Out-of-plane polarization modulated band alignments in β-In₂X₃/α-In₂X₃ (X = S and Se) vdW heterostructures

The detection of graphene [1] has sparked a growing fascination with the investigation of two-dimensional (2D) materials, leading to the swift advancements of 2D materials [2–4]. Since then, a large number of 2D materials have been experimentally synthesized or theoretically predicted, including but not limited to silicene [5], black phosphorene (BP) [6, 7], transition metal carbides/nitrides [8–11], transition metal chalcogenides [12–15], and group III metal chalcogenides (MX and M₂X₃) [16–21]. In addition, the construction of van der Waals (vdW) heterostructures enlarges the 2D material families [22–24]. Compared with single-layer 2D materials, the properties of 2D vdW heterostructures can be efficiently designed according to particular requirements, which makes them more useful and flexible in the applications of photovoltaic and optoelectronic devices [25, 26]. There are three types of band edge alignments in 2D semiconductor/semiconductor vdW heterostructures, each of which can be applied in different fields. The type-I heterostructures are commonly used in light-emitting diodes [27] and lasers [28], type-II heterostructures are suitable for photocatalysis [29, 30] and photodetectors [31], and type-III can be employed in tunneling field-effect transistors [32]. Therefore, designing vdW heterostructures with specific functions has become an emerging platform for developing innovative photovoltaic and optoelectronic devices.

Recently, a category of semiconducting 2D In₂X₃ (X = S, Se, and Te) monolayers has gained increasing attention due to their fascinating properties [33–35]. The 2D In₂X₃ monolayers have two distinct phases known as α-In₂X₃ and β-In₂X₃, and several of them have been successfully synthesized in experiments, such as α-In₂Se₃ [36–38] and β-In₂X₃ (X = S, Se and Te) [18, 39–41]. Among them, the β-In₂X₃ monolayers exhibit high photoresponsivity and light absorption and show great potential applications in photodetectors [42] and solar cells [43, 44]. On the other hand, the out-of-plane electric polarization arising from the asymmetric structure of α-In₂X₃ monolayers provides an efficient way to tune the electronic structures of the α-In₂X₃-based vdW heterostructures [45–52]. Therefore, systematic investigation of the out-of-plane electric polarization effect on the electronic structures of the β-In₂X₃/α-In₂X₃ vdW heterostructures and unraveling their further applications in photovoltaic and optoelectronic fields are meaningful and of great interest.

This work comprehensively studies the structural, electronic, band alignments, and optical properties of the β-In₂X₃/α-In₂X₃ vdW heterostructures based on density functional theory (DFT) calculations. The ab initio molecular dynamics (AIMD) simulations demonstrate the thermal dynamic stabilities of the β-In₂X₃/α-In₂X₃ vdW heterostructures. By switching the out-of-plane polarization direction of α-In₂X₃ layers, the band edge alignment features of β-In₂S₃/α-In₂S₃, β-In₂Se₃/α-In₂Se₃, and β-In₂Se₃/α-In₂S₃ heterostructures can be tuned from type-I to type-II. Moreover, the β-In₂Se₃/α-In₂S₃↓ heterostructure is suitably used as a solar cell with a PCE of 17.9%. Our results shed light on the effect of out-of-plane polarization direction of α-In₂X₃ layers on the electronic structures of the β-In₂X₃/α-In₂X₃ vdW heterostructures and offer new insight into designing advanced electronic devices based on 2D In₂X₃-based vdW heterostructures.

The present works were based on the DFT implemented in the Vienna ab initio Simulation Package (VASP) [53]. The projector-augmented wave method [54] was used to describe the interactions between valence and core electrons, while the generalized gradient approximation, as parameterized by Perdew-Burke-Ernzerhof [55] was used to treat exchange-correlation functional effects. The cutoff energy for the plane-wave basis set was set to 500 eV, and an 11 × 11 × 1 k-points grid was used for the Brillouin zone sampling. The energy and force tolerance convergence criteria were set to 10⁻⁶ eV and 0.01 eV Å⁻¹. The Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) [56] was used to correct the band structures and band edge alignments of heterostructure and monolayer systems. A vacuum layer of at least 20 Å were added along the z-direction. For the α-In₂X₃ monolayers and β-In₂X₃/α-In₂X₃ vdW heterostructures, using the center of mass as the reference point, the Berry phase [57, 58] expressions with dipole correction are applied to these systems, which allows for the adjustment of the misalignment between the vacuum levels on the different sides of the structures. The DFT-D3 approach [59, 60] was utilized to account for vdW interactions. AIMD [61] simulations were performed using a canonical ensemble regulated by the Nose-Hoover thermostat [62] to examine the thermodynamic stability of heterostructures. The VASPKIT code [63] was used to post-processing some VASP calculation results.

