaCuSe2 are present at 6.6, 6.7 and 5.4 eV, respectively.

Figure 4
figure 4

Calculated a real component, b imaginary component of dielectric function, c absorption coefficient I(ω), d energy loss function L(ω), for ACuSe2, respectively

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As we can see that the three curves remain constant from ε1(− 1.4) at 8.9 eV. The peaks observed mainly in the three materials are due to the transitions from the valence band top to the conduction band bottom. It is deduced that these values of the dielectric constants for the three materials have an inverse relationship with their corresponding band gaps due to the negative energy ranges.

In Fig. 4b, the imaginary part of dielectric function, ε2(ω), is shown. The principal peaks of the optical critical point are 13, 11.8 and 12 for ScCuSe2, YCuSe2 and LaCuSe2, respectively, which correspond to their energy values of 1.8, 2.3 and 3 eV. From the imaginary part of dielectric constant, we calculated the frequency-dependent absorption coefficient I(ω), as shown in Fig. 4c. For all the three materials, the absorption spectra start increasing nearly at 1 eV to reach some maxima at 9.1, 7.7 and 7.2 eV for ScCuSe2, YCuSe2 and LaCuSe2, respectively, and, then, it continues increasing hardly in the range from 9.4 to 13.5 eV.

One of the most important optical parameters is the electron energy loss function L(ω), because it permits to measure the propagation loss of energy inside the solid materials. As shown in Fig. 4d, L(ω) starts increasing from 1 eV for the three materials to get the maximum peaks at 12.4, 11.8 and 11.7 eV for ScCuSe2, YCuSe2 and LaCuSe2, respectively. These peaks in the L(ω) spectra are called to be plasma resonance, and they are due to the lost energy when the incident photon energy is higher than Eg of material. After that, the energy loss function declines rapidly for the three materials.

In Fig. 5a, the calculated reflectivity spectra R(ω) are displayed for the three materials. The zero frequency limits are about 0.31, 0.25 and 0.29 for ScCuSe2, YCuSe2 and LaCuSe2, respectively. The maximum of peaks for the three materials is situated in the energy range of 7.2–9.4 eV. The sharp peaks of ScCuSe2 and YCuSe2 show that these materials can be used as shielding materials against UV radiations. In Fig. 5a, a sharp increase is seen from 12 to 13.5 eV for the three materials. In Fig. 5b, the real part of the optical conductivity is plotted. It starts increasing at 0.9 eV for the three materials and reach high peak values at 6.6, 5.5 and 5.3 eV in ScCuSe2, YCuSe2 and LaCuSe2, respectively. A sharp decrease of the spectra in the range of 10 to 13.5 eV can be also seen. The refractive index n(ω) is plotted in Fig. 5c. It starts increasing from the values of 3.5, 3.1 and 3.3 to get the maximum value at the energy values of 1.3, 2 and 2.1 eV for ScCuSe2, YCuSe2 and LaCuSe2, respectively. Then, it decreases to get a value below 1.0 for the energy range of 8.0–13.5 eV. That is due to the group velocity of the incident radiation (vg = c/n), which was greater than the speed of light, c (3.108 m/s), for all the three materials. In addition to the mentioned parameters, we calculated the extinction coefficient k(ω), as shown in Fig. 5d. We observe a sharp increasing for the three spectra from 1 eV to reach the maximum value of k (0), begins at 1.2 eV. Then, it starts increasing when the energy increases until reached the maximum value of 2.2 (3.8 eV), 2.0 (5.5 eV) and 2.2 (5.5 eV) for ScCuSe2, YCuSe2 and LaCuSe2, respectively. After that, the spectra decrease hardly.

Figure 5
figure 5

Calculated a reflectivity R(ω), b optical conductivity σ(ω), c refractive index n(ω) and d extinction coefficient k(ω) for ACuSe2

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Thermoelectric properties

The electron transport properties are based on the band structure, and they were calculated with the use of semiclassical transfer theory of Boltzmann and solid rigid band theory, as applied in the BoltzTrap code. Thermoelectric compounds are of great importance for renewable energy device applications because heat energy is lost in many energy generation and consumption devices. The best thermoelectric material has a small electrical to thermal conductivity ratio in order to enhance the efficiency of the devices. Indeed, devices based on these materials exhibit a variety of advantages, including low level of noise, high reliability, lack of moving parts and long service life. To understand the thermoelectric behavior for the examined materials, we have calculated their thermoelectric proprieties in the temperature range of 0–800 K. In Fig. 6, we show Seebeck coefficient S, figure of merit ZT, specific heat capacity Cv, power factor S2σ/τ, thermal conductivity κe/τ and susceptibility χ as a function of temperature. The temperature-dependent Seebeck coefficient is shown in Fig. 6a. For YCuSe2, it increases with temperature to reach the maximum value of 1.38 × 10–4 V/K at room temperature, and then it starts decreasing to attain the minimum value of 0.43 × 10–4 V/K at 800 K. For ScCuSe2, the Seebeck coefficient increases hardly to get the maximum value of 0.6 × 10–4 V/K at 800 K. In LaCuSe2, it increases to reach the maximum at 0.84 × 10–3 V/K at 400 K and then starts decreasing slowly until 800 K. The maximum of the Seebeck coefficient moves to high temperatures on the change of Y to La and Sc, respectively.

