#### 3.1. Structural Properties

**TlBiS**_{2}: TlBiS

_{2} has rhombohedral structures with space group D5 3d (R-3m, Space group No. 166), similar to Bi

_{2}Te

_{3}. There are four atoms, namely, 1-Tl, 1-Bi, and 2-S positioned in layers normal to the three-fold axis in the arrangement Tl-S-Bi-S, as shown in

Figure 1a. Each Tl/Bi layer is placed between two S layers, which indicates a strong interlayer coupling so that the crystal structure is substantially three-dimensional. Tl, Bi, and S are placed at the (0, 0, 0), (0.5, 0.5, 0.5), and (±u, ±u, ±u) sites, respectively. This structure has inversion symmetry where both Bi and Tl represent inversion centers. The basic TlBiS

_{2} structure is a simple NaCl-type lattice. It is similar to

ABQ_{2}-type compounds (

A,

B and

Q are monovalent atom, trivalent atom and chalcogen respectively). The TlBiS

_{2} structure is rhombohedral along the cubic [111] direction and matching to the

c axis of the primitive hexagonal arrangement. The sum of the ionic radii for a coordination number (CN) of 6 is 2.87 Å for Bi

^{3+}/S

^{2−} and 3.34 Å for Tl

^{+}/S

^{2−} [

26]. The experimentally-determined and theoretically-derived bond length matches well for Bi-S, but for the Tl-S distance the value is about 5.3% smaller. Our calculated lattice parameters and the positional parameters all fitted well with the experimental findings (see

Table S2) [

26].

**Ba**_{3}BiN: (Ba

_{3}N)Bi crystallizes in a hexagonal anti-perovskite variant of the BaNiO

_{3} structure type (

P63/

mmc, space group No. 194,

Z = 2). This phase consists of Ba

_{6}N octahedral units sharing faces formed with three Ba ions according to a rod-like structure along [001] (see

Figure 1b). The calculated Ba-N distance is 2.677 Å which compares well with those in sub-nitrides with nitrogen species in octahedral coordination [

27,

28]. Hexagonal perovskite crystal structures can only be expected for compounds containing alkaline-earth metal species with large radii. Hence, the resulting distance

d (N-N) and the Coulomb repulsion between N

^{3−} in face-sharing octahedra has to be formed. The resulting distance N-N is 3.3218 Å is sufficiently large.

**Ag**_{2}BaS_{2}: Ag

_{2}BaS

_{2} crystallises in the trigonal CaA1

_{2}Si

_{2}-type structure, a = 4.386 (1) A,

c = 7.194 (2) A, space group P3m1,

Z = 1, where S and Ag atoms are arranged in the chemically ordered double-corrugated hexagonal layers and Ca atoms are intercalated between them [

29], as shown in

Figure 1c. These layers can, in turn, be described as being made up from two stacked AgS layers, with each layer being a two-dimensional infinite net of chair-like six-membered rings. Every atom in the Ag

_{2}S

_{2} layer is four-coordinate, but the coordination environment is very different for Ag and S. Each Ag is surrounded by four S atoms, forming a distorted tetrahedron. The S is also four-coordinate in Ag, but the environment is most unusual, a flipped tetrahedron or umbrella shape.

**ZrSO:** ZrSO crystalizes in cubic (P2

_{1}3, space group No. 198) and tetragonal (P4/nmm, space group No. 129) form [

30,

31]. Tetragonal ZrSO crystallises in the PbFCl-type structure. The form of this phase has not yet been synthesized [

31]. All preparations techniques proved that the tetragonal phase was always accompanied by considerable proportions of cubic ZrSO, and in some cases, even by ZrO

_{2}. It seemed likely, therefore, that the tetragonal phase also contains oxygen [

31]. According to our theoretical energy volume curve (see

Figure S1), the cubic form is more stable than the tetragonal form and the energy difference between these two structures is very small (36 meV/f.u.). Moreover, it is clear from

Figure S1 that the energy minima for these two structures are well separated, and the energy well is deep enough to stabilize the individual phases. We can thus conclude that this phase can be experimentally stabilized using a high-pressure technique.

