Single-Layer GaInSe3: Promising Water-Splitting Photocatalyst with Solar Conversion Efficiency over 30% from Theoretical Calculations

Hydrogen energy from solar water-splitting is known as an ideal method with which to address the energy crisis and global environmental pollution. Herein, the first-principles calculations are carried out to study the photocatalytic water-splitting performance of single-layer GaInSe3 under biaxial strains from −2% to +2%. Calculations reveal that single-layer GaInSe3 under various biaxial strains has electronic bandgaps ranging from 1.11 to 1.28 eV under biaxial strain from −2% to +2%, as well as a completely separated valence band maximum and conduction band minimum. Meanwhile, the appropriate band edges for water-splitting and visible optical absorption up to ~3 × 105 cm−1 are obtained under biaxial strains from −2% to 0%. More impressively, the solar conversion efficiency of single-layer GaInSe3 under biaxial strains from −2% to 0% reaches over 30%. The OER of unstrained single-layer GaInSe3 can proceed without co-catalysts. These demonstrate that single-layer GaInSe3 is a viable material for solar water-splitting.


Introduction
With the development of human society and economy, the demand and consumption of energy have become unprecedented, and the environmental problems caused by the exploitation and consumption of traditional energy are prominently increasing.Therefore, it is urgent to seek a new source of energy which does not produce pollution in the production and utilization processes.Hydrogen with high calorific value and pollution-free combustion has become the first choice for future energy.Hydrogen energy from efficient and environmentally friendly solar water-splitting has become an ideal method by which to solve our energy-related problems.During the early years of this research, semiconductor oxides such as TiO 2 , ZnO, and SnO 2 were focused on [1][2][3].However, the solar-to-hydrogen (STH) conversions of these semiconductor oxides are very limited due to the high recombination rates of charge carriers and low sunlight absorption [4,5].Hence, reducing the recombination rates of carriers and enhancing solar absorption are two main approaches to improving STH conversions.As compared with traditional semiconductor oxides, two-dimensional (2D) materials possess low carrier recombination rates due to shorter carrier migration distances [6][7][8].Meanwhile, for 2D materials, since the active potential is rich, the performance of sunlight absorption is strong, and the structural characteristics (such as planar, curved, vertically symmetric, or asymmetrical structures) are diverse, two-dimensional layered materials have been rapidly developed in photocatalytic water-splitting for hydrogen production [6][7][8].Since solar absorption is directly related to the bandgap, reducing the bandgap is an effective approach to enhancing optical absorption.After the inclusion of electric dipoles (fields), the bandgaps of 2D photocatalysts may decrease to ~0.5 eV [9].Hence, the inside electric field not only enhances the optical absorption but also effectively reduces the recombination of carriers [10,11].
Single-layer M 2 X 3 (M = Ga, In; X = S, Se, Te) are a typical group of 2D semiconductors with inside electric fields [12,13].In 2 X 3 (X = S, Se, Te) is an indirect bandgap semiconductor with good performance in photocatalytic water-splitting.Single-layer Ga 2 X 3 (X = S, Se, Te) has better performance of in water-splitting and higher hydrogen production.It has been found that single-layer M 2 X 3 (M = Ga, In; X = S, Se, Te) has excellent solar water-splitting performance owing to separated charge carriers [10,11].After the anions in single-layer M 2 X 3 are replaced, the STH conversion is enhanced [9].Moreover, the bandgap, band edge position, and sunlight absorption of single-layer chalcogenide can be adjusted via biaxial strain, thereby improving the solar energy conversion efficiency [1][2][3].In recent years, due to the successful synthesis of vertically asymmetric monolayers Janus MoSSe [14], Janus 2D materials have become a new research hotspot in the field of photocatalysis.The structural, electronic, and transport properties of single-layer GaInX 3 (X = S, Se, Te) were predicted via first-principles calculations [15].Single-layer GaInX 3 were found to have structural stability, highly directional isotropic elasticity, a distinct vacuum level difference, and highly directional isotropic mobility; hence, they are candidates for applications in nanoelectronic devices [15].In particular, we should note that a direct bandgap of 1.20 eV for single-layer GaInSe 3 indicates its visible and infrared light absorption [15].Moreover, as a sulfur-free compound, single-layer GaInSe 3 is environmentally friendly.Consequently, it would be meaningful to investigate the solar water-splitting performance of single-layer GaInSe 3 .
In this work, the solar water-splitting property of single-layer GaInSe 3 is calculated by the first-principles method.Since water-splitting can be effectively affected via biaxial strain [16,17], we also apply biaxial strains from −2% to 2% to tune the solar water-splitting property.Theoretical calculations suggest that under biaxial strains from −2% to 0%, singlelayer GaInSe 3 has appropriate band edges for water-splitting and visible optical absorption up to ~3 × 10 5 cm −1 .Impressively, the STH efficiency of single-layer GaInSe 3 under biaxial strains from −2% to 0% surpasses 30%, demonstrating the potential applications of single-layer GaInSe 3 as a solar water-splitting material.

