3.1. Structural details:
GaInS2 monolayer has a hexagonal structure with four atomic layers stacked in the order of S-In-Ga-S. The geometric design is the same as Ga2S2, which is a layered structure having broken inversion and mirror symmetry. The GaInS2 structure is modeled from the structure Ga2S2 by replacing one Ga atom with In atom. This Janus structure belongs to P3m1 (C3v) with space group number 156. The side view and top view of a 4×4×1 supercell have been shown in Fig. 1. It shows that the Ga atom and S atom are exactly vertically above the In and S atom, respectively. The lattice constant of the hexagonal unit cell was optimized by relaxing the structure, gives a value a = b = 3.78 Å, which matched well as reported previously35. The distance between Ga-In, Ga-S, In-S, and S-S atoms are 2.66 Å, 2.41 Å, 2.52 Å, and 4.95 Å, respectively. The atoms make the following angles between them: ∠Ga-In-S, ∠ In-Ga-S, ∠ In-S-S, and ∠ Ga-S-S and their corresponding values are 119.98°, 115.29°, 60.01°, and 64.7°, respectively.
3.2. Electronic properties:
The electronic band structure and density of states (DOS) of the relaxed system have been calculated as shown in Figs. 2 (a) and (b). Band structure of monolayer has been studied along the high symmetric k-path K-Γ-M-K of hexagonal Brillouin zone. The GaInS2 monolayer shows indirect band gap of 1.79 eV. The conduction band minima (CBM) rests at Γ point, while the valance band maxima (VBM) rests between K and Γ point and Fermi level rests between VBM and CBM. The valance band has two nearly degenerate states, rest between K- Γ and Γ-M point, which is because two S atoms contributed to the valance band.
To know the contribution of atoms in the VBM and CBM, total DOS and partial DOS have been studied. Figure 2 (b) indicates that VBM is contributed mainly by the p orbital of the S atoms. CBM is contributed by P orbitals of Ga and In atom. Further insight revealed that the VBM is contributed primarily by Px orbital and some of Py and Pz orbitals of S atom and minor contribution of Px orbitals of Ga and In atom. The CBM contributed by Py, Pz orbital of Ga and In atom and Px orbital of S atom. Two dimensional charge density of the monolayer has been plotted to know about charge distribution. The different color shows the amount of charge accumulation as shown in the scale bar. It can be easily concluded that the maximum charge is accumulated on S atoms and less amount of charge is on Ga and In atoms. The more accumulation of charge on S atom can be understood by its electronegativity property. Sulfur is more electronegative than Ga and In atom and that causes more charge accumulation on it.
3.3. Effect of biaxial strain:
The effect of biaxial strain on the band structure of GaInS2 monolayer was studied. We have applied up to 10% tensile and compressive strain with an increment of 2% to the studied band structure. The band structure and projected density of states (PDOS) with 4% and 8% tensile strain and compressive strain (-4% and − 8%) are shown in Figs. 3 (a)-(d). At 4% tensile strain, the band remained indirect, as shown by the red arrow. The band gap was found to be decreasing to 0.98 eV. At 8% strain as shown in Fig. 3(b), the band gap decreases to 0.47 eV, which means phase transition from semiconductor to semimetal has occurred at 8% tensile strain. Compressive strain results indirect to direct type semiconducting properties and little increase in the band gap. At -4% strain, monolayer changes its semiconducting state from indirect to direct type. The VBM and CBM rest at Γ point, and the bandgap for the monolayer at that particular strain is 2.22 eV. The position of VBM remains at Γ point for further increase in compressive strain, but CBM shifted from high symmetric Γ to M point at -6% strain, and again CBM shifted from M to K point at -8% strain. The bandgap at -6% and − 8% strain is 2.08 eV and 1.73 eV, respectively.