The formation energy (E_f) and binding energy (E_b) of the β-In₂X₃/α-In₂X₃ heterostructures can be defined as follows:

$$\begin{align}{E_{\text{f}}} = {E_{\beta {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_{\text{3}}}{\text{/}}\alpha {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_3}}} - {E_{\beta {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_{\text{3}}}}} - {E_{\alpha {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_3}}}\end{align} \tag{ 1 }$$

$$\begin{align}{E_{\text{b}}} = ({E_{\beta {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_{\text{3}}}{\text{/}}\alpha {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_3}}} - E_{\beta {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_{\text{3}}}}^{{\text{fix}}} - E_{\alpha {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_3}}^{{\text{fix}}})/A\end{align} \tag{ 2 }$$

where ${E_{\beta {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_{\text{3}}}{\text{/}}\alpha {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_3}}}$, ${E_{\beta {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_{\text{3}}}}}$, ${E_{\alpha {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_3}}}$ are the total energies of β-In2X3/α-In2X3, β-In2X3 and α-In2X3, respectively. $E_{\beta {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_{\text{3}}}}^{{\text{fix}}}$ and $E_{\alpha {\text{-I}}{{\text{n}}_{\text{2}}}{{\text{X}}_3}}^{{\text{fix}}}$ represent the total energies of β-In₂X₃ and α-In₂X₃ monolayer fixed in the corresponding heterostructures lattice constant, respectively, and A is the cross-sectional area. In 2D vdW heterostructure, the formation energy refers to the energy changing of forming certain vdW heterostructure, which represents the energy difference when stacking two independent 2D materials to create the heterostructure. The formation energy can be used to assess the stability of the heterostructure, where a more negative value of formation energy indicates the structure is more stable. On the other hand, the binding energy reflects the binding strength between two layers of the heterostructure. Thus, the more negative value of binding energy indicates the stronger interaction between two layers in heterostructures. Herein, the formation energy includes the binding energy and additional lattice distortion energy.

Prior to constructing the β-In₂X₃/α-In₂X₃ (X = S and Se) heterostructures, we initially investigated the corresponding monolayers. The atom structures of α-In₂X₃ and β-In₂X₃ monolayers are shown in figures S1(a) and (b), respectively. Both the β-In₂X₃ and α-In₂X₃ monolayers belong to the trigonal 2D lattice with space groups of $P\overline 3 m1$ and $P3m1$, respectively, and possess 5 atomic layers with the X-In-X-In-X stacking configuration. After the structures are fully relaxed, the lattice constants are estimated to be 3.84, 4.01, 3.93, and 4.10 Å for β-In₂S₃, β-In₂Se₃, α-In₂S₃, and α-In₂Se₃ monolayers, respectively. Due to the asymmetric structure of the α-In₂X₃ monolayers, the out-of-plane electric polarization will be produced, pointing from the upper chalcogenide atoms to the bottom chalcogenide atoms, as marked by the red arrow in figure S1(b). On the one hand, the electric polarization in the α-In₂X₃ monolayers will lead to inhomogeneous charge distribution, generating a built-in electric field, which can facilitate the separation of photogenerated carriers. On the other hand, the built-in electric field will result in the existing electrostatic potential difference between the two surfaces of the α-In₂X₃ monolayers. Therefore, two different band alignments of the energy bands will be generated with respect to the vacuum levels on the different surfaces of α-In₂X₃ monolayers. It should be noted that the polarization orientation of α-In₂X₃ monolayers can be reversed via the application of external electric field. The band structures illustrated in figures S1(c) and (d) were used to understand the electronic structures of β-In₂X₃ and α-In₂X₃ monolayers. It can be seen that all of them pertain to indirect band gap semiconductors. The calculated band gaps of β-In₂S₃, β-In₂Se₃, α-In₂S₃, and α-In₂Se₃ monolayers by using HSE06 functional are 1.97 eV, 1.2 eV, 2.15 eV, and 1.41 eV, respectively. It should be noted that the obtained lattice constants and band gaps of the α-In₂X₃ and β-In₂X₃ monolayers match the previous experimental and theoretical results very well [33, 39, 40, 64], providing evidence of the validity of these theoretical predictions.