There are four S and three O atoms surround each Zr atom in the cubic phase. The S atoms form a stretched tetrahedron with one Zr-S separation of 2.61 Å and three of 2.63 Å. The O atoms form an equilateral triangle centered at the Zr site, in such a way that there is an O atom in each of the stretched faces of the S atoms tetrahedron. At a distance of 2.13 Å from each oxygen atom, the Zr atom is slightly out of the plane of the oxygen triangle. The configuration may be regarded as a distorted octahedron consisting of three S atoms and three O atoms, with an extra sulfur atom above the center of the face of the octahedron driven by the oxygen atoms. From this point of view, the co-ordination of zirconium is quite similar to that observed in K

_{3}ZrF

_{7}. The separations of S-S and S-O are 3.59 and 2.96 Å, respectively. These separations are shorter than the sum of the radii (3.68 Å for S-S and 3–24 Å for S-O) owing to the sharing of edges between the coordination polyhedra [

30].

#### 3.2. Electronic Properties

In order to investigate the potential applicability of non-silicon semiconductors as a light-harvesting medium, the band gap of these materials is a crucial factor that needs to be further explored. Both short-circuit current and open-circuit voltage is regulated by the band gap of the photoactive semiconductors. Broader band gap leads to higher open-circuit voltage but fewer excited electrons, which results in lower short-circuit current. Narrower band gap leads to low open circuit voltage but more excited electrons, which result in larger short-circuit current. The ideal solar cell is theoretically shown to have a maximum of 32% efficiency with an optimal band gap E

_{g} = 1.4 eV is [

2]. In real cells, the solar spectrum is a broad energy spectrum and it does not match the band gap well where thermalisation loss occurs, which eventually results in efficiencies below the detailed-balance limit [

2]. Band gap calculation using electronic band structures gives a promising opportunity to identify suitable PV materials. The calculated band structure of trigonal-TlBiS

_{2}, hexagonal-Ba

_{3}BiN, trigonal-Ag

_{2}BaS

_{2} and tetragonal-ZrSO crystals along a high-symmetry path in the first Brillouin zone are presented in

Figure 2. For electronic structure calculations, we employ hybrid functional (HSE06) and estimate the band gap values for these materials. The results of electronic structure calculations, listed in

Table 1, span the in the range from 1.10 to 2.6 eV. The four compounds exhibit a direct band gap at the Г

**k**-point, as shown in

Figure 2.

The HSE06 band structure of TlBiS

_{2} is shown in

Figure 2a. Both the valence band maximum (VBM) and the conduction band minimum (CBM) are well placed at the Г

**k**-point. This clearly shows that TlBiS

_{2} is a direct band gap semiconductor with valence bands derived from Bi-s, S-p, and Tl-s states, and conduction bands derived from S-s, Bi-p, and Bi-d states (

Figure 3). The HSE06 band gap is 1.1 eV for TlBiS

_{2}, which is nearly equal to that of Silicon. Bahadur Singh et al., showed that TlBiS

_{2} has GGA band gap of 0.64 eV with direct band gap type at Г

**k**-point [

32]. It is important to note that we employed a more accurate HSE06 method compared to the GGA calculation method used by Bahadur Singh et al. [

32] and the difference is approximately 0.46 eV. This is as expected because it is well known that calculations using GGA underestimate the band gap value, while the HSE06 screened hybrid functional is very successful in precisely calculating the band gap value. Our calculation shows that we have a band gap of 1.42 eV at F

**k**-point TlBiS

_{2} as shown in

Figure 2a. This shows that TlBiS

_{2} well suited for PV applications as optimal band gap for the best performance is 1.4 eV as mentioned earlier [

12].

