Photocatalytic water splitting properties of GeC/InS van der Waals heterostructure: first-principles calculations

Before studying the GeC/InS heterostructure, we first discussed the structure and properties of single-layer GeC and InS in detail to verify the accuracy of the calculation method we used. The lattice constants of unit cells GeC and InS are 3.23 eV and 3.91 Å respectively after optimization, which is consistent with previous research results [44–46]. In order to meet the requirements of the lattice mismatch degree of the heterostructure, we expanded the unit cell GeC by 2 × 2 and InS by √3 × √3. Figures 1(a) and (b) show the fully optimized geometries of a 2 × 2 × 1 GeC monolayer and a √3 × √3 × 1 InS monolayer, with lattice constants of 6.55 Å and 6.77 Å, respectively, which are honeycomb structures. During the structure optimization process, the Ge–C bond length was tuned to 1.88 Å for GeC and 2.56 Å for In–S and 2.80 Å for In–In in InS, and these values are in agreement with the results of previous studies [44, 45]. By using the HSE06 functional, we obtained the energy band diagrams (c) and (d), indicating that GeC and InS belong to semiconductor materials with band gaps of 2.89 eV and 2.66 eV, respectively. These results are consistent with the findings of other researchers and further demonstrate the reliability of our computational approach [45, 47]. In the Brillouin zone, the former is a direct band gap where both the CBM and the VBM are located at the high-symmetry K point, and the latter is an indirect band gap where the CBM and the VBM are located at the high-symmetry Г and K–Г points, respectively.

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The heterostructure is formed by vertically stacking two single-layer materials, and its lattice mismatch needs to be within 5% to form a stable state. The lattice constant of GeC is 6.55 Å, while that of InS is 6.77 Å, resulting in a mismatch of only about 3.2%, indicating that heterostructures of GeC and InS can be successfully prepared experimentally. After optimization, the geometric structure of the heterostructure is shown in figure 2(a), with a lattice constant of 6.59 Å and a layer spacing of 3.51 Å. In the heterostructure, the Ge–C bond length is 1.90 Å, the In–S bond length is 2.53 Å, and the In–In bond length is 2.79 Å. After analysis, we found that the interatomic bond lengths changed very little before and after the formation of the heterostructure, indicating that almost no atomic rearrangement occurred, thanks to the small lattice mismatch. Binding energy is a key parameter to evaluate the thermodynamic stability of the system. For GeC/InS heterostructure, we can use the following formula to calculate the binding energy:

$$\begin{equation}{E_{\text{b}}} = {E_{{\text{GeC/InS}}}} - {E_{{\text{GeC}}}} - {E_{{\text{InS}}}}.\end{equation} \tag{ 1 }$$

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In the above formula, E_b represents the binding energy, E_GeC/InS, E_GeC and E_InS represent the total energy of the heterostructure GeC/InS, GeC and InS respectively. In the definition of binding energy, negative values indicate the stability of the structure, and the smaller the value, the more stable it is. The binding energy of the GeC/InS heterostructure is calculated by formula (1) to be equal to −0.39 eV, which is similar to the binding energy of other heterostructures g-C₃N₄/SnS₂ and HfS₂/SiSe [48, 49]. It shows that there is a stable interaction in the heterostructure, and this effect is the weak van der Waals force. In addition, vertical strain can be used to tune the electronic properties of the heterostructure, so we tried to introduce vertical strain by adjusting the distance between the GeC and InS layers. Generally speaking, heterostructures with different layer spacing have different stabilities. As shown in figure 2(b), when the layer spacing is 3.51 Å, the binding energy is the smallest, that is, the stability is the highest. Whether the layer spacing is gradually increased or decreased, the binding energy gradually increases, that is, the stability decreases. Especially when the layer spacing decreases to 2.81 Å, the binding energy becomes positive, indicating that the heterostructure is no longer stable. Therefore, we selected a heterostructure with a layer spacing of 3.51 Å for subsequent studies, which is consistent with the optimization results. To further prove the stability of this structure, we calculated its phonon spectrum. The results are shown in figure S1. The absence of negative values indicates that the GeC/InS heterostructure is stable. In order to verify the thermodynamic stability of the GeC/InS heterostructure, we performed AIMD calculations and constructed a 2 × 2 GeC/InS supercell, as shown in figure 2(c). The energy and temperature eventually stabilize throughout the process, and the interpolated plots show that the heterostructure shows no obvious structural distortion after 6 ps of 300 K simulation, which further proves that the GeC/InS heterostructure is dynamically stable.

