First-Principle Calculations to Investigate Structural , Electronic , Elastic , Mechanical , and Optical Properties of K 2 CuX ( X = As , Sb ) Ternary Compounds

E cient materials with good optoelectronic properties are required for the good performance of photovoltaic devices. In this work, we present ndings of a theoretical investigation of the structural, electronic, elastic, mechanical, and optical properties of K2CuX (X As, Sb) ternary compounds. e computations were carried out by using the density functional theory (DFT) formalism as implemented in the quantum espresso (QE) software package. e calculated lattice constants of 19.1414 a.u (K2CuAs) and 20.0041 a.u (K2CuSb) are in agreement with the experimental results from the literature.e materials under study were found to have bandgaps of 1.050 eV (K2CuAs) and 1.129 eV (K2CuSb). e valence band was majorly formed by Cu-3d, As2p, and Cu-4s states while the conduction band was majorly dominated by Cu-5p in K2CuAs, whereas in K2CuSb, the valence band was mainly formed by Cu-3d, Cu-4s, and Sb-3p states while the conduction band was majorly formed by Sb-3p and Cu-5p states. e investigated materials were found to be mechanically stable at zero pressure, ductile, and ionic. e optical absorption coe cient curves were found to cover the ultraviolet to visible (UV-Vis) regions, thus making K2CuAs and K2CuSb good UV-Vis absorbers hence their suitability for photovoltaic applications.


Introduction
Photovoltaic technology depends on various materials for photon-to-electron conversion, which mainly includes organic, inorganic, or organic-inorganic blends [1]. e contemporary photovoltaic industry is currently dominated by the inorganic class with high incidence photon-electron conversion e ciency (IPCE) materials comprising mainly silicon, gallium arsenide, and cadmium telluride, among others [1]. e commercial applicability of these inorganicbased materials in the photovoltaic industries is, however, limited by the high costs and level of purity required during production [2]. Most of these materials, when applied in photovoltaics, work in the principle of light absorption, charge separation, and transport of minority carriers, all happening in the same material [3]. In the recent past, perovskite material has gained traction due to the rapid growth in its conversion e ciency from as low as 3% to the current 25% [4] in a short span of time. Despite their rapid growth in their e ciency, these types of materials are dogged by the instability of the photovoltaic systems produced by using such materials, and this has begged to cast the search wider [4][5][6][7]. e search for alternative classes of materials has led to the realization of cheap and abundant ternary semiconductor compounds [8].
ese ternary semiconductor compounds have been found to have excellent electronic as well as optical properties [9]; they also do not require sophisticated technologies when performing thin lm deposition since they work on the principle of majority carrier transport, where light absorption and charge separation take place in the absorber, while charge transport takes place in other layers of the optoelectronic device [10]. Among the desirable properties of ternary semiconductor compounds is the considerable wide bandgap range of 1-3 eV [11], which covers the UV-Vis region of the electromagnetic spectrum.
is property makes these compounds promising candidates for application in the fields of solar cells, lasers, and photodetectors [12]. Among the ternary semiconductor compounds, ternary compounds adopting the ABC 2 -type configuration are most studied due to their wide range of photovoltaic applications [12]. e ABC 2 -type ternary compounds comprise chalcogenides with type I-III-VI 2 anions and pnictides with type II-IV-V 2 anions [13]. Experimental studies, as well as theoretical predictions of properties of the ABC 2 -type ternary compounds, have been performed for application in the optoelectronic and photovoltaic fields. Among the ABC 2 -type ternary compounds, the copper-based ternary compounds have been principally investigated for potential solar cell application [11,[14][15][16][17][18]. Other studies have been carried out on ABC 2 -type ternary compounds by replacing the copper element with other elements. For instance, Khan et al. [19] investigated the structural, electronic, and optical properties of ternary CaCN 2 compound by using the DFT, where in their deduction from the study, the results showed that the compound had properties that are suitable for utilization in the photovoltaic applications. Additionally, optical properties of SrSiP 2 and CaSiP 2 compounds by using the DFT approach have been investigated [20]. Elements such as gold, silver, and aluminum, among others, have also been investigated as a replacement for copper for potential optoelectronic and photovoltaic applications . Ternary chalcogenides are more promising among the multinary compounds because the majority possess properties such as low toxicity semiconductors and tunable bandgaps allowing improvement of intrinsic properties [43]. In a study for such compounds performed by Regulacio Han, a CuInS 2 compound was studied for application as a light-emitting diode due to its unique properties for bright light emission and nondegradability on thermal exposure. In a different study of ABX compounds, the combination of A � Cu, Ag; B�Zn; and X � Ge, Sn was found suitable for application in photocatalysis [43]. Extensive DFT work of similar compounds is available in the literature, such as ABiX 2 and ABiX 3 (A � Na, K; X � O, S) and the references therein [44]. ABC 2 -type ternary compounds based on potassium elements have been less studied as compared to their copper, silver, gold, and aluminum analogues [45]. Specifically, ternary compounds with K-Cu-X (X � N, P, As, Sb, Bi) formula have been less reported apart from a few exceptions [46][47][48]. To the best of our knowledge, there are a few experimental reports in the literature on the crystal structure of K-Cu-X (X � As; Sb) [48] hence the motivation for this study.
e objective of this work is to investigate the structural, electronic, elastic, mechanical, and optical properties of K 2 CuAs and K 2 CuSb ternary compounds using the DFT method for potential applications in the photovoltaic field.