In the case of Ba

_{3}BiN, the VBM and CBM are located at Г

**k**-point. Thus, the calculated HSE06 band structure in

Figure 2b shows that Ba

_{3}BiN is a direct band gap semiconductor with a band gap of 1.29 eV. From the

Figure S3 of Supplementary Materials, the valence band derived from Bi-p and hybridized Ni-d states and conduction bands are mainly composed of Bi-s and Ba-s states. According to Imran Ullah et al., Ba

_{3}BiN is a direct band gap semiconductor with a band gap of 0.64 eV at Г

**k**-point [

33]. The comparison between the present results using HSE06 with the previous results using GGA [

33] reveals that the band gap of Ba

_{3}BiN is previously underestimated by 0.79 eV. To the best of our knowledge, no HSE06 or experimental studies have been previously reported on Ba

_{3}BiN. For Ag

_{2}BaS

_{2}, the calculated HSE06 band structure in

Figure 2c shows that the VBM and CBM are located at the Г

**k**-point. Thus, Ag

_{2}BaS

_{2} is a direct band gap semiconductor with a band gap of 1.95 eV. Note that our direct band gap value of 1.95 eV calculated with HSE06 closely matches the previous HSE06 band gap value of 2.01 eV [

34] calculated by Aditi Krishnapriyan et al. To the best of our knowledge, no experimental study has been previously reported on Ag

_{2}BaS

_{2}. From

Figure 2b and the

Figure S4 of Supplementary Materials, the valence band maximum is derived from S-p states and conduction band derived from Ag-s states. In the case of ZrSO, both the VBM and the CBM are located at Г. Thus, ZrSO is a direct band gap semiconductor with valence bands derived from S-p states, and conduction bands derived from O-s states (

Figure S5). The calculated HSE06 band gap between VBM and CBM is 2.60 eV. To the best of our knowledge, no HSE06 and experimental study have previously reported on ZrSO.

#### 3.4. Lattice Dynamic Stability

Lattice dynamic calculations have also been performed on TlBiS

_{2}, Ba

_{3}BiN, Ag

_{2}BaS

_{2}, and ZrSO under ambient conditions. To validate the dynamical stability of these compounds, the total phonon density of states is calculated at the equilibrium volume. In

Figure 4, we displayed their total phonon density of states. No imaginary frequencies were observed, revealing that TlBiS

_{2}, Ba

_{3}BiN, Ag

_{2}BaS

_{2}, and ZrSO are dynamically stable. We present the site projected phonon density of states for TlBiS

_{2}, Ba

_{3}BiN, Ag

_{2}BaS

_{2}, and ZrSO in

Figure 5. The vibrational modes spread over the 2 to 50 THz range in the case of TlBiS

_{2} phase. In the low frequencies region, Tl is dominant over Bi and S. The lattice vibrational modes for the Tl, S, and Bi are present between 3–10, 5–21, and 18–50 THz, respectively. In the case of Ba

_{3}BiN, the vibrational modes spread over 0 to 52 THz. The lattice vibrational modes for N, Bi, and Ba are present between 0–19, 2–20 and 3–51 THz, respectively. For Ag

_{2}BaS

_{2}, the vibrational modes spread over 0.3 to 7.5 THz. The Ag-S and Ba-S stretching modes are present between 1–3 THz. Other Ag-Ba, S-Ag, and S-Ba stretching modes are present between 5–6.5, 5–6.5 and 4.3–7 THz, respectively. The lattice vibrational modes for Ag, Ba, and S are present between 0.3–3.6, 0.3–3.6, and 4.3–7.5 THz, respectively.

In the case of ZrSO, the vibrational modes spread over 2 to 74 THz. The Zr-O and S-O stretching modes are present between 5–30 THz. Other O-Zr and O-S stretching modes are present at 40–65 and 47–60 THz, respectively. The lattice vibrational modes for Zr, S, and O are present between 2–35, 2–35, and 35–74 THz, respectively. The calculated zero-point energy (ZPE) for the TlBiS

_{2}, Ba

_{3}BiN, Ag

_{2}BaS

_{2}, and ZrSO phases varies from 0.10 to 1.3 eV/f.u. (see

Table 3), and the following ZPE sequence are: Ag

_{2}BaS

_{2} < TlBiS

_{2} < ZrSO < Ba

_{3}BiN.