Figure 6
figure 6

Calculated thermoelectric proprieties as a function of temperature: a the Seebeck coefficient, b figure of merit, c specific heat capacity, d power factor, e thermal conductivity and f susceptibility

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The efficiency of thermoelectric materials is estimated also by the parameter of the thermoelectric figure of merit ZT = S2σT/κ, where S is the Seebeck coefficient, σ is the electrical conductivity and κ is the thermal conductivity. As clear from Fig. 6b for ScCuSe2, ZT starts increasing with temperature to reach the maximum value of 0.18 at 800 K. In YCuSe2, we observe a sharp increasing from 100 to 200 K where ZT attain a high value of 0.45 at 300 K and then starts decreasing. In LaCuSe2, the maximum value of ZT is at 100 K and then we see a small decreasing with temperature. The maximum of ZT moves to high temperatures on the change of Y to La and Sc. The evolution of ZT with temperature is similar to that of the Seebeck coefficient in agreement with the formula of ZT which is proportional S2.

Moreover, to understand the behavior of the lattice vibrations, we calculated the specific heat capacity Cv, as shown in Fig. 6c. As seen, Cv increases with temperature for the three materials except for the ScCuSe2, which show a hard increasing. Thus, YCuSe2 and LaCuSe2 have greater specific heat capacity values, as compared to ScCuSe2 material. In fact, the heat capacity increases as T3 at low temperatures until 400 K, reaches the inflexion point at 500 K and then moves toward the saturation at 25 J mol−1 K−1 (Dulong and Petit law). To study the materials efficiency, we calculated the power factor (PF). The used formula is PF = S2 σ/τ, and it identifies the capability of a compound to yield electricity. As shown in Fig. 6d for LaCuSe2, the PF is still constant, while ScCuSe2 shows a strong increase with temperature to reach the high value of 2.1 × 1011 W/mk2s at 800 K, and it indicates that it is a valuable material for thermoelectric applications. For YCuSe2, the PF increases with increasing temperature to attain the maximum value of 1.17 × 1011 W/mk2s at 300 K and then it starts decreasing. Thermal conductivity is the ability of a material to conduct heat, and it is determined by phonons and charge carriers. As seen in Fig. 6e, the thermal conductivity increases with rise in temperature. The calculated values of S at 800 K are 9 × 1014, 7 × 1014 and 7.2 × 1014 (W/mKs) for ScCuSe2, YCuSe2 and LaCuSe2, respectively. The susceptibility of the three examined materials is plotted in Fig. 6f, which describes the material magnetic response. In the case of ScCuSe2, it increases with increasing temperature to reach the maximum value of 1.3 × 10–11 m3 mol−1, and then it starts decreasing until 800 K. For YCuSe2, it starts increasing from 0.16 × 10–11 at 100 K to 0.6 × 10–11 m3 mol−1 at 800 K, while for LaCuSe2, it nearly remains constant at the value of 10–11 m3 mol−1. When the temperature increases, the susceptibility increases from an initial value to a maximum and then it decreases which is the classical evolution as in the case of ScCuSe2. This investigation of thermoelectric proprieties shows that the examined materials exhibit large Seebeck coefficients, especially of ScCuSe2 and YCuSe2, and it confirms that they are highly efficient thermoelectric materials.

Conclusion

In the present work, the electronic, optical and thermoelectric properties of rare-earth copper selenides chalcogenides ACuSe2 (A = Sc, Y and La) compounds have been explored using FP-LAPW method implemented in the Wien2k code. The electronic band structures calculated with the accurate mBJ approach show that the studied materials are wide band gap semiconductors. Their obtained fundamental gaps are 0.7, 1.2 and 0.8 eV for ScCuSe2, YCuSe2 and LaCuSe2, respectively. ScCuSe2 and YCuSe2 have indirect band gaps along Γ-M direction, whereas LaCuSe2 displays a direct band type transition at the Γ-point. We computed the total density of states (TDOS) and atomic-projected density of states (PDOS) for the studied compounds. The TDOS for the three materials is nearly similar due to the similarity in chemical bonding. The values of the dielectric constants for the three materials have an inverse relationship with their corresponding band gap because of the negative energy ranges. The computed reflectivity spectra R(ω) for the three materials have the zero frequency limits at 0.31, 0.25 and 0.29 for ScCuSe2, YCuSe2 and LaCuSe2, respectively, with the maximum peaks in energy range of 7.2–9.4 eV. The sharp peaks of ScCuSe2 and YCuSe2 show that these materials can be used as shielding materials against the UV radiations. The dispersive refractive index n(ω) increases from 3.5, 3.1 and 3.3 to the maximum values of 4.4,4 and 4.1 at the short infrared and visible wavelengths in ScCuSe2, YCuSe2 and LaCuSe2, respectively. The increasing values of power factor and the figure of merit ZT with temperature show that the ScCuSe2 and YCuSe2 materials are suitable for thermoelectric applications. Finally, we conclude that ScCuSe2 and YCuSe2 are expected to be potential candidates for thermoelectric device applications. This work sheds light on the optoelectronic and thermoelectric properties of ACuSe2 ternary compounds and could offer valuable guidance for the thermal management of copper selenides chalcogenides-based nanoelectronics devices.