3.1 Structural properties

Single crystal X-ray diffraction analysis revealed that the ternary compounds AGaQ₂ (A=K, Cs; Q=S, Se, Te) ^[1,2,4,5] crystallize in space group C2/c (No. 15) of the monoclinic system and adopt the TiGaSe₂ structure type with sixteen formula units per unit-cell (Z=16) as illustrated in Fig. 1. The studied compounds have a layered structure with fundamental building blocks of [Ga4Q10] supertetrahedra. These supertetrahedra form two dimensional (2D) [GaQ₂]^— anionic layers which are separated by K⁺ or Cs⁺ cations. The conventional cell of AGaQ₂ (A=K, Cs; Q=S, Se, Te) contains 9 atoms, 2K, 2A and 5Q, the atomic positions and the structural data are given in Tables I, II.

Figure 1 Schematic diagram of the monoclinic unit-cell of AGaQ₂ (A=K,Cs; Q=S, Se, Te) with 16 formula. 

By fully geometry optimization, the calculated equilibrium structural parameters of the studied compounds are summarized in Table I and compared with the available experimental data ^[1,2,4,5]. The results show that the lattice parameters (a,b,c), angle β, and unit cell equilibrium volume V are in good agreement with the experimental values; the maximum relative error of the calculated equilibrium lattice constants a, b, and c is about 0.24% (2.14%), 0.16% (1.99%) and 0.9% (2.29%) for KGaS₂ (KGaSe₂), respectively, and, about 1.82% (2.77%), 1.72% (2.46%) and 2.09% (2.52%) for KGaTe₂ (CsGaTe₂) respectively compared with the respective reported data. There is a slight overestimation of the calculated lattice constants relative to the experimental values. This is not surprising, it is due to the use of the GGA-PBE approximation. The relative deviation of the calculated angle β and unit cell equilibrium volume V deviate by 0.21% (0.42%) and 0.54% (6.7%) for KGaS₂ (KGaSe₂), and by 0.01% (0.07%) and 5.74% (7.94%) for KGaTe₂ (CsGaTe₂), respectively, are compared with the experimental data.

It can be seen that the unit cell volume increases from KGaS₂, KGaSe₂ to KGaTe₂, which is assigned to the increasing of the atomic radius from S, Se to Te respectively.

Also, CsGaTe₂ unit cell volume is larger than that of KGaTe₂, which is also assigned to the small atomic radius of K compared to that of Cs. The calculated atomic positions, listed in Table II, agree well with the experimental data.

3.2 Electronic properties

It is known that GGA-PBE overestimates the band gap of low band gap materials and underestimates the one of high band gap materials ^[16]. Nevertheless, GGA functionals are known to underestimate Kohn-Sham band gaps (compared to experimentally determined values), whereas the Hartree-Fock (HF) method overestimates them ^[17]. Therefore, hybrid functionals were introduced by Becke ^[18] which describe exchange using a combination of the exact nonlocal Hartree-Fock-exchange interaction with GGA. But the computational evaluation became difficult due of the slowly decaying Fock exchange with distance ^[19]. Heyd et al., suggested an hybrid-functional scheme for extended systems ^[19,20], giving rise to the Heyd-Scuseria-Ernzerhof (HSE) functional by substituting the exchange energy in the conventional GGA-PBE functional ^[8] by a functional which mixes HF exchange energy and PBE, the spatial decay of the HF exchange interaction is accelerated by substitution of the full 1/r Coulomb potential with a screened one, the exchange energy term is split into short-range (SR) and long-range (LR) components and the long-range exchange energy is given only by the PBE term, and the correlation energy comes entirely from PBE.

where w is the parameter which determines the screening, defining the range of the Hartree-Fock correction ^[19]. This is handled by decomposing the Coulomb kernel into a short-range and a long-range term ^[21], according to:

in which erf and erfc are, respectively, the error and the complementary error function. Choosing w = 0.3Å^-1 leads to the HSE03 functional, whereas w = 0.2Å^-1 leads to the HSE06 functional ^[20,22].

In general, the GGA-PBE functional method does not produce the exact value for the band gap; it underestimates the band gap value ^[25] while the nonlocal hybrid functional method developed by Heyd, Scuseria, and Enzerhof (HSE06) gives better estimation of electronic properties of solids. This method yields a band gap comparable with the experimental one by including the exact contribution of the Hartree-Fock exchange ^[21-24,26], the HSE06 functional hybrid density replaces the slowly decomposing long-range part of the exchange Fock by the density functional counterpart. For this reason, to better estimate the electronic bandgap value, we have also used HSE06 in addition of GGA-PBE for comparison.