In addition to the dynamic stability, we employ the vibrational density of states to compute the specific heat capacity (C

_{v}) of TlBiS

_{2}, Ba

_{3}BiN, Ag

_{2}BaS

_{2}, and ZrSO at constant volume and pressure. The C

_{v} as a function of temperature presented in

Figure 6 in the temperature range from 0 K to 1000 K. For TlBiS

_{2}, the specific heat capacity increases rapidly below 500 K. The value of C

_{v} is almost constant at 90 J/K/mol for above 500 K. In the case of Ba

_{3}BiN, the C

_{v} increases rapidly up to 1000 K. The specific heat capacity increases rapidly below 100 K for Ag

_{2}BaS

_{2}. The C

_{v} is almost constant at 125 J/K/mol for above 100 K. For ZrSO, the C

_{v} increases from 100 K to 1000 K. The following C

_{v} sequence are: TlBiS

_{2} < ZrSO < Ag

_{2}BaS

_{2} < Ba

_{3}BiN.

#### 3.5. Mechanical Stability

The mechanical stability of a system is an essential condition to validate the existence of a compound in a given crystalline structure. The elastic constants are typically used to describe the mechanical properties of a system and to estimate its hardness. To validate the mechanical stability of TlBiS

_{2}, Ba

_{3}BiN, Ag

_{2}BaS

_{2}, and ZrSO, we calculated the single-crystal elastic constant tensor using the finite strain technique. The elastic constants describe the ability of materials to deform, or conversely, the stress required to maintain a given deformation. Both stresses and strains have three tensile and three shear components. The linear elastic constants form a 6 × 6 symmetric matrices, with 27 independent components, so that (

s_{i} only)

s_{ij} = C

_{ij} ε_{j} (

s_{i} is stress tensor, C

_{ij} is elastic constant matrix,

ε_{j} (

j = 1, 6 in Voigt index) is the strain tensor, and

i = 1, …, 6) for small stresses and strains [

37].

The stiffness of a crystal against an externally applied strain can be determined from its elastic constants. Any symmetry present in the structure may make some of these components equal, while others may be fixed to zero. The calculated elastic constants of four non-silicon materials listed in

Table 4. The elastic constant C

_{44} is a crucial parameter, indirectly describing the indentation hardness of the materials. As shown in

Table 4, all the examined compounds have a small C

_{44} value, indicating these materials possess a relatively weak shear strength.

For trigonal structures, the mechanical stability criteria at zero pressure are as follows [

38]:

In this study, trigonal TlBiS_{2} and Ag_{2}BaS_{2} have six independent elastic constants. All the three mechanical stability conditions given in Equations (1)–(3) are satisfied for the TlBiS_{2} and Ag_{2}BaS_{2} phases. Hence, this indicates that these two trigonal phase materials are mechanically stable.

For the hexagonal system, the Born stability criteria are [

38]:

The hexagonal-Ba_{3}BiN has five independent elastic constants. All three conditions for the mechanical stability given in equations 4 to 6 are satisfied for this structure, and this finding indicates that hexagonal-Ba_{3}BiN phases are mechanically stable.

The mechanical stability criteria for the tetragonal phase are given by [

38]:

ZrSO has a tetragonal structure, and thus, six independent elastic constants. All three conditions for mechanical stability given in Equations (7)–(10) are satisfied for this structure. Hence, the tetragonal-ZrSO phase is mechanically stable at ambient conditions. Equations (1)–(9) and

Table 4 validated the mechanical stability criteria for the crystal under ambient conditions. This outcome is consistent with the phonon calculations presented in

Section 3.4.