Based on the four monolayers mentioned above, we have constructed four different heterostructures by perpendicularly combing one β-In₂X₃ layer and one α-In₂X₃ layer together, namely β-In₂S₃/α-In₂S₃, β-In₂Se₃/α-In₂Se₃, β-In₂Se₃/α-In₂S₃ and β-In₂S₃/α-In₂Se₃ heterostructures, the corresponding lattice mismatches are 2.12%, 2.21%, 2.11%, and 6.31%, respectively. It is noted that β-In₂S₃/α-In₂Se₃ heterostructure can be fabricated despite the relatively large lattice mismatch between β-In₂S₃ and α-In₂Se₃, since both the β-In₂S₃ and α-In₂Se₃ monolayers exhibit outstanding mechanical characteristics [65, 66]. Due to the different out-of-plane polarization states of the α-In₂X₃ monolayers, we denote the heterostructure formed by the connection of β-In₂X₃ and α-In₂X₃ in the downward polarization direction as the β-In₂X₃/α-In₂X₃↓ heterostructure, while connection in the upward polarization direction as the β-In₂X₃/α-In₂X₃↑ heterostructure. Three distinct stacking configurations were employed as an initial criterion to examine how the stacking patterns influence the stability of heterostructures with different polarization states, as depicted in figures 1(a) and (b). The calculated lattice constants (a), interlayer distances (d_layer), formation energies (E_f), and binding energies (E_b) for all the heterostructures with different stacking configurations are listed in table 1. One can see that the interlayer distances of heterostructures range from 2.820 to 3.737 Å, which belong to the typical vdW gap [67, 68]. On the other hand, except for β-In₂S₃/α-In₂Se₃↓ heterostructure with stacking-I configuration possesses a positive value of formation energy, all of the other formation energies and binding energies show negative values, indicating the β-In₂X₃/α-In₂X₃ heterostructures are stable from thermodynamic point of view. Furthermore, the binding energies of the β-In₂X₃/α-In₂X₃ heterostructures fall within the range from −9.99 to −18.51 meV Å⁻², providing further evidence that they belong to vdW heterostructures [67]. It is noted that the more negative the formation and binding energy values, the more stable the structure. Considering this factor, we can conclude that the most stable stacking patterns are the stacking-III for both the β-In₂X₃/α-In₂X₃↓ and β-In₂X₃/α-In₂X₃↑ vdW heterostructures. Therefore, the subsequent calculations are focused on stacking-III configuration.

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The AIMD simulations were carried out to further understand the thermal dynamics stabilities of the β-In₂X₃/α-In₂X₃ vdW heterostructures. The time-dependent energy curves shown in figures 2(a)–(h) indicate that the total energies of the heterostructures remain stable within a reasonable range throughout the simulation periods. Additionally, the final structures obtained after 10 ps of simulation display minimal changes compared to their initial configurations, with atoms exhibiting only slight vibrational movements around their respective equilibrium positions. The AIMD simulation results have demonstrated that the β-In₂X₃/α-In₂X₃ vdW heterostructures exhibit exceptional thermal dynamic stability. Furthermore, the phonon dispersion curves of β-In₂S₃/α-In₂S₃↓ and β-In₂Se₃/α-In₂Se₃↓ heterostructures are also calculated and shown in figures S2(a) and (b), respectively. Unfortunately, there are existing significant imaginary frequencies in both heterostructures. To understand the origin of the significant imaginary frequencies in heterostructures, we calculated the phonon dispersion curves of β-In₂X₃ and α-In₂X₃ monolayers, and the corresponding results are plotted in figures S2(c)–(f). Clearly significant imaginary frequencies for β-In₂X₃ monolayers can be found in phonon dispersion curves, which agree with previous work [34]. This might be attributed to the strong displacive characters resulting from temperature-induced α to β phase transition, leading to the disappearance of polarization in β-In₂X₃ [69]. Previous studies have demonstrated that the imaginary frequency of β-In₂X₃ monolayers can be eliminated by making a small in-plane displacement of the middle X layer [70]. This small displacement of the middle X layer for β-In₂X₃ monolayers will generate in-plane polarization, which might guarantee the stability of β-In₂X₃ monolayers [70]. Therefore, the significant imaginary frequencies can also be eliminated by moving the middle X layer of β-In₂X₃ layers in β-In₂X₃/α-In₂X₃ vdW heterostructures. Although the β-In₂X₃/α-In₂X₃ vdW heterostructures are unstable from the perspective of the phonon dispersion curves. It is interesting to note that the 2D β-In₂X₃ has been successfully synthesized experimentally, which can be transformed from the 2D α-In₂X₃ at high temperature, and kept stable at room temperature [39–41, 71, 72]. Moreover, the Graphene/β-In₂S₃ vdW heterostructure has also been synthesized through the dry transfer method [42]. Therefore, combining the formation energy and AIMD simulation results, we believe the β-In₂X₃/α-In₂X₃ heterostructures can also exist at room temperature and be synthesized in the near future.