The band structures calculated along the high symmetry directions: Z(0,0,1/2), Γ(0,0,0), Y(0,1/2,0), A(1/2,1/2,0), B(1/2,0,0), D(1/2,0,1/2), E(1/2,1/2,1/2), C(0,1/2,1/2), F(0,1/2,0) and Q(0,1/2,1/2) in the irreducible Brillouin zone, using the pseudo-potential plane wave (PP-PW) method with the two approaches GGA-PBE and HSE06 for the four materials KGaS₂, KGaSe₂, KGaTe₂ and CsGaTe₂, at their equilibrium structural parameters, are plotted in Figs. 2, 3. The estimated band gap values using both GGA-PBE and HSE06 approximations are summarized in Table III with experimental and theoretical data available in the literature for comparison.

Figure 2 Band structure of KGaQ₂ (Q=S, Se) calculated along high symmetry lines. 

Figure 3 Band structure of KGaTe₂ and CsGaTe₂ calculated along high symmetry lines. 

The obtained band diagrams have the same form, since the conduction band minima and valence band maxima of all studied materials appear at the center of the Brillouin zone, the Γ point, indicating that all the studied compounds are direct band gap semiconductors according to the calculations by either GGA-PBE or HSE06. We notice also that the HSE06 functional gives notably larger band gaps than the GGA-PBE functional. There is a notable difference between the band gap values yielded by these two functionals. The difference is 0.73 eV, 0.82 eV, 0.55 eV and 0.70 eV for KGaS₂, KGaSe₂, KGaTe₂ and CsGaTe₂ respectively. From the Table III, it is clear that the GGA-PBE functional method underestimates the electronic bandgap in the case of KGaSe₂ while HSE06 functional method is in better agreement with the experimental value. There are no reported experimental or theoretical results on the band gaps for the remaining compounds, so this is their first theoretical prediction.

On the other hand, it can be observed that the fundamental band gap reveals a decreasing trend on increasing size of chalcogen from S, Se to Te in the case of KGaQ₂ (Q=S, Se, Te). The larger band gap is observed in the case of KGaS₂, we explain that S is more electronegative than Se and Te, this is consistent with the conclusions of the previously reported study which indicate that the band-gap is inversely proportional to the cell volume ^[26].

It can be seen that, in GGA-PBE calculations, the band gap value was equal to 2.588 eV and 1.781 eV for KGaS₂ and KGaSe₂ respectively. The gap energy value in the case of KGaSe₂ compound is much smaller than the experimental value ^[2]. However, the use of the HSE06 hybrid functional gives 3.321 eV and 2.604 eV for KGaS₂ and KGaSe₂ respectively. We notice that the calculated band gap value by HSE06 is basically consistent with the experimental value in th case of KGaSe₂. We have also calculated the band gap of AGaTe₂ (A=K, Cs), finding that the values increase by the use of HSE06, from 1.227 eV (KGaTe₂) and 1.280 eV (CsGaTe₂) to 1.777 eV and 1.979 eV respectively. However, by changing from S, Se to Te in KGaQ₂ (Q=S, Se, Te) an important difference is noticed in gap energy value almost of 1 eV and on the other hand the replacement of K-atom by Cs-atom in AGaTe₂ (A=K, Cs) induces only a small difference of almost 0.2 eV. There are no reported experimental or theoretical results on the band gaps for AGaTe₂ (A=K, Cs), so this is their first theoretical prediction. However, the hybrid density functional theory has improved the band gap, where we find larger band gap values than those obtained by GGA- PBE functional. According to the literature, as mentionned before, GGA-PBE calculations underestimate the band gaps and HSE06 calculations gives better estimation of band gaps, therefore the obtained gap value using the HSE06 confirms that the studied compounds are direct band-gap semiconductors with wide band-gaps.

Figure 4 represents the total and partial densities of states (TDOS and PDOS) of KGaQ₂ (Q=S,Se) and AGaTe₂ (A=K, Cs) compounds using the HSE06 approximation. Fermi level is set as zero energy. The valence states are regrouped in three divisions. First, we notice that at lower energies (< -15 eV) there are bands centered on -28.9 eV (-29.4 eV, --30 eV) for KGaS₂ (KGaSe₂, KGaTe₂) respectively and on -21.45 eV for CsGaTe₂, formed only of K-s and Cs-s states, which are not shown in the figure for clarity purpose, they do not form chemical bonding to other. And, the bands centered around -13.7 eV for KGaS₂, -14 eV for KGaSe₂ and -14.4 eV for KGaTe₂ (CsGaTe₂) are mainly formed of gallium Ga-d states. Next, the K-p states are located at -12.5 eV, -12.19 eV, -13.6 eV for KGaQ₂ (Q=S, Se, Te) and the Cs-p states at -8.2 eV for CsGaTe₂. As well as the chalcogen-s states are located at -10.6 eV and -11.3 eV for KGaS₂ and KGaSe₂ successively and at -9.8 eV for KGaTe₂ (CsGaTe₂). The K-p (Cs-p) and the chalcogen-s states exhibit very low hybridization, indicating weak chemical bonding between the K (Cs) atom and the chalcogen atoms. Finally, the upper valence band ranging from about -3.3 eV to 0 eV is mainly formed by the chalcogen-s,p and small contribution from Ga-s,p states. We notice a fairly large mixture of the Ga states and the chalcogen states for the title materials, and no contribution of potassium or cesium states around the Fermi level in this region, indicating that the band gap is primarily determined by the [GaQ2]^— layer (Q=S, Se, Te).