From the calculated elastic constants, the bulk (B

_{v}, B

_{R}) and shear moduli (G

_{v}, G

_{R}) are calculated from Voigt–Reuss–Hill approximations [

39,

40]. The bulk and shear moduli contain information related to the hardness of the material under various types of deformation. Generally, very hard materials hold very large bulk and shear moduli to support the volume decrease and to restrict deformation, respectively [

41]. From

Table 4, it can be identified that the listed TlBiS

_{2}, Ba

_{3}BiN, and Ag

_{2}BaS

_{2} phases have a smaller bulk modulus than ZrSO (181.403 GPa). This indicates that ZrSO is more difficult to compress than the other three materials. Among these compounds, the bulk modulus sequence is: ZrSO > Ag

_{2}BaS

_{2} > TlBiS

_{2} > Ba

_{3}BiN. As we know, the shear modulus is more closely-connected to hardness than the bulk modulus. From

Table 4, the shear modulus of ZrSO is higher than the other three compounds. Hence, the hardness of the tetragonal-ZrSO phase is higher than trigonal-TlBiS

_{2}, hexagonal-Ba

_{3}BiN, and trigonal-Ag

_{2}BaS

_{2}. Among these compounds, the shear modulus trend is ZrSO > TlBiS

_{2} > Ag

_{2}BaS

_{2} > Ba

_{3}BiN. Seemingly, the bulk and shear moduli of Ba

_{3}BiN are smaller than other compounds. Thus, Ba

_{3}BiN is easy to compress and is the softest of the examined materials.

The parameter G/B can be introduced, in which G indicates the shear modulus and B the bulk modulus. The low/high of G/B value is connected with the ductility or brittleness of the materials. The critical G/B value that separates the ductile and brittle materials is 0.5 [

41]. If the G/B value of materials is smaller than 0.5, then those materials are ductile; otherwise they are brittle. From

Table 4, the calculated G/B values of all four materials are greater than 0.5, indicating that these materials are ductile. Next, the value of Poisson’s ratio is indicative of the degree of directionality of the covalent bonding. Among these compounds, the small Poisson’s ratio (0.13) for hexagonal-Ba

_{3}BiN indicates a high degree of covalent bonding. All these phases present a very scattered Young’s (varying from 29 to 339 GPa). The compressibility value of these phases suggests that these compounds, with the exception of ZrSO, are very soft materials. The compressibility sequence is ZrSO < Ag

_{2}BaS

_{2} < TlBiS

_{2} < Ba

_{3}BiN.

#### 3.6. Optical Properties

The optical behavior of a compound has a major impact on its properties for photovoltaic applications. Optical dielectric function ε(ω) = ε

_{1}(ω) + ιε

_{2}(ω) is the fundamental quantity that describes the optical properties of a material. It describes the response of a material to a radiated electromagnetic field and the propagation of the field inside the material. The dielectric function is dependent on the frequency of electromagnetic field, and it is connected to the interaction between photons and electrons. The absorption coefficient of the material is dependent on the imaginary part, ε

_{2}(ω), and it can be derived from the inter-band optical transitions by summing over the unoccupied states, using the equation [

42,

43],

where the indices

α,

β are the Cartesian components,

$\Omega $ is the volume of the primitive cell,

q denotes the Bloch vector of the incident wave,

c and

v are the conduction and valence band states respectively,

k is the Bloch wave vector,

w_{k} denotes the

**k**-point weight,

$\delta $ is a Dirac delta function,

u_{ck} is the cell periodic part of the orbital at

**k**-point k,

${\u03f5}_{ck}$ refers to the energy of the conduction band, and

${\u03f5}_{vk}$ refers to the energy of the valence band. The real part ε

_{1}(ω) of the dielectric function can be derived from ε

_{2}(ω) using the Kramer-Kronig relationship [

42,

43]

where P indicates the principal value, η is the complex shift. All the frequency dependent linear optical properties, such as the absorption coefficients α(ω), can be calculated from ε

_{1}(ω) and ε

_{2}(ω) [

42,

43].