Computational Details
First-principles calculations are performed within the framework of the density functional theory (DFT) via the Vienna ab initio simulation package (VASP) [18][19][20].The interaction between the core and valence electrons is treated via the projector-augmented wave (PAW) method [21,22], and the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE) is used to describe the exchange and correlation potential [23].The k-point mesh size with 11 × 11 × 1 and a cutoff energy of 500 eV are employed to optimize the structural parameter.The vacuum thickness of ~20 Å is added to avoid the interlayer reaction between repeated images, and the van der Waals (vdW) interlayer interaction is described using the semi-empirical DFT-D3 method [24].The energy and force convergences are set as 10 −7 eV and 0.01 eV/Å, respectively.Because the GGA-PBE usually underestimates the band gaps, the HSE06 method [25][26][27] and the G 0 W 0 method [28] are both used to calculate band structures.Phonon dispersion curves are computed within a 4 × 4 × 1 supercell approach using the density functional perturbation theory [29] and extracted via the Phonopy code [30,31].In ab initio molecular dynamics (AIMD) simulations, the initial configuration within the 3 × 3 × 1 supercell is annealed at 300 K in the NVT ensemble, and each AIMD simulation lasts for 8 ps, with a time step of 1 fs.The elastic constants are calculated using the finite different method [29].
To obtain the optical absorption, the 8 × 8 × 1 k-point grid is sampled in the G 0 W 0 + BSE calculations [32][33][34][35].(1) GGA-PBE calculations with an energy convergence criterion of 1 × 10 −8 eV are performed; (2) GGA-PBE calculations are restarted to allow for full optical transitions, where ~200 empty bands are added; (3) G 0 W 0 calculations are Molecules 2023, 28, 6858 3 of 12 carried out to obtain quasi-particle excitations, where an energy cutoff of 150 eV for the response functions, the spectral method, and 72 frequency points are adopted; (4) The Bethe-Salpeter equation is solved, where 22 highest occupied valence bands and 22 lowest unoccupied conduction bands are included.

Structural and Electronic Properties
Single-layer GaInSe 3 displays a 2D hexagonal unit cell with an in-plane constant of a = 3.95 Å (Figure 1a).Single-layer GaInSe 3 is stacked by Se-In-Se-Ga-Se atomic layers along the z-direction (Figure 1a), with an effective thickness of ~1 nm (Figure S1).Due to the asymmetric structure, single-layer GaInSe 3 exhibits an inside electric dipole (field) along the z-direction.The calculated electric dipoles are 9.36, 9.29, 9.26, 9.07, and 9.00 (10 −2 e•Å) for −2%, −1%, 0%, +1%, and +2% biaxial strains, respectively.Hence, the inside dipole will play an important role in separating the generated holes and electrons.The large inside electric dipole produces a significant vacuum level difference ∆Φ between the bottom Ga-Se layer and the top Se-In-Se layer (Figure 1b and Figure S2 and Table S1).
a time step of 1 fs.The elastic constants are calculated using the finite different method [29].
To obtain the optical absorption, the 8 × 8 × 1 k-point grid is sampled in the G0W0 + BSE calculations [32][33][34][35].(1) GGA-PBE calculations with an energy convergence criterion of 1 × 10 −8 eV are performed; (2) GGA-PBE calculations are restarted to allow for full optical transitions, where ~200 empty bands are added; (3) G0W0 calculations are carried out to obtain quasi-particle excitations, where an energy cutoff of 150 eV for the response functions, the spectral method, and 72 frequency points are adopted; (4) The Bethe-Salpeter equation is solved, where 22 highest occupied valence bands and 22 lowest unoccupied conduction bands are included.