Figure 4 Calculated total (TDOS) and partial density of states (PDOS) of KGaQ₂ ( Q=S, Se, Te) and CsGaTe₂. 

The conduction band is formed by chalcogen -s,p states with a small contribution of the Ga-s,p and K(Cs)-s,p states. For all studied compounds, we note that the contribution of potassium states is negligible. Our results of the DOS spectra are approximately similar with those reported in literature ^[2].

In summary, and as we have seen above, Table III shows the band-gap values of the KGaQ₂ (Q=S,Se,Te) and CsGaTe₂ materials, where the band-gap value 3.321 eV for KGaS₂ is the largest among the four, followed by 2.604 eV for KGaSe₂ and 1.777 eV for KGaTe₂, and increases to 1.979 eV for CsGaTe₂ when replacing the potassium atom by cesium atom. This is agreeing with the previously reported results, which indicate that the value of the band gap is inversely proportional to the cell volume ^[26]. The anionic radius increasing (1.84 Å, 1.98 Å, 2.21 Å for S, Se and Te successively) increases the cell volume (Table I) leading to a decrease in the minimum conduction band which causes the observed decrease in the band gap. Thus, the existence of direct and wide bandgaps of the studied materials indicates their their possible photovoltaic applications.

3.3 Optical properties

Energy gap and refractive index are two important physical parameters that characterize the electronic and optical properties of semiconductors. Since the energy gap determines the photon absorption threshold in semiconductor and the refractive index is a measure of it transparency with respect to incident spectral radiation, they are useful for determining the suitable applications of semiconductor in electronic, optical and optoelectronic devices.

In the literature, several studies on known semiconductors have been carried out to understand the correlation between the energy gap and the refractive index in order to generate the relation between them. Different empirical models have been proposed, among the best known which have been applied successfully to direct band gap semiconductors are the model proposed by Ravindra in 1979 ^[13] and that proposed by Moss in 1985 ^[14]. It was seen that where Ravindra’s relation fails Moss’s relation approaches experimental results. On the other hand, in 1995, Heve and Vandamm claimed that their model is accurate for wide band gap semiconductors and for most materials used in optoelectronic devices ^[15].

In this work, the refractive index n was estimated using the three different empirical models, cited above, based on the previously calculated fundamental energy gap E_g, estimated from the HSE06 approximation, using the following equations:

• Ravindra relation ^[13]:

with α = 4.084 and β = -0.62 eV^-1.

• Moss relation ^[14]:

where k is a constant with a value of 95 eV.

• Herve-Vandamme relation ^[15]:

with A = 13.6 eV and B = 3.4 eV. The high-frequency dielectric constant ϵ_∞ can be calculated using the relation

The reflection coefficient R for high frequencies is given by the relation

We have calculated the reflection coefficient R and the high-frequency dielectric constant ϵ_∞ of the studied materials using (6) and (7) as function of the refractive index n. The results are presented in Table IV. It can be observed, and whatever the empirical model used, that the refractive index decreases with increasing the energy gap. It is clear that both values of n and ϵ_∞ increase from KGaS₂ to KGaSe₂ then to KGaTe₂ and also decrease from KGaTe₂ to CsGaTe₂. It can be concluded that the smaller the energy band gap, the higher the refractive index and dielectric constant and the smaller the reflection coefficient, the more useful the material to optoelectronic devices.

We recall that the results presented in the actual paper on n and ϵ_∞ are not available from experimental data or other theoretical studies for comparison, and this results remain as predictions. Moreover, we can remark that the obtained results show that the Herve and Vandamme model represents, generally, the average values between the results using Moss and that of Ravindra. According to the literature, Heve and Vandamme based their equation on oscillatory theory, assuming the UV resonance energy has a constant difference with the band energy. However, they found that their model produces results very close to the experimental data for energy gaps greater than 1.43 eV ^[27]. Therefore, we believe that the results that could be considered a good prediction are those obtained by the use of the model proposed by Herve and Vandamme.