Computational Details
e structural, electronic, elastic, mechanical, and optical properties of K 2 CuAs and K 2 CuSb ternary compounds were computed within the DFT [49,50]. Formalism is implemented in the QE code [51]. e K 2 CuAs and K 2 CuSb crystal structure input files were downloaded from the materials project database [52,53]. and the   [54] for DFT calculations. e electron-ion interaction was denoted by using the projector augmented-wave function (PAW) method [55].
e Generalized Gradient Approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) [56,57] was chosen to define the exchange-correlation effect of the electrons. e optimized cutoff energy of 150 Ry and 8 × 8 × 8 Monkhorst-Pack grid for Brillouin zone integration were used. Geometry optimization was performed by computing the total energy per unit cell at several lattice constant values to obtain the ground state structural properties. Based on the optimized lattice constants, the elastic, electronic, and optical properties were calculated.

Results and Discussion
3.1. Structural Properties. Both K 2 CuAs and K 2 CuSb ternary compounds adopt an orthorhombic crystal system with the space group Cmcm as reported elsewhere [48]. ese crystal structures have K atoms bonded in 4 coordinate geometries to 4 equivalent As and Sb atoms, whereas the Cu atoms are bonded in a linear geometry to 2 equivalent As and Sb atoms [48], as shown in Figure 1.
From Table 1, the As-Cu bond is stronger than the As-K bond in K 2 CuAs, while the Sb-Cu bond is stronger than the Sb-K bond in K 2 CuSb. e K-K bond is the weakest with long bond lengths; this difference is attributed to the fact that the volume per atom tends to increase with the increase in atomic radius and therefore affects crystal lattice basis [13]. e total energy at various lattice constant values is computed and presented in Figure 2.
e computed total energy as a function of lattice constant values was fitted in the Birch ̶ Murnaghan equation of state [58] 19 19.5 20 20.5 21  properties. e computed ground state lattice parameters in Table 2 are consistent with the ones reported by Eisenmann et al. [48].