Structural and Electronic Properties
Single-layer GaInSe3 displays a 2D hexagonal unit cell with an in-plane constant of a = 3.95 Å (Figure 1a).Single-layer GaInSe3 is stacked by Se-In-Se-Ga-Se atomic layers along the z-direction (Figure 1a), with an effective thickness of ~1 nm (Figure S1).Due to the asymmetric structure, single-layer GaInSe3 exhibits an inside electric dipole (field) along the z-direction.The calculated electric dipoles are 9.36, 9.29, 9.26, 9.07, and 9.00 (10 −2 e • Å) for −2%, −1%, 0%, +1%, and +2% biaxial strains, respectively.Hence, the inside dipole will play an important role in separating the generated holes and electrons.The large inside electric dipole produces a significant vacuum level difference ∆ between the bottom Ga-Se layer and the top Se-In-Se layer (Figures 1b and S2, and Table S1).The stability of unstrained single-layer GaInSe3 has been investigated [15], and here we study its stability under biaxial strains.From −2% to 2% biaxial strains, only very small imaginary frequencies appear near the Γ point, and they could be eliminated gradually by enlarging the supercell [36], expressing good dynamic stability (Figure S3).Single-layer GaInSe3 under biaxial strains from −2 to +2% exhibits free energy fluctuating around −160 eV and maintains the integrity of structural frames throughout the simulation process (Figures S4 and S5), proving thermal stability at 300 K.The theoretical elastic constants Cij (Table S2) satisfy the Born-Huang mechanical stability criterion of lattice The stability of unstrained single-layer GaInSe 3 has been investigated [15], and here we study its stability under biaxial strains.From −2% to 2% biaxial strains, only very small imaginary frequencies appear near the Γ point, and they could be eliminated gradually by enlarging the supercell [36], expressing good dynamic stability (Figure S3).Single-layer GaInSe 3 under biaxial strains from −2 to +2% exhibits free energy fluctuating around −160 eV and maintains the integrity of structural frames throughout the simulation process (Figures S4 and S5), proving thermal stability at 300 K.The theoretical elastic constants C ij (Table S2) satisfy the Born-Huang mechanical stability criterion of lattice dynamics [37-39]: C 11 > 0 and C 11 − C 12 > 0 suggesting mechanical stability.What is more, the total energy under the considered strain is very close to that at the ground state (Figure S6).Therefore, the dynamic stability, mechanical stability, and thermal stability under the considered biaxial strains have shown that single-layer GaInSe 3 can be realized in experiments.
A quasi-direct energy gap appears for single-layer GaInSe 3 under biaxial strains from −2% to +2%.Bandgaps are 1.28 eV for −2%, 1.25 eV for −1%, 1.21 eV for 0%, 1.16 eV for +1%, and 1.11 eV for +2% strain (Figure 2a-d and Figure S7).Notably, the moderate bandgaps of single-layer GaInSe 3 are very helpful for visible light absorption.Furthermore, due to the vacuum level difference between the top and bottom surfaces, the VBM of unstrained single-layer GaInSe 3 is located at the top Se-In-Se trilayer-more precisely, at the highest Se atomic layer.Its CBM is mainly seated in the bottom Ga-Se bilayer (Figure 1b).Thus, the CBM and VBM are spatially separated, thus decreasing the electron-hole binding rate and improving the photocatalytic performance.The separated CBM and VBM also appear in single-layer GaInSe 3 under biaxial strains from −2% to 2% (Figure S8). the total energy under the considered strain is very close to that at the ground state (Figure S6).Therefore, the dynamic stability, mechanical stability, and thermal stability under the considered biaxial strains have shown that single-layer GaInSe3 can be realized in experiments.
A quasi-direct energy gap appears for single-layer GaInSe3 under biaxial strains from −2% to +2%.Bandgaps are 1.28 eV for −2%, 1.25 eV for −1%, 1.21 eV for 0%, 1.16 eV for +1%, and 1.11 eV for +2% strain (Figures 2a-d and S7).Notably, the moderate bandgaps of single-layer GaInSe3 are very helpful for visible light absorption.Furthermore, due to the vacuum level difference between the top and bottom surfaces, the VBM of unstrained single-layer GaInSe3 is located at the top Se-In-Se trilayer-more precisely, at the highest Se atomic layer.Its CBM is mainly seated in the bottom Ga-Se bilayer (Figure 1b).Thus, the CBM and VBM are spatially separated, thus decreasing the electron-hole binding rate and improving the photocatalytic performance.The separated CBM and VBM also appear in single-layer GaInSe3 under biaxial strains from −2% to 2% (Figure S8).In addition, the spin-orbital coupling (SOC) has little influence on the electronic property of single-layer GaInSe3 (Figure S9).The bandgaps at the G0W0 level are presented in Figure S10.Since the bandgaps of 2D materials are underestimated by PBE and overestimated by G0W0 [16], only the HSE06 bandgaps are focused on herein.In addition, the spin-orbital coupling (SOC) has little influence on the electronic property of single-layer GaInSe 3 (Figure S9).The bandgaps at the G 0 W 0 level are presented in Figure S10.Since the bandgaps of 2D materials are underestimated by PBE and overestimated by G 0 W 0 [16], only the HSE06 bandgaps are focused on herein.