In figure 3(a), the energy band structure of the GeC/InS heterostructure is depicted. Our calculations reveal an indirect band gap of 1.54 eV for the heterostructure. The band gap is smaller than that of single-layer GeC and InS, which means that electrons can more easily cross the band gap from the valence band (VB) to the conduction band (CB) to form free electrons. After comparing with the band structure of the single-layer material (figures 1(c) and (d)), we found that the CB of the GeC/InS heterostructure is closest to that of InS, while the VB is similar to that of GeC. This can be attributed to the smaller lattice mismatch in the heterostructure and the weakening of van der Waals interactions. The GeC/InS heterostructure displays an indirect band gap of 1.54 eV, with the band gap extending from the blue K point at the VBM to the orange Г point at the CBM. Subsequently, we conducted an analysis of the heterostructure's partial wave state density, as illustrated in figure 3(b). Notably, the CBM primarily comprises InS, while GeC significantly contributes to the VBM. This is consistent with the comparison results of the band structure diagrams, indicating that the heterostructure has a typical type-II band alignment. For the GeC monolayer, the states in CBM and VBM are dominated by Ge-p and C-p orbitals together. In a single layer of InS, the contribution to the CBM state primarily arises from In-s and S-p orbitals, whereas the VBM state results from In-p and S-p orbitals. Consequently, upon the formation of the heterostructure, the CBM state predominantly comprises In-s and S-p orbitals, while GeC significantly contributes to the VBM state through its C-p orbitals. This result is also clearly reflected in figure 3(b). Therefore, by combining these two single-layer materials to form a heterostructure, their respective properties are well preserved. The band structure of the heterostructure indicates the existence of type-II band alignment, which means that the GeC/InS heterostructure has potential application prospects in optoelectronic devices.

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The coupling between single layers will lead to the redistribution of charge at the heterostructure interface, and new electronic properties may appear. This phenomenon depends on the work function of the material, and is defined as follows:

$$\begin{equation}\Phi = {E_{{\text{vac}}}} - {E_{\text{F}}}.\end{equation} \tag{ 2 }$$

Among them, E_vac represents the vacuum energy level, and E_F represents the Fermi energy level. Through calculation, we obtained that the work functions of single-layer GeC, InS and GeC/InS heterostructures are 5.12, 6.72 and 5.3 eV respectively, as shown in figure 4, which is basically consistent with previous research results [47]. When the GeC and InS monolayers are in contact, since the work function of the GeC monolayer is lower than that of the InS monolayer, the Fermi level of the GeC monolayer is higher than that of InS. This leads to an automatic transfer of electrons from the GeC monolayer to the InS monolayer on contact, so that the Fermi energy levels of GeC (InS) move down (up) until they reach the same position, giving the GeC/InS heterostructure a work function of 5.3 eV. Electrons accumulate in the InS monolayer, while holes accumulate in the GeC monolayer. This accumulation creates a built-in electric field directed from GeC to InS.

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Due to the charge rearrangement at the interface, the electronic properties of the heterostructure will be somewhat different from those of the original single layer. In order to see the charge distribution at the GeC/InS heterostructure interface in more detail, we calculated the average charge density difference Δρ(z) in the Z-axis direction through the formula [50, 51],

$$\begin{align}\Delta \rho \left( z \right) &= {\mathop \int \nolimits}{\rho _{\text{T}}}\left( {x,y,z} \right){\text{d}}x{\text{d}}y - {\mathop \int \nolimits}{\rho _{{\text{GeC}}}}\left( {x,y,z} \right){\text{d}}x{\text{d}}y \nonumber\\ &\quad - {\mathop \int \nolimits}{\rho _{{\text{InS}}}}\left( {x,y,z} \right){\text{d}}x{\text{d}}y.\end{align} \tag{ 3 }$$