Electronic
Properties. e electronic band structures and projected density of states (PDOSs) of the K 2 CuAs and K 2 CuSb ternary compounds were computed by using the optimized crystal structures and presented as follows.
e K 2 CuAs and K 2 CuSb compounds have narrow bandgaps of 1.050 eV and 1.129 eV (see Figures 3(a) and 4(a)), respectively. e maxima of the valence bands and the minima of the conduction bands occur at different symmetry points (Y-Γ) in the Brillouin zone, implying that K 2 CuAs and K 2 CuSb ternary compounds are indirect bandgap semiconductors. e projected density of states describes the available states for electrons to occupy when projected on atomic orbitals. By carrying out the projected density of states calculation, we can tell which electronic states (shell (s, p, d, f ) and orbital) for a particular atom contribute to the formation of the band edges. Suppose two orbitals lying in the same energy range hybridize, the value of the projected density of states increases. In this way, these orbitals are said to have contributed to the band structure edges. e orbital contributions to the formation of valence bands and conduction bands are described by the PDOS in the energy region −3.5 eV to 5 eV. As illustrated in Figure 3(b), the upper valence band in the region −1.4 to the Fermi level was majorly formed by As-2p and Cu-3d with a small contribution from the other states while the middle valence band in the energy region −2.6 eV to −1.5 eV was majorly formed by Cu-3d with few contributions from the other states. e lower valence band in the energy region −3 eV to −3.4 eV was formed mainly by As-2p, Cu-3d, and Cu-4s, whereas the conduction band was majorly formed by Cu-5p and As-1s with little contribution from the other states. In the case of the K 2 CuSb compound (Figure 4(b)), the valence band was majorly formed by the Cu-3d, Sb-3p, and Cu-4s orbitals, while the conduction band was majorly formed by the Cu-5p and Sb-3p orbitals.

Elastic and Mechanical Properties.
e K 2 CuAs and K 2 CuSb ternary compounds adopt an orthorhombic crystal structure featuring 9 independent elastic constants [59] given as C 11 , C 12 , C 13 , C 22 , C 23 , C 33 , C 44 , C 55 , and C 66 . e necessary and sufficient conditions for elastic stability of the orthorhombic crystal system [59,60] are given as Cu-4s Cu-5p As-1s As-2p pdos-tot  Advances in Materials Science and Engineering C 11 > 0; C 11 C 22 > C 2 12 , C 11 C 22 C 33 + 2C 12 C 13 C 23 , e computed elastic constants in Table 3 satisfy the conditions for elastic stability of the orthorhombic system; thus, the K 2 CuAs and K 2 CuSb ternary compounds are mechanically stable. Other mechanical properties are shown in Table 4.
Bulk modulus measures the resistance against volume change resulting from applied external pressure [61]. Large B    [61]. From the structural properties, the obtained bond lengths in K 2 CuAs are shorter than those in the K 2 CuSb crystal structure thus the higher value of B in the K 2 CuAs compound. e ductile (ionic) and brittle (covalent) nature of materials is determined by using Pugh's ratio B/G and Poisson's ratio, n [62]. e restriction for brittleness is B/ G < 1.75; otherwise, the material is said to be ductile [63].
Poisson's ratio n � 0.1, 0.25, and 0.33 for pure covalent, ionic, and metallic bonds, respectively, [64]. us, we can conclude that K 2 CuAs and K 2 CuSb ternary compounds are ductile and strongly dominated by ionic character. ese findings are in agreement with the other reports in the literature for compounds with similar stoichiometry [47]. e stiffness of a material is determined by applying Young's modulus value [65]. e higher the value of E, the stiffer the material [65]; therefore, K 2 CuAs compound is stiffer than K 2 CuSb.