Band Alignment for Water-Splitting
As for water-splitting by sunlight, the most important prerequisite is that the aligned CBM level is higher than the H + /H 2 reduction potential (E H + /H 2 ), while the aligned VBM level is below the O 2 /H 2 O oxidation potential (E O 2 /H 2 O ).Herein, the band edges are aligned using the method proposed by Toroker et al. [40] and revised by Yang et al. [10].Considering the charge location of CBM and VBM (Figure 1b), the hydrogen evolution reactions (HERs) of single-layer GaInSe 3 take place on the Ga atomic layer, and the oxygen evolution reactions (OERs) mainly occur on the highest Se atomic layer.From −2% to +2% biaxial strains, the band edges of single-layer GaInSe 3 are favorable for OERs (Figures 2a-d and S7).However, the band edges under +1% and +2% tensile biaxial strains are unfavorable for HERs.As a result, single-layer GaInSe 3 under biaxial strains from −2% to 0% can accomplish both HERs and OERs.
The overpotentials, including χ(H 2 ) and χ(O 2 ), which, respectively, scale the watersplitting abilities of HERs and OERs, are further studied (Table 1).The χ(H 2 ) (χ(O 2 )) is the potential difference between the aligned CBM (VBM) level and the H + /H 2 (O 2 /H 2 O) potential.Under biaxial strains of −2% and −1%, the χ(H 2 ) values are larger than 0.2 eV, and the χ(O 2 ) values are larger than 0.6 eV, indicating good HER and OER ability [10].In addition, the potential energies of photogenerated carriers (U e and U h ) are also calculated to reveal the HER and OER abilities [41,42], also suggesting good HER and OER ability for single-layer GaInSe 3 at −2% and −1% biaxial strain.

Optical Absorption and Exciton Binding Energy
For water-splitting, the photocatalysts should absorb as much as solar energy.Herein, the G 0 W 0 -BSE method is applied to study the absorption coefficient [32][33][34][35]43].Due to the existence of a vacuum, the effective thickness is considered [44].Calculations reveal that the first absorption peaks of GaInSe 3 under biaxial strains of −2%, −1%, and 0% are located at 1.57, 1.50, and 1.39 eV, respectively (Figure 3a).From −2% to 0% biaxial strains, single-layer GaInSe 3 exhibits multiple visible optical absorption peaks, and the maximum optical absorption reaches up to ~3 × 10 5 cm −1 , apparently stronger than that of other typical 2D photocatalysts such In 2 Se 3 [45].It is not difficult to find that the maximum optical absorption is increasing with the decreasing compressive biaxial strain.Moreover, the optical absorption spectra are red-shifted with the decreasing compressive biaxial strain.On the other hand, the square transition dipole matrix elements (P 2 ) between the top VB and the bottom CB are calculated via HSE06 [46].From −2% to 0% biaxial strains, the maximum P 2 appears near the Γ point and approaches 20 Debye 2 (Figure 3b).The allowed optical transition further promises infrared and visible light absorption.
Water-splitting reactions require separated photogenerated electrons and holes.The exciton binding energy is an important parameter by which to measure the separation efficiency of charge carriers.Considering the quasi-direct bandgap of single-layer GaInSe 3 , the exciton binding energy E b is the energy difference between the first optical bandgaps (Figure 3a) and quasi-direct bandgaps (Figure S10).The quasi-direct bandgaps of single-layer GaInSe 3 under biaxial strains from −2%, −1%, and 0% are 2.03, 2.07, and 2.13 eV, respectively.The corresponding E b values are 0.46, 0.57, and 0.74 eV, and lower than those of 2D photocatalysts, e.g., In 2 Se 3 (0.69 eV) [10] and WSSe (0.82 eV) [17].First, the relatively smaller exciton binding energies could be attributed to the separated CBM and VBM.Secondly, the inside electric field could play a big role in reducing the exciton binding energies.