3.4. Effect of the electric field:
A finite electric field is applied to the monolayer along the perpendicular direction of the surface (Z-direction) using the modern theory of polarization as implemented in QE. The effect of finite electric field on electronic band structure has been studied. Upon application of a finite electric field, the structural change is negligible. An external electric field is applied up to 4 V/nm and − 4 V/nm with an increment of 2 V/nm, which can be considered as a perturbation on a stable system or application of gate voltage on field effect transistor (FET) device. The field dependent modification of band structures is shown in Figs. 4 (a)-(d). For the 2 V/nm applied field, the band type remains the same, although the bandgap reduced to 1.54 eV. With 4 V/nm, the material becomes a direct band gap semiconductor with a gap of 1.03 eV. The negative electric field does not affect the band structure much, and the band gap also does not change significantly. At the − 2 eV/nm field, the band gap decreases slightly to 1.77 eV, while at -4 V/nm, the gap increases to 1.82 eV. Band structure remains indirect for both negative electric fields.
The variation of VBM and CBM with applied strain and finite electric field along the transverse direction is shown in Fig. 5 (a) and (b). At the positive strain region, the band gap decreases linearly with applied strain. Along the negative region non-linear variation of band gap has been occurred. The band gap increases linearly up to -4% compressive strain and then decreases linearly. We have divided the variation of VBM and CBM energy into four regions, namely A, B, C, and D. The VBM energy varies linearly with applied positive strain and at -2% strain, as shown in region A. As the negative strain increases, a jump in VBM energy has been observed, and again a linear variation with a different slope has been occurred (region B). In CBM, linear variation of energy was observed from − 4–10% strain (region C). A non-linear region (region D) was observed from − 6% to -10% applied strain. In the whole A region, the VBM rests between Γ and K, but at -4% strain, VBM shifts to Γ point. This is why a certain jump in VBM energy has occurred. In the whole B region, VBM rests at Γ point. Similarly, CBM rests at Γ point throughout the C region, which causes linear variation there. At -6%, CBM shifted to M point, and at -8% of strain again shifted to K point, resulting in a non-linear region. To understand the linear trend of the states, PDOS has been studied as shown in Fig. 3 (a)-(d). The region A is dominated by the Px orbital of the S atom, while the Pz orbital of the S atom dominates region B. Region C is dominated by as expected Py and Pz orbital of Ga atom as CBM rests at same Γ point. In the region D, a change of high symmetric point from M to K has been occurred and caused non-linear characteristics. At -6% strain, CBM energy moves to the M point, and for − 8% and − 10% strain, it remains at K point.
External electric field-dependent variation of VBM and CBM has three different regions E-F. The slope of linearly variant F and G are almost the same, which is responsible for not changing the bandgap. At 4 V/nm electric field, VBM changes to Γ point and causes a band gap reduction.
3.5. Optical properties:
The optical properties of the GaInS2 monolayer have been calculated along the transverse direction of the material. The real part of the dielectric function (ϵ1), the imaginary part of the dielectric function (ϵ2), and absorption coefficient (α) varying with photon energy is shown in Figs. 6 (a)-(c). The zero-point value of ϵ1 for the unstrained structure is 3.16. Although with 4% tensile and compressive strain, the value does not change remarkably but at 8% strain, the value increases to 3.39. The highest value peak arises at 3.49 eV with a value of 5.28 for the unstrained layer. The peak value increases slightly at -4% strain but again decreases for − 8% applied strain. The lowest peak has been observed at 8% tensile strain with a value of 4.37 at 3.59 eV. Some negative value for ϵ1 has been observed with the highest value of -0.81 for − 4% strain. All the ϵ1 values were found almost stable on a high photon energy range. The value of ϵ2 has started to increase at the visible energy region at 1.25 eV for all strained and relaxed layers. Only for 8% applied strain, it starts increasing at 0.3 eV. The peak value for the unstrained layer is 4.6 at 4.25 eV. The peak value is highest for 4% applied strain and decreases for compressive strain. The monolayer has been seen to be a very good absorber of light with high absorption coefficient. Although the strain does not have much effect on peak position, it changes the absorption value. The absorption starts in the visible region at 1.3 eV for unstrained monolayer, which starts decreasing for all strains except − 4% compressive strain. The absorption started at 0.65 eV for 8% strain. The first peak arises at 4.84 eV for the relaxed system, and the highest peak arises at 7.09 eV with a value of 8.54×105 cm− 1. The absorption decreases with tensile strain, but the first peak is getting prominent with tensile strain, while it started vanishing with applied compressive strain. The material shows absorption near visible and ultraviolet region.