1 Introduction

Ever since the 1970’s global energy crisis, the gap between the energy demand and supply chain has persisted to widen. Due to the global transition to industrialization and urbanization, the need for renewable and clean solar energy has been on the rise [1]. In photovoltaic fabrication, one of the most important components is the light absorbing layer also known as the active layer. The solar cells’ active layer may be derived from either organic, inorganic, or organic–inorganic blended materials with a suitable bandgap for photon absorption [2]. Among the materials used as absorbers, ternary compounds have been extensively investigated owing to their potential applications in the field of photovoltaics [3, 4]. The ternary compounds with the ABC2- type crystal structure are grouped into chalcogenides (AIBIII \({\mathrm{C}}_{2}^{\mathrm{VI}}\)) and pnictides (AIIBIV \({\mathrm{C}}_{2}^{\mathrm{V}}\)) [5]. These materials have narrow bandgaps in the range of the solar spectrum thus they are appropriate for solar cell applications [5].

Irfan et al. [6] studied the structural, electronic, thermophysical, and optical properties of K3Cu3P2 and K3Ni2P2 pnictide materials by employing DFT for potential application in thermophysical and optoelectronic fields. Li et al. [7] investigated the thermoelectric properties as well as electronic structures of ABX2 materials by using first-principles calculation and reported their prospective applications in thermoelectric devices. Iqbal et al. [8] carried out a first-principles study on the electronic and optical properties of AlGaAs2 and AlGaSb2 compounds for possible applications in thermoelectric and optoelectronic. A similar study was carried out on CdSiAs2 chalcopyrite [9]. Mechanical properties of BeSiP2 and BeGeP2 chalcopyrites have also been investigated [5]. In a different study by Fai et al., DFT + U approach was employed to study the vibrational, thermoelectric, and electronic properties of AgInTe2 and AgGaTe2 chalcopyrite [10]. Recently, the physical properties of SrPd2Sb2 chalcopyrite have been studied using the DFT approach [11]. Further DFT studies on the properties of ABC2- type ternary compounds have been reported in several treatise [12,13,14,15,16,17,18,19]. Indeed, employing the computational-based tools could help in gaining insightful information for proposing photo-related applications of the investigated structures [20,21,22]. In fact, in the field of photovoltaic solar cells, the solid-state type of absorber has been dominant ever since first device was unveiled by Chapin et al. [23] in 1954. Recently, computational designs using the density-functional theory have played an important role in the development of the metal halide perovskite from its debut in 2012 to its current high conversion efficiency of 25% in the shortest period in history of photovoltaics [24]. The use of insilico techniques may enable further advancement toward the Shockley–Queisser limit [25, 26] through bandgap engineering, as well as search for new compounds via combination of different elements.

Despite many first-principles investigations existing in the literature on ABC2- type ternary semiconductor compounds, the full potential for applications of these materials has not yet been realized. Therefore, there is a need to do more investigations on the properties of these materials for the full realization of their applications. In this study, we have employed a density-functional theory to analyze the properties of X3Sb2Au3 (X = K, Rb) ternary compounds. The properties that have been emphasized on are the structural, electronic, elastic, and optical properties in order to get insights into their optoelectronic potential. The motivation for this investigation came from the fact that there are few theoretical and experimental studies reported in the literature [27, 28] on the crystal structure of K3Sb2Au3 and Rb3Sb2Au3 ternary compounds. Furthermore, apart from the proposed crystal structure, studies on their properties had not been done previously.

2 Computational methods

In this investigation, a plane wave pseudopotential method within the Kohn–Sham framework DFT as implemented in the QE package [29] was used to perform the theoretical calculations [30, 31]. Generalized gradient approximation with Perdew–Burke–Ernzerhof (PBE) [32] functional was used. The K3Sb2Au3 and Rb3Sb2Au3 crystal structure input files were obtained from the materials project database [28] and the materials cloud input generator implemented in quantum espresso was used to generate the PWscf input file as well as the pseudopotentials [33] for DFT calculations. The lattice parameters of the K3Sb2Au3 crystal used were a = b = c = 15.3919 a.u and α = β = γ = 45.985°, while the lattice parameters of the Rb3Sb2Au3 crystal were a = b = c = 15.665 a.u and α = β = γ = 47.353°. These lattice parameters are of the rhombohedral crystal structure representation. The cutoff energy of 160 Ry and Monkhorst–Pack k-point mesh of 8 × 8 × 8 with offset of 1 were used in this work. These values were arrived at by minimizing the total energy with respect to the cutoff energy and k-points. Geometry optimization was performed by minimizing the total energy with respect to the lattice constants. The optimized lattice constants, cutoff energy and k-points thus obtained were used in doing further calculations of electronic, elastic, mechanical, and optical properties.