Transport Mobility
The recombination rate of charge carriers is further investigated by including carrier mobility.Expressly, higher mobility means a lower recombination possibility.Herein, The carrier mobility is calculated using the deformation potential (DP) theory [47,48]: Here, e is the electron charge, h is the reduced Planck constant, C 2D is the elastic constant, k B is the Boltzmann constant, and T is the room temperature (300 K). m * is the effective mass and can be written as 1

Transport Mobility
The recombination rate of charge carriers is further investigated by including carrier mobility.Expressly, higher mobility means a lower recombination possibility.Herein, The carrier mobility is calculated using the deformation potential (DP) theory [47,48]: Here, e is the electron charge, ℏ is the reduced Planck constant, C2D is the elastic constant,  is the Boltzmann constant, and T is the room temperature (300 K).  * is the effective mass and can be written as * = ℏ ( ) .The average effective mass  is expressed as  =  *  * .The symbol  represents the deformation potential constant.The fittings for effective masses  * and deformation potential constants  are listed in Figures S11-S16.Herein, the  values are obtained with the consideration of VBM and CBM distribution (Figure 1b).Specifically, the  value of generated holes is taken as the slope of the linear fitting between ( −  ) and strain ε.Ed of generated electrons is taken as the slope of the linear fitting between ( −  ) and ε.The calculated effective masses  * , elastic constants  , and deformation potential constants  are summarized in Table 2.
For unstrained single-layer GaInSe3, the electron effective masses (0.15 me) and elastic constants (78.36 N/m) along the x and y directions are both isotropic.Due to the small electron-effective masses, a relatively large mobility (~1250 cm 2 •V −1 •s −1 ) of photogenerated electrons is obtained.For comparison, recently discovered 2D photocatalysts display electron mobility: 1049 cm 2 •V −1 •s −1 for GeS [49] and 601 cm 2 •V −1 •s −1 for CoGeSe3 [50].Moreover, apparent hole mobility anisotropy between the x and y directions appears owing to various effective masses and deformation potential constants.Another promi-  For unstrained single-layer GaInSe 3 , the electron effective masses (0.15 m e ) and elastic constants (78.36 N/m) along the x and y directions are both isotropic.Due to the small electron-effective masses, a relatively large mobility (~1250 cm 2 •V −1 •s −1 ) of photogenerated electrons is obtained.For comparison, recently discovered 2D photocatalysts display electron mobility: 1049 cm 2 •V −1 •s −1 for GeS [49] and 601 cm 2 •V −1 •s −1 for CoGeSe 3 [50].Moreover, apparent hole mobility anisotropy between the x and y directions appears owing to various effective masses and deformation potential constants.Another prominent characteristic is that the generated electrons run much faster than the generated holes, which promises an effective separation of photogenerated electrons and holes.Under −2% and −1% biaxial strains, effective separation of photogenerated carriers inside single-layer GaInSe 3 still exists, and then ensures the occurrence of photocatalytic reactions.

