Strain and Electric Field Modulated Indirect to Direct Band Transition of Monolayer GaInS2

Strain and electric eld dependent electronic and optical properties have been calculated using density functional theory (DFT) and time-dependent DFT (TD-DFT) for GaInS 2 monolayer. GaInS 2 monolayer shows an indirect band gap of 1.79 eV where valance band maxima (VBM) and conduction band maxima (CBM) rest between K and Γ point and at Γ point, respectively. Under a particular tensile strain (8%), a phase change from semiconductor to semimetal has been noticed. While at 4% compressive strain, the material changes from indirect to direct band gap of 2.22 eV having VBM and CBM at Γ point. With further increase in compressive strain, CBM shifted from Γ to M point, which leads to an indirect band gap again. The electric eld also affects the band structure of monolayer GaInS 2 and shows the transition between indirect to direct band gap at positive electric eld of 4 V/nm, which acts normal to the surface. The strain-dependent optical properties are also calculated, which suggests that the absorption coecient increases with compressive strain.


Introduction
The discovery of graphene 1,2 , a two-dimensional (2D) material, has made researchers curious about further research on new low-dimensional materials due to their magni cent physical, electronic, and optical properties. Owing to their excellent electronic, optical, mechanical, thermoelectric properties 2D materials can be used as nanoelectronic and optical devices [3][4][5][6][7] . Enormous research has been gone on developing the properties, for that 2D materials such as arsenic sul de 8,9 silicene 10,11 black phosphorus are being studied. Along with these materials, other novel 2D materials like transition metal dichalcogenides (TMDC) 12,13 , metal diiodides 14 , metal chalcogenides 15,16 are now at the peak of interest for the researchers. Low dimensional TMDCs were used for various applications such as transistors, gas sensors, photodetectors, solar cells, and light-emitting diode [17][18][19][20][21][22] . Metal diiodides monolayers with hexagonal honeycomb structures can be used as thermoelectric devices due to their ultra-low lattice thermal conductivity 23,24 . Metal monochalcogenides are being studied for their recent application in nanoelectronic devices. However, band modulation is required for various device applications, which can be done by applying strain or external electric eld on 2D materials. By strain engineering, enhancement of materials properties has been reported 25 .
Two-dimensional group-III monochalcogenides are being studied because of their exceptional optoelectronic properties, and these properties can be modi ed by applying strain and external electric elds. Gallium and indium chalcogenides are the well known 2D materials with indirect band gap and are layered hexagonal structures 26 . Among them, gallium chalcogenides are being focused on due to their possibilities to be utilized in experiments. Experimental synthesis of the monolayers has been done to study their properties 27,28 . Along with these materials, different types of designed materials are required for the development of nanoscale devices. Because of that, different types of material structures have been designed, Janus structure is one of them. It is well known that optical and electronic properties depend on the structure and symmetry of the materials. Janus structure gives different and improved results because of its symmetry breaking to meet the requirement of applications. Bui et al. designed Janus Ga 2 SSe, Ga 2 STe, Ga 2 SeTe monolayer and shows its optical and electronic properties 29 . In this study, they showed that Ga 2 SSe has indirect band gap while Ga 2 STe and Ga 2 SeTe show direct band gap.
The optical absorptions peaks are observed in the UV region.
In this work, we have designed Janus GaInS 2 monolayer by replacing one Ga atom with In atom from the Ga 2 S 2 monolayer structure. After atomic relaxation, its electronic and optical properties have been studied. Unstrained structure exhibits an indirect band gap of 1.79 eV. Application of biaxial tensile strain results a semiconductor to semimetal phase change while compressive strain causes indirect to direct band gap for the material. The external electric eld also causes indirect to direct band gap transition for the material. It shows a high absorption coe cient starting at the visible range and has a peak at the UV region. This work indicates that the material can be used in optoelectronic device as a photovoltaic absorber layer, UV photodetection and many more applications.

Computational Details
First principle calculations of monolayer GaInS 2 were performed using DFT. Structural optimization, electronic properties were calculated using ultrasoft pseudopotential 30  Where, ε 1 , ε 2 , ω, C are the real and imaginary functions of dielectric, frequency of incident light, and speed of light respectively.

Structural details:
GaInS 2 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 Ga 2 S 2, which is a layered structure having broken inversion and mirror symmetry. The GaInS 2 structure is modeled from the structure Ga 2 S 2 by replacing one Ga atom with In atom. This Janus structure belongs to P3m1 (C 3v ) 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 previously 35 . 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.

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 kpath K-Γ-M-K of hexagonal Brillouin zone. The GaInS 2 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 P x orbital and some of P y and P z orbitals of S atom and minor contribution of P x orbitals of Ga and In atom.
The CBM contributed by P y , P z orbital of Ga and In atom and P x 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.

Effect of biaxial strain:
The effect of biaxial strain on the band structure of GaInS 2 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.

Effect of the electric eld:
A nite electric eld is applied to the monolayer along the perpendicular direction of the surface (Zdirection) using the modern theory of polarization as implemented in QE. The effect of nite electric eld on electronic band structure has been studied. Upon application of a nite electric eld, the structural change is negligible. An external electric eld 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 The variation of VBM and CBM with applied strain and nite electric eld 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 P x orbital of the S atom, while the Pz orbital of the S atom dominates region B. Region C is dominated by as expected P y and P z 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 eld-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 eld, VBM changes to Γ point and causes a band gap reduction.

Optical properties:
The optical properties of the GaInS 2 monolayer have been calculated along the transverse direction of the material. The real part of the dielectric function (ϵ 1 ), the imaginary part of the dielectric function (ϵ 2 ), and absorption coe cient (α) varying with photon energy is shown in Figs. 6 (a)-(c). The zero-point value of ϵ 1 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 ϵ 1 has been observed with the highest value of -0.81 for − 4% strain. All the ϵ 1 values were found almost stable on a high photon energy range. The value of ϵ 2 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 coe cient. 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 rst peak arises at 4.84 eV for the relaxed system, and the highest peak arises at 7.09 eV with a value of 8.54×10 5 cm − 1 . The absorption decreases with tensile strain, but the rst peak is getting prominent with tensile strain, while it started vanishing with applied compressive strain. The material shows absorption near visible and ultraviolet region.

Conclusion
We have studied the electronic and optical properties of the GaInS 2 monolayer. GaInS 2 monolayer has an indirect band gap which changes to direct band gap with applied 4% compressive strain. 8% tensile strain causes a change of the state of the material from semiconducting to semi-metallic state. An external electric eld with a value of 4 V/nm, applied normal to the plane, also changes the semiconducting property from indirect to direct band gap type. However, the band gap does not change effectively with the electric eld. The Janus monolayer also possesses exciting optical properties with a high value of absorption coe cient and dielectric function. Figure 1 The side view and top view of GaInS2 monolayer 4×4×1 supercell. It forms a hexagonal honeycomb structure.