Among them, ${\rho _{\text{T}}}\left( {x,y,z} \right)$, ${\rho _{{\text{InS}}}}\left( {x,y,z} \right)$and ${\rho _{{\text{GeC}}}}\left( {x,y,z} \right)$ represent the charge density of GeC/InS heterostructure, single layer GeC and InS at point (x, y, z) respectively. It can be seen from figure 5(a) that there is a certain rearrangement of charges near the heterostructure interface. Generally speaking, electrons are concentrated in the InS layer and there is a certain amount of electron consumption on the GeC layer, which was visually observed through the 3D isosurface map. Figure 4(b) shows the electrostatic potential of the heterostructure in the Z-axis direction. It can be seen that there is a potential drop between the single layer GeC and InS. A large potential drop plays an important role in the separation of photoexcited electrons and holes. Bader charge analysis of the GeC/InS heterostructure is shown in figure 5(c). Compared with independent GeC monolayers and InS monolayers, the sum of electrons transferred by Ge and C atoms is 0.058e, while In and S atoms get 0.058e, indicating that 0.058e diffused from the GeC single layer to the InS layer. The specific electron transfer numbers are in figure S2.

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In order to use the GeC/InS heterostructure as a photocatalyst for water splitting, it is necessary to ensure that it has appropriate band gap and band edge positions to span the redox potential of water. According to the literature, we know that under vacuum conditions at pH = 0, the redox potential of water (E_H+/H2/E_O2/H2O) is −4.44 eV and −5.67 eV respectively (shown as the red dotted line in figure 6) [52]. To further evaluate the suitability of the GeC/InS heterostructure, we calculated E_H+/H2 and E_O2/H2O in the range of pH = 0–7, as shown in the following formulas:

$$\begin{equation}E_{{{\text{H}}^{{ + }}}{\text{/}}{{\text{H}}_{\text{2}}}}^{{\text{red}}} = - 4.44\,{\text{eV}} + {\text{pH}} \times 0.059\,{\text{eV}}\end{equation} \tag{ 4 }$$

$$\begin{equation}E_{{{\text{O}}_{\text{2}}}{\text{/}}{{\text{H}}_{\text{2}}}{\text{O}}}^{{\text{oxd}}} = - 5.67\,{\text{eV}} + {\text{pH}} \times 0.059\,{\text{eV}}.\end{equation} \tag{ 5 }$$

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The calculation results at different pH values show that as the pH value increases, the values of E_H+/H2 and E_O2/H2O also gradually increase. At a pH of 7, the E_H+/H2 and E_O2/H2O values stand at −4.03 eV and −5.26 eV, respectively. Figure 6 illustrates the band edge positions and band gaps, calculated with HSE06, for single-layer GeC, InS, and GeC/InS heterostructures. The staggered energy bands in the GeC/InS heterostructure on the right once again clearly prove that the heterostructure is a type-II band alignment. The CBM of GeC monolayer is higher than the reduction potential of H⁺/H₂ (−4.44 eV), which is suitable for the reduction reaction of water splitting. The CBM and VBM of InS satisfy the band edge positions of the redox of water splitting.

Figure 6 further provides an illustrative depiction of the migration of photoexcited electron-hole pairs at the interface of the GeC/InS heterostructure. Since the work function of InS is higher than that of GeC, when they come into contact, electrons migrate from the GeC side to the InS side until the equilibrium state of the electron Fermi level is reached. This results in the formation of a space charge region at the GeC/InS heterostructure interface, which generates an electric field E_int in the direction from GeC to InS, consistent with the previous electron interlayer transfer. The existence of E_int causes the electrons in the space charge region to have additional potential energy at every point, causing the energy band near the interface to bend. As a consequence, the positive charges present on the InS side experience repulsion from the holes in GeC, resulting in a reduction of potential energy. This interaction causes the energy band of the InS layer to bend downward at the interface. Conversely, the potential energy of the GeC layer increases, causing its energy band to bend upward. When subjected to light, electrons within the VB of both GeC and InS get excited into their respective CB, generating photogenerated electrons and leaving photogenerated holes in the VB. Under the influence of E_int, these photogenerated electrons efficiently recombine with photogenerated holes within the VB of GeC (Path-1). Due to the influence of band bending, the electron flow from CB in GeC to CB of InS and the flow of holes from VB in InS layer to VB of GeC are hindered (Path-2 is blocked). Finally, the photogenerated electrons stay in the GeC layer, while the photogenerated holes accumulate in the InS layer, which helps to effectively separate the photogenerated electron-hole pairs. This direct Z-scheme transfer path promotes redox reactions between different layers: the hydrogen production reaction mainly occurs in the GeC layer, while the oxygen production reaction occurs in the InS layer. Therefore, the GeC/InS heterostructure has the potential to serve as a highly efficient photocatalyst.