3 Results and discussion

3.1 Structural properties

Both the K3Sb2Au3 and Rb3Sb2Au3 compounds crystallize in the trigonal crystal symmetry belonging to the space group orientation R3m (space number 166) [28]. The total energies of the crystal structures were calculated at different values of the lattice constants in the rhombohedral representation as shown in Fig. 1.

Fig. 1
figure 1

Schematic showing the convergence of the total energy with the lattice constants of (a) K3Sb2Au3 and (b) Rb3Sb2Au3 ternary compounds

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The computed total energy and lattice constant values were fitted in the Birch–Murnaghan equation of state [34] to obtain the ground state structural properties as shown in Table 1. The ground state lattice parameters obtained for the K3Sb2Au3 and Rb3Sb2Au3 are consistent with the ones reported in other works [27, 28].

Table 1 Ground state lattice and structural Properties of the rhombohedral lattice of the K3Sb2Au3 and Rb3Sb2Au3 ternary compounds obtained by fitting the total energy and lattice constant values in the Birch–Murnaghan equation of state
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The crystal structures of K3Sb2Au3 and Rb3Sb2Au3 compounds are presented in Fig. 2 below.

Fig. 2
figure 2

Rhombohedral representation of the crystal structures of (a) K3Sb2Au3 and (b) Rb3Sb2Au3 ternary compounds

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The bond lengths and bond angles of the K3Sb2Au3 and Rb3Sb2Au3 crystal structures were obtained as shown in Table 2.

Table 2 The bond lengths and bond angles as calculated for the K3Sb2Au3 and Rb3Sb2Au3 crystal structures
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The calculated bond lengths shown in Table 2 corroborates similar values to the ones reported in the literature for the crystal structures of K3Sb2Au3 and Rb3Sb2Au3 [27, 28] as previously proposed.

3.2 Electronic properties

The optimized K3Sb2Au3 and Rb3Sb2Au3 structures were used in the calculation of electronic properties. The band structure and the projected density of states (PDOS) of the K3Sb2Au3 and Rb3Sb2Au3 compounds are presented in Figs. 3 and 4 below.

Fig. 3
figure 3

The calculated electronic properties (a) Band structure and (b) PDOS of K3Sb2Au3 ternary compound

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Fig. 4
figure 4

The calculated electronic properties (a) Band structure and (b) PDOS of Rb3Sb2Au3 ternary compound

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From the band structure plots of the two compounds, (see Figs. 3(a) and 4(a)); it can be observed that the K3Sb2Au3 and Rb3Sb2Au3 ternary compounds are semiconductors with bandgaps of 1.236 eV and 1.353 eV, respectively. The calculated bandgaps lie within the range of 1–3.5 eV for ternary compounds covering the electromagnetic spectrum in the visible region [5, 35]. The computed bandgaps are slightly below the values reported elsewhere [28], this may be attributed to the known fact that GGA-PBE tends to underestimate bandgap. The valence band maxima and the conduction band minima of both materials are at the same symmetry points in the first Brillouin zone which implies the direct nature of the bandgaps. The Fermi level lies between the valence band maxima and conduction band minima and it governs the possibility of electrons occupying different energy levels. The magnitude of the Fermi level predicts the rate of electronic transitions. The nearer the Fermi level is to the conduction band, the lesser the energy required for the electrons to transition from the valence to the conduction bands. In the considered − 6 to 6 eV energy region, the valence bands were majorly formed by the Au-4d state in the region − 6 to − 3.5 eV with little contribution by the other states. The valence band in the energy regions − 3 eV to the Fermi level is mainly formed by Sb-3p with a small contribution from the other states. The conduction band was formed mainly by Sb-3p states with low contribution from the other states. From the band structure and projected density of states plots, it is noticed that there are dense and close-packed energy bands that are intermediate in the energy region − 6 eV to − 4 eV of the valence band implying interband transitions, thus suggesting the materials may find applications in solar cells [6].