OER and HER
Although single-layer GaInSe 3 exhibits a suitable band edge for redox reaction under biaxial strains from −2% to 0%, it is necessary to calculate the Gibb free energy in OER and HER.Combined with the analysis of the charge density of VBM and CBM, the HER will take place on the bottom Ga-Se bilayer, and the OER will take place on the highest Se atomic layer.The OER is divided into four steps, as follows in Equations ( 1)-( 4) [2,51]: The two steps of HER are given by Equations ( 5) and ( 6) [2,51]: where * indicates the adsorbed material (single-layer GaInSe 3 ), and OH * , O * , OOH * , and H * represent the adsorbed intermediates.For all the calculations, spin polarization is taken into account.By considering the zero-point energy and entropy corrections, the expression of ∆G can be written as ∆G = ∆E ads + ∆E ZPE -T∆S, where ∆E ads is the adsorption energy, and ∆E ZPE and ∆S are the difference of zero-point energy and entropy difference between the adsorbed state and the gas phase, respectively.T represents the room temperature of 300 K.
For each reaction of oxidation generation, ∆G can be written as follows in Equations ( 7)- (10) [2,51]: where ∆G U (∆G U = −eU) denotes extra potential bias provided by an electron in the electrode, and U is the electrode potential relative to the standard hydrogen electrode (SHE).∆G pH ∆G pH = k B T × ln10 × pH represents the free energy contributed in different pH.
Because of the required large computational resources, only the OER and HER of unstrained single-layer GaInSe 3 are studied.The optimized configurations of the OH * , O * , and OOH * intermediates are given (Figure S17).The formation of OH * , O * , and OOH * are all endothermic in the absence of solar light (Figure 4a).The third step is a rate-limiting step involving the OOH * formation and has a Gibbs free energy change of 2.25 eV.Therefore, the minimum external potential for OER converted into exothermic heat is 2.25 V.As shown in Table 1, for unstrained single-layer GaInSe 3 , the external electric potential supplied by the photogenerated hole is 2.50 V U , h = U h /e ), indicating that OER can proceed smoothly without a co-catalyst.
where ∆ (∆ = −) denotes extra potential bias provided by an electron in the electrode, and U is the electrode potential relative to the standard hydrogen electrode (SHE).∆ (∆ =   × 10 × pH) represents the free energy contributed in different pH.
Because of the required large computational resources, only the OER and HER of unstrained single-layer GaInSe3 are studied.The optimized configurations of the OH * , O * , and OOH * intermediates are given (Figure S17).The formation of OH * , O * , and OOH * are all endothermic in the absence of solar light (Figure 4a).The third step is a rate-limiting step involving the OOH * formation and has a Gibbs free energy change of 2.25 eV.Therefore, the minimum external potential for OER converted into exothermic heat is 2.25 V.As shown in Table 1, for unstrained single-layer GaInSe3, the external electric potential supplied by the photogenerated hole is 2.50 V ( , =  /), indicating that OER can proceed smoothly without a co-catalyst.On the other hand, the Gibbs free energy profile of HER is displayed in Figure 4b.The optimized configurations of the H * intermediate are given (Figure S18), where the HER active site is the Ga atomic layer.In the absence of light irradiation, the first and second steps are endothermic and exothermic with ΔG = 0.88 eV, respectively.Therefore, the minimum external potential for a successful HER is 0.88 V.However, the external electric potential of unstrained single-layer GaInSe3 (Table 1) supplied by the photogenerated electrons is 0.12 V.In this condition, the first step involving the H* formation is also endothermic, with ΔG = 0.76 eV, indicating that HER cannot proceed successfully without a cocatalyst.Fortunately, in previous experiments, the HER energy barrier could be lowered to less than 0.1 eV [52].On the other hand, the Gibbs free energy profile of HER is displayed in Figure 4b.The optimized configurations of the H * intermediate are given (Figure S18), where the HER active site is the Ga atomic layer.In the absence of light irradiation, the first and second steps are endothermic and exothermic with ∆G = 0.88 eV, respectively.Therefore, the minimum external potential for a successful HER is 0.88 V.However, the external electric potential of unstrained single-layer GaInSe 3 (Table 1) supplied by the photogenerated electrons is 0.12 V.In this condition, the first step involving the H* formation is also endothermic, with ∆G = 0.76 eV, indicating that HER cannot proceed successfully without a cocatalyst.Fortunately, in previous experiments, the HER energy barrier could be lowered to less than 0.1 eV [52].