In order to evaluate the photocatalytic activity of the GeC/InS van der Waals heterostructure, the intermediate reactants and Gibbs free energy of the HER and OER processes were calculated respectively. The HER process reaction is as follows:

$$\begin{equation}{}^{*} + {{\text{H}}^ + } + {{\text{e}}^ - } \to {\text{H}}^{*}\end{equation} \tag{ 6 }$$

$$\begin{equation}{\text{H}}^{*} + {{\text{H}}^ + } + {{\text{e}}^ - } \to {{\text{H}}_2} + {}^{*}.\end{equation} \tag{ 7 }$$

Among them, * represents the active site on the GeC/InS heterostructure. From reaction equations (6) and (7), it can be concluded that the intermediate in the HER process is H*. We know that the necessary condition for becoming a high-efficiency hydrogen evolution photocatalyst is ΔG_H ∼ 0 [53]. As shown in figure 7(a), the ΔG_H of the GeC/InS van der Waals heterostructure is 0.35 eV. Therefore, this heterostructure is suitable as a HER photocatalyst. The most stable adsorption position of H* on the GeC/InS layer is shown in figure 7(c).

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The OER process is as follows (8)–(11):

$$\begin{equation}{}^{*} + {{\text{H}}_{\text{2}}}{\text{O}} \to {\text{OH}}^{*} + {{\text{H}}^ + } + {{\text{e}}^ - }\end{equation} \tag{ 8 }$$

$$\begin{equation}{\text{OH}}^{*} \to {\text{O}}^{*} + {{\text{H}}^ + } + {{\text{e}}^ - }\end{equation} \tag{ 9 }$$

$$\begin{equation}{\text{O}}^{*} + {{\text{H}}_{\text{2}}}{\text{O}} \to {\text{OOH}}^{*} + {{\text{H}}^ + } + {{\text{e}}^ - }\end{equation} \tag{ 10 }$$

$$\begin{equation}{\text{OOH}}^{*} \to {}^{*} + {{\text{O}}_2} + {{\text{H}}^ + } + {{\text{e}}^ - }.\end{equation} \tag{ 11 }$$

Reaction equations (8)–(11) indicate that the intermediate products of the OER process are OH*, O* and OOH*. The calculated results of ΔG_OH*, ΔG_O* and ΔG_OOH* of the GeC/InS van der Waals heterostructure are 2.67, 2.56, and 5.49 eV, as shown by the black line in figure 7(b). The third step has the largest free energy difference change of 2.94 eV, which is a potentially decisive step in the OER process. Under the external potential of U = 0 V and U = 1.23 V, the second and fourth steps decrease, and the remaining steps increase. Considering the maximum free energy difference, an appropriately increased external potential is needed to promote OER, so we explore the free energy change trend under an external potential of U = 2.95 eV. The orange line in figure 7(b) indicates that the free energy of all reaction steps shows a downward trend, which further illustrates that OER is an exothermic process under an applied potential of 2.95 eV.

Optical properties are an important indicator of photocatalyst performance, and light absorption coefficient is an important indicator of optical properties. The expression of light absorption coefficient α(ω) is [54],

$$\begin{equation}\alpha \left( \omega \right) = \frac{{\sqrt 2 \omega }}{c}{\left\{ {{{\left[ {\varepsilon _1^2\left( \omega \right) + \varepsilon _2^2\left( \omega \right)} \right]}^{1/2}} - {\varepsilon _1}\left( \omega \right)} \right\}^{1/2}}.\end{equation} \tag{ 12 }$$

Among them, ₁(ω) and ₂(ω) are the real part and imaginary part of the dielectric constant respectively, α, ω, and c are the absorption coefficient, angular frequency and speed of light under vacuum conditions, respectively [55, 56]. As shown in figure 8(a), we calculated the absorption spectra of the GeC layer, InS layer and GeC/InS heterostructure. Both single layers absorb visible light to a certain extent, and at 345 nm, the heterostructure light absorption reaches a maximum of 4.57 × 10⁵ cm⁻¹. The GeC/InS heterostructures exhibit a wider light absorption range, extending to 1000 nm. At this time, the GeC/InS heterostructure can absorb more visible light photon energy, which is related to the overlap of electronic states caused by the interlayer coupling of the two monolayers. As can be seen from figure 8(b), when +1% or +2% strain () is applied, the two absorption peaks have a certain red shift near visible light, indicating that more photons can be absorbed at this time; when strains of −1% and −2% are applied, the overall light absorption undergoes a blue shift, and the absorption peak near 472 nm increases to a certain extent, indicating that the light absorption capability has been enhanced to a certain extent. Therefore, applying strain to the GeC/InS heterostructure is an effective measure to change the light absorption capability.