3.3 Elastic properties

Both K3Sb2Au3 and Rb3Sb2Au3 crystal structures belong to the rhombohedral (I) class representation which features six independent elastic constants C11, C12, C13, C14, C33, and C44 [36]. The four necessary and sufficient conditions for elastic stability of the rhombohedral (I) class representation [36] include;

$$ \left\{ {\begin{array}{*{20}c} {{\text{C}}_{11} > \left| {{\text{C}}_{12} } \right|; {\text{ C}}_{44} > 0,\quad \quad \quad \quad \quad } \\ { {\text{C}}_{13}^{2} < \frac{1}{2}{\text{C}}_{33} \left( {{\text{C}}_{11} + {\text{C}}_{12} } \right)\quad \quad \quad \;\;{\kern 1pt} } \\ { {\text{ C}}_{14}^{2} < \frac{1}{2} {\text{C}}_{44} \left( {{\text{C}}_{11} - {\text{C}}_{12} } \right) = {\text{C}}_{44} {\text{C}}_{66} } \\ \end{array} } \right. $$
(1)

The conditions in Eq. (1) are satisfied using the independent elastic constants in Table 3. This implies that both K3Sb2Au3 and Rb3Sb2Au3 are mechanically stable. From these elastic constants, other mechanical properties are extracted and shown in Table 4.

Table 3 Computed elastic Constants Cij (GPa) of K3Sb2Au3 and Rb3Sb2Au3 ternary compounds
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Table 4 Voigt-Reuss-Hill Approximations of bulk modulus B (GPa), Young’s modulus E (GPa), Shear modulus G (GPa), Pugh’s ratio B/G, Poisson’s ratio n, average Debye sound velocities Vm (m/s) and Debye temperatures θD (K) of K3Sb2Au3 and Rb3Sb2Au3 ternary compounds
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The ductile (ionic) and brittle (covalent) nature of materials is determined by Pugh’s ratio B/G and Poisson’s ratio, n [37]. A material is said to be ductile if B/G > 1.75 whereas materials are said to be brittle if B/G < 1.75. Thus, in our study, both K3Sb2Au3 and Rb3Sb2Au3 are ductile. The Poisson’s ratio n = 0.1, 0.25, and 0.33 for pure covalent, ionic, and metallic bonds, respectively [38]. Therefore, we can conclude that the materials studied are strongly dominated by the metallic character. The stiffness of a material is measured by Young’s modulus [39]. A high value for Young’s modulus implies more stiffness while a low value for Young’s modulus implies less stiffness [39]. Debye temperature is a measure linked to a material’s physical properties [40]. It is obtained from the elastic constants [41] as well as the average Debye sound velocities according to the relation [42]:

$$ \theta_{D} = \frac{{\text{h}}}{{{\text{K}}_{{\text{B}}} }}\left( {\frac{3}{{4\pi V_{a} }}} \right)^{\frac{1}{3}} V_{m} $$
(2)

where Va is the volume of a unit cell while h and KB refer to the Planck’s and Boltzmann’s constants, respectively [42].

3.4 Optical properties

To explore the material’s prospects for optoelectronic applications, it is important to investigate its optical properties. The optical properties describe the electromagnetic frequency response of solid-state materials to the incident photon energies [43]. In this work, the frequency-dependent dielectric constants were computed using the Sternheimer equation by neglecting the local fields without summing up the empty bands [44] in the rhombohedral representation. The complex dielectric wave function which describes the electron response in a material is given as [45];

$$ \varepsilon \left( \omega \right) = \varepsilon_{1} \left( \omega \right) + i\varepsilon_{2} \left( \omega \right) $$
(3)

where \({\varepsilon }_{1} \left(\omega \right)\) refers to the real part and \({\varepsilon }_{2} \left(\omega \right)\) imaginary part of the dielectric wave function. From the computed \({\varepsilon }_{1} \left(\omega \right)\) and \({\varepsilon }_{2} \left(\omega \right)\) values, the other optical constants including the refractive index n(ω), absorption coefficients α(ω), energy loss spectrum L(ω), and reflectivity R(ω) were calculated using the equations [17, 46, 47].