Solar-to-Hydrogen Efficiency
Here, we investigate the solar conversion efficiency of single-layer GaInSe 3 in water-spitting reactions.Generally, the η STH value is obtained by timing the light absorption efficiency ( η abs ) and the carrier utilization efficiency (η cu ).The η abs value is calculated via Equation (11): where P( w) is the AM1.5G solar energy flux at the photon energy w, and E g is the bandgap by HSE06 (Figures 2 and S6).Due to the narrower bandgap, the η abs values of single-layer GaInSe 3 under biaxial strains from −2% to 0% exceed 70%.The η cu is calculated via Equation ( 12): where ∆G = 1.23 eV is the potential difference for photocatalytic water-splitting.The symbol E is the actual energy barrier for photocatalytic water-splitting and is determined via Equation ( 13): ( The overpotentials of χ(H 2 ) and χ(O 2 ) values are listed in Table 1.Furthermore, the η cu values from −2% to 0% are found to surpass 70%.Herein, the STH efficiencies ( η STH ) under biaxial strains of −2%, −1%, and 0% via Equation ( 14) are 44.18%,45.85%, and 44.00%, respectively.
The η STH is further corrected by the inside electric field via Equation ( 15 where ∆Φ is the potential difference between the vacuum level at the top and bottom surfaces (Table 1).Very excitingly, the corrected STH efficiency (η , STH ) of single-layer GaInSe 3 under biaxial strains of −2%, −1%, and 0% reaches 31.20%,32.50%, and 31.29%,respectively, far surpassing the commercial standard (10%) [10] and showing glorious prospects for commercial application (Table 3).It should be pointed out that we assume the external quantum efficiency of 100% for the overall water-splitting reaction; that is, the obtained STH efficiency is a theoretical limitation.Lower SHT efficiency was obtained in the experiments.Specifically, the low U e of single-layer GaInSe 3 under zero strain suggests possible photo corrosion, which will reduce the STH efficiency.

Conclusions
The solar water-splitting properties of single-layer GaInSe 3 have been studied.Under biaxial strains from −2% to +2%, single-layer GaInSe 3 exhibits stable structures and moderate bandgaps from 1.11 to 1.28 eV.Under biaxial strains from −2% to 0%, single-layer GaInSe 3 holds suitable band edges for photocatalytic water-splitting at pH = 0, and the strong visible optical absorption is up to ~3 × 10 5 cm −1 .Meanwhile, the charge carriers can be effectively separated due to the strong inside electric field and high electron mobility.More attractively, the solar energy conversion efficiency surpasses 30%.The OER of

Figure 1 .
Figure 1.(a) The top and side views, where the rhombus denotes the primitive cell and the rectangle represents the orthogonal supercell.(b) The vacuum level difference of unstrained single-layer GaInSe3, where the inset is the charge distribution of CBM (red) and VBM (blue).The red dashed horizontal lines are auxiliary lines.The solid blue curve shows the change in vacuum level.

Figure 1 .
Figure 1.(a) The top and side views, where the rhombus denotes the primitive cell and the rectangle represents the orthogonal supercell.(b) The vacuum level difference of unstrained single-layer GaInSe 3 , where the inset is the charge distribution of CBM (red) and VBM (blue).The red dashed horizontal lines are auxiliary lines.The solid blue curve shows the change in vacuum level.

Figure 2 .
Figure 2. The electronic band structure (Left) and water-splitting property (Right) of single-layer GaInSe3 under biaxial strains of (a) −2%, (b) −1%, (c) 0%, and (d) +1% obtained via HSE06.The upper and lower red columns denote the CBM and the VBM, respectively.The blue and red lines represent the redox potentials of H + /H2 and H2O/O2, respectively.

Figure 2 .
Figure 2. The electronic band structure (Left) and water-splitting property (Right) of single-layer GaInSe 3 under biaxial strains of (a) −2%, (b) −1%, (c) 0%, and (d) +1% obtained via HSE06.The upper and lower red columns denote the CBM and the VBM, respectively.The blue and red lines represent the redox potentials of H + /H 2 and H 2 O/O 2 , respectively.

Figure 4 .
Figure 4. (a) The Gibbs free energy change of OER on the Se atomic plane; (b) that of HER on the bottom Ga atomic layer of unstrained single-layer GaInSe3 at pH = 0.The * indicates the adsorbed material.

Figure 4 .
Figure 4. (a) The Gibbs free energy change of OER on the Se atomic plane; (b) that of HER on the bottom Ga atomic layer of unstrained single-layer GaInSe 3 at pH = 0.The * indicates the adsorbed material.