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In order to further quantify the photocatalytic efficiency of GeC/InS heterostructure. Using the approach developed by Fu et al [57], we performed calculations to determine the efficiency of STH conversion. The detailed calculation is as follows:

$$\begin{equation}{\eta _{{\text{STH}}}} = {\eta _{{\text{abs}}}} \times {\eta _{{\text{cu}}}}.\end{equation} \tag{ 13 }$$

The calculation involved evaluating the light absorption efficiency (η_abs) and the carrier utilization efficiency (η_cu).

η_abs definition:

$$\begin{equation}{\eta _{{\text{abs}}}} = \frac{{{{\mathop \int \nolimits}}_{{E_{\text{g}}}}^\infty P\left(h\omega \right){\text{d}}\left(h\omega \right)}}{{{{\mathop \int \nolimits}}_0^\infty P\left(h\omega \right){\text{d}}\left(h\omega \right)}}\end{equation} \tag{ 14 }$$

where P(hω) is the AM1.5G solar flux at the photon energy hω, E_g is the band gap of GeC/InS (1.54 eV), ${\mathop \int \nolimits}_0^\infty P(h\omega ){\text{d}}(h\omega )$ is the total power density of incident simulated sunlight (AM1.5G), ${\mathop \int \nolimits}_{{E_{\text{g}}}}^\infty P(h\omega ){\text{d}}(h\omega )$ indicates the power density that can be absorbed by the GeC/InS heterostructure.

η_cu definition:

$$\begin{equation}{\eta _{{\text{cu}}}} = \frac{{\Delta G{{\mathop \int \nolimits}}_E^\infty \frac{{P\left(h\omega \right)}}{{h\omega }}{\text{d}}\left(h\omega \right)}}{{{{\mathop \int \nolimits}}_{{E_{\text{g}}}}^\infty P\left(h\omega \right){\text{d}}\left(h\omega \right)}}.\end{equation} \tag{ 15 }$$

Among these parameters, ΔG represents a constant value of 1.23 eV, which signifies the potential difference required for water splitting. E corresponds to the photon energy that can effectively drive the water splitting process, while $\Delta G{\mathop \int \nolimits}_E^\infty \frac{{P(h\omega )}}{{h\omega }}{\text{d}}(h\omega )$ denotes the effective photocurrent density. The value of E can be determined using the following formula:

$$\begin{equation}E = \left\{ {\begin{array}{*{20}{l}} {{E_{\text{g}}},\left(\chi \left({H_2}\right) \unicode{x2A7E} 0.2,\chi \left({O_2}\right) \unicode{x2A7E} 0.6\right)} \\ {{E_{\text{g}}} + 0.2 - \chi \left({H_2}\right),\left(\chi \left({H_2}\right) \lt 0.2,\chi \left({O_2}\right) \unicode{x2A7E} 0.6\right)} \\ {{E_{\text{g}}} + 0.6 - \chi \left({O_2}\right),\left(\chi \left({H_2}\right) \unicode{x2A7E} 0.2,\chi \left({O_2}\right) \lt 0.6\right)} \\ {{E_{\text{g}}} + 0.8 - \chi \left({H_2}\right) - \chi \left({O_2}\right),\left(\chi \left({H_2}\right) \lt 0.2,\chi \left({O_2}\right) \lt 0.6\right)} \end{array}} \right.\!\!\!\!\!\!\!\!.\end{equation} \tag{ 16 }$$

Among them, χ(H₂) and χ(O₂) are the potentials of hydrogen evolution and oxygen evolution respectively. It is calculated that χ(H₂) = 1.937 eV and χ(O₂) = 1.397 eV of GeC/InS, so E_g = 1.54 eV. The calculated STH conversion efficiency of the GeC/InS heterostructure is 44.39%. As shown in figure 9, compared with the heterostructures studied previously, the STH efficiency of GeC/InS is higher. For example, PtS₂/g-C₃N₄ (31.64%) [58], GeS/GeSe (31.21%) [59], g-C₆N₆/InP (27.31%) [60], GaN/InS (26.33%) [45], SeGe/SnS (26.45%) [61].

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