A description of optical absorption is represented by the imaginary parts of the dielectric function. The absorption onsets in the \({\varepsilon }_{2} \left(\omega \right)\) curve was found to be 1.23 eV and 1.34 eV for K3Sb2Au3 and Rb3Sb2Au3, respectively [see Fig. 5(a)]. These values are consistent with the ones obtained from the band structure. The refractive index of materials is related to the real part of the dielectric function according to the relation \(=\sqrt{{\varepsilon }_{1}(0)}\). Starting from the zero-frequency limit, \({\varepsilon }_{1} \left(\omega \right)\) curves attain maximum values at 2.1 eV and 2.6 eV for K3Sb2Au3 and Rb3Sb2Au3, respectively. At 4–6 eV energy regions, the \({\varepsilon }_{1} \left(\omega \right)\) curves go below zero. This is the region where the incident photon radiation is said to be fully attenuated [48] and the materials depict a metallic behavior [49]. The number of photons absorbed by a particular medium is measured by the absorption coefficient [17]. Both K3Sb2Au3 and Rb3Sb2Au3 materials show strong absorption in the energy regions 2.8 eV to 9 eV. These regions cover the UV–Vis part of the electromagnetic spectrum thus the materials under study can be used as UV–Vis absorbers. The refractive index plays a key role in the optical and electronic properties of materials [17]. The major refractive index peaks of both K3Sb2Au3 and Rb3Sb2Au3 materials lie within the visible region; < 3.1 eV. The obtained values of refractive indices are 3.41 and 3.11 for K3Sb2Au3 and Rb3Sb2Au3, respectively. These values are in agreement with the ones obtained for compounds with similar crystal structures [6, 50, 51]. The surface behavior of materials is characterized by reflectivity [17]. Small bandgap materials yield large reflectivity values as compared to large bandgap materials [17]. The reflectivity of K3Sb2Au3 and Rb3Sb2Au3 materials is high in the visible region and it decreases at higher energies. An energy loss function L(ω), represents the energy lost by fast electrons getting into a medium [17]. There were no distinct peaks of L(ω) in the lower energy regions. The major peaks appeared in the regions 9.66–10.09 eV for both K3Sb2Au3 and Rb3Sb2Au3 materials.

Fig. 5
figure 5

Computed optical properties of K3Sb2Au3 and Rb3Sb2Au3 ternary compounds showing (a) imaginary dielectric functions, (b) real dielectric functions, (c) refractive indices, (d) absorption coefficients, (e) energy loss spectra, and (f) reflectivity

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4 Conclusions

In this work, we have investigated the structural, electronic, elastic, and optical properties of K3Sb2Au3 and Rb3Sb2Au3 ternary compounds by using density-functional theory. The band structure predicted K3Sb2Au3 and Rb3Sb2Au3 as semiconductor compounds with direct bandgaps of 1.236 eV and 1.353 eV, respectively. The valence bands have been found to be majorly formed by the Au-4d state in the region − 6 to − 3.5 eV with a small contribution from the other states whereas the valence band in the energy regions − 3 eV to the Fermi level was mainly formed by Sb-3p with a small contribution from the other states. The conduction band was formed mainly by Sb-3p states with low contribution from the other states. The calculated elastic properties show that the materials are mechanically stable and exhibit both ductile and near metallic behaviors. High refractive indices of 3.41 and 3.11 for K3Sb2Au3 and Rb3Sb2Au3 were obtained, respectively. The optical onsets of the imaginary curves are consistent with the values obtained from the band structure. The calculated optical absorption spectra show that both materials absorb strongly within the UV–Vis (2–9 eV) regions. In summary, K3Sb2Au3 and Rb3Sb2Au3 ternary compounds were found to be thermodynamically and mechanically stable which makes them easy to synthesize experimentally. Furthermore, direct bandgaps lying between 1 and 1.6 eV are optimal for solar cell applications. High refractive indices and absorption coefficients within the UV–Vis regions make the studied compounds good UV–Vis absorbers hence their suitability for photovoltaic applications.