Optical Properties.
To explore the prospect of any material for solar cell and optoelectronic applications, an investigation of the materials' frequency response of various optical constants to the incident photon radiation is key [66]. A complex dielectric wave function describes the electron response in a material [67], and it is described in the form of an equation as where ε 1 (ω) and ε 2 (ω) refer to the real and imaginary parts of the complex dielectric wave function. All the other optical constants including the absorption coefficients α (ω), refractive index n (ω), extinction coefficient K(ω), reflectivity R (ω), and energy loss function L (ω) presented in Figure 5 are computed by using the equations presented as follows [41], [63,68,69]: e imaginary part of the dielectric wave function describes the photon absorption in crystalline materials [61]. e peaks in ε 2 (ω) curves result from the electronic transitions from the valence to the conduction bands. e absorption onsets in ε 2 (ω) curves refer to the materials bandgaps which lie within the visible region, <3.1 eV for K 2 CuAs and K 2 CuSb compounds, an implication of strong interband transitions.
is makes K 2 CuAs and K 2 CuSb promising candidates for solar cell applications [41]. Additionally, narrow bandgaps facilitate faster electron transitions as opposed to wide bandgaps [61]. e key feature of the ε 1 (ω) curve is ε 1 (Energy � 0), also referred to as the static value [70].
is static value is correlated to the material's refractive index as n � ������ (ε 1 (0)). Starting from Energy � 0, the ε 1 (ω) plot attained major peaks at low energy regions, 1.786 eV and 1.652 eV for K 2 CuAs and K 2 CuSb, respectively. e photon transmission persisted until the ε 1 (ω) values became negative at energy regions 5.681-9.173 eV. At this energy region, the incident photon radiations are assumed to be fully attenuated [71] and the compounds assert a metallic behaviour [72]. e calculated refractive indices for K 2 CuAs and K 2 CuSb were 2.49 and 2.55, respectively. e major refractive index peaks reside within the visible region. e optical absorption coefficients of K 2 CuAs and K 2 CuSb compounds cover the UV-Vis regions in the range of 2.82-11.71 eV; this demonstrates that these compounds can be utilized for photovoltaic applications. e materials' surface behaviour and energy loss by fast electrons entering a medium are determined by reflectivity and energy loss function, respectively [41]. e main peaks of the reflectivity curves were observed in the regions 2.217-5.345 eV. e reflectivity decreased beyond this region.
ere was no significant absorption in the visible regions, as depicted in the loss spectrum. e major absorption peak occurred at higher energy regions >10 eV. e optical properties' results obtained in this work are in agreement with the results obtained previously on the related materials [46,47].

Conclusions
In summary, we have studied the structural, electronic, elastic, mechanical, and optical properties of K 2 CuAs and K 2 CuSb ternary compounds by using the DFT method as implemented in the QE package. Equilibrium lattice constants of 19.1414 and 20.0041 a.u were obtained for K 2 CuAs and K 2 CuSb, respectively. e bandgaps obtained were 1.050 eV and 1.129 eV for K 2 CuAs and K 2 CuSb, respectively. e formation of the valence band was primarily by Cu-3d, As-2p, and Cu-4s states, while the conduction band was majorly formed by Cu-5p and As-1s states in K 2 CuAs whereas, in K 2 CuSb, the valence band was majorly formed by Cu-3d, Cu-4s, and Sb-3p states while the conduction band was mainly formed by Sb-3p and Cu-5p states. Both K 2 CuAs and K 2 CuSb were found to be mechanically stable at zero pressure, ductile, and ionic, thus their potentiality for resilient materials application. e calculated bandgaps, high refractive indices, high absorption coefficients, and wide energy coverage of the absorption coefficients spectra, mainly in the UV-Vis regions of the electromagnetic spectrum, make K 2 CuAs and K 2 CuSb ternary compounds suitable for photovoltaic applications.
Data Availability e source files and raw data supporting the conclusions in this paper will be assessed upon request from the authors.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Supplementary Materials
e results and conclusions of this work are supported by a supplementary file submitted along with the manuscript. e File comprises the K 2 CuX (X = As, Sb) crystal structure data, the pseudopotentials, Plane Wave selfconsistent field (PWscf ) input files, output files for structural properties, output files for elastic and mechanical properties as well as the Equations used for calculating optical properties. Also, the references to K 2 CuX (X = As, Sb) source files are provided. Supplementary Materials (Supplementary Materials)