Experimental absorption spectra are in agreement with the inclusion of excitonic effects treated within the Bethe-Salpeter equation (BSE) in general [

44,

45,

46]. By averaging multiple grids using BSE, the calculated the dielectric function of these four materials can be further improved. The calculated ε

_{2}(ω) of the dielectric function and the absorption coefficients of TlBiS

_{2}, Ba

_{3}BiN, Ag

_{2}BaS

_{2}, and ZrSO are presented in

Figure 7 and

Figure 8, respectively. From the directional dependency of ε

_{1}(ω) and ε

_{2}(ω), trigonal-TlBiS

_{2} is highly isotropic, whereas hexagonal-Ba

_{3}BiN, trigonal-Ag

_{2}BaS

_{2}, and tetragonal-ZrSO are less anisotropic. We present the average of the real and imaginary parts of the dielectric function for the four examined compounds.

In

Figure 7, we plotted both real and imaginary part of the dielectric function of (a) TlBiS

_{2}, (b) Ba

_{3}BiN, (c) Ag

_{2}BaS

_{2}, and (d) ZrSO is plotted against the photon energy. The optical absorption coefficients of all these materials were calculated using BSE and plotted in

Figure 8. For a comparison, we have also plotted both the experimentally-verified [

47] and the BSE-calculated [

19] values for the optical absorption coefficient of silicon in the same

Figure 8. Absorption in a material take place only when the incident photon has more energy than energy band gap of the material. Since TlBiS

_{2} is a direct band gap material with a band gap of 1.10 eV, we notice the absorption to occur when the energy of the photon is around 1.08 eV, as shown in

Figure 8. It is clearly seen in

Figure 8 that there are absorption peaks at 1.32 eV, 1.93 eV, 2.45 eV, and 3.6 eV. The absorption coefficient of TlBiS

_{2} has a maximum value when the photon energy is about 3.6 eV. For silicon, the absorption coefficient becomes appreciably different from zero after 2.5 eV, and it is still not very large up to 3 eV. This phenomenon can be attributed to the indirect band gap of silicon that leads to low absorption in the visible region. We observe that the absorption coefficient of TlBiS

_{2} is superior to silicon in the visible region. This is due to the direct band gap at Г and F

**k**-points that prevails in the TlBiS

_{2}.

In the case of hexagonal-Ba

_{3}BiN, the dielectric function is calculated at the BSE level (

Figure 7b). Due to the narrow band gap, Ba

_{3}BiN can absorb photons mostly in the visible region. The HSE06 band gap is 1.29 eV, and it is direct. Therefore, Ba

_{3}BiN exhibits an absorption which rapidly increases after 1.26 eV. It can be observed that the absorption peaks of Ba

_{3}BiN are at 2.08 eV, 3.21 eV, and 3.5 eV (

Figure 8). The absorption coefficient of Ba

_{3}BiN reaches its maximum when the photon energy is about 3.5 eV. From

Figure 8, it can be noted that the optical absorption of Ba

_{3}BiN occurs in the visible region, with higher values compared to other materials considered. The reason behind the high optical absorption of Ba

_{3}BiN is due to the direct band gap of 1.29 eV. From

Figure 8, it can be observed that absorption peaks of Ag

_{2}BaS

_{2} are at 1.94 eV, 2.24 eV, 2.7 eV, and 3.6 eV. Ag

_{2}BaS

_{2} seems to have a lower absorption coefficient than TlBiS

_{2} and Ba

_{3}BiN in the visible region. However, Ag

_{2}BaS

_{2} exhibits better optical absorption than silicon in the visible region. In the same

Figure 8, we notice that absorption peaks of ZrSO start at 2.5 eV. The absorption peaks of ZrSO are also observed at 2.6 eV, 3 eV, and 3.59 eV. Among these four non-silicon materials, the absorption coefficient of ZrSO is smaller than those of the other three; this is due to the wideband gap of ZrSO.