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To reach insights into the electronic structures of the β-In₂X₃/α-In₂X₃ vdW heterostructures, the projected HSE06 band structures showing the contributions from individual layers are represented in figure 3. As described in figures 3(a), (c), (e) and (g), all the β-In₂X₃/α-In₂X₃↓ vdW heterostructures preserve the indirect band gap characteristic. Additionally, except for β-In₂S₃/α-In₂Se₃↓ heterostructure exhibit type-II band alignment feature, the others belong to type-I heterostructures. This is attributed to the fact that the conduction band minimum (CBM) and valence band maximum (VBM) of β-In₂S₃/α-In₂S₃↓, β-In₂Se₃/α-In₂Se₃↓, and β-In₂Se₃/α-In₂S₃↓ heterostructures are contributed by the β-In₂X₃ layer. Whereas in β-In₂S₃/α-In₂Se₃↓ heterostructure, CBM and VBM are from the β-In₂S₃ and α-In₂Se₃ layers, respectively. The calculated band gap of β-In₂S₃/α-In₂S₃↓, β-In₂Se₃/α-In₂Se₃↓, β-In₂Se₃/α-In₂S₃↓, and β-In₂S₃/α-In₂Se₃↓ heterostructures are 1.87, 1.12, 1.04 and 1.26 eV, respectively. It is worth noting that the type-II band alignment nature and suitable band gap of β-In₂S₃/α-In₂Se₃↓ heterostructure indicate its great application potential in photovoltaic devices [73]. On the other hand, for the β-In₂X₃/α-In₂X₃↑ vdW heterostructures shown in figures 3(b), (d), (f) and (h), it is interestingly seen that all of them are type-II heterostructures, as the CBM and VBM are contributed by the β-In₂X₃ and α-In₂X₃ layer, respectively. Compared with type-I heterostructures, spatial separation of carriers in type-II heterostructures effectively extends the recombination time of photogenerated carriers, which is more beneficial for the application in optoelectronic and photovoltaic fields [74–76]. Moreover, owing to the formation of type-II band alignment tailoring the band gap [29], the band gaps of the β-In₂X₃/α-In₂X₃↑ vdW heterostructures ranging from 0.55 to 0.78 eV are smaller than the β-In₂X₃/α-In₂X₃↓ vdW heterostructures.

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To unravel the physical origin of how the out-of-plane polarization of α-In₂X₃ monolayers modulates the band edge alignments, figure 4(a) illustrates a schematic diagram of the band energy levels in the β-In₂X₃/α-In₂X₃ vdW heterostructures. It is worth noting that the emergence of electrostatic potential differences (Δϕ) between the two surfaces of α-In₂X₃ leads to different band alignments of the energy bands with respect to the vacuum levels on the different surfaces. Herein, the calculated Δϕ is 1.27 and 1.18 eV for α-In₂S₃ and α-In₂Se₃ monolayers, respectively, which agree well with previous studies [33, 35]. The significant Δϕ of α-In₂X₃ monolayers give an excellent opportunity to achieve different types of band alignment in α-In₂X₃-based heterostructures as the surface of the α-In₂Se₃ layers in contact with the β-In₂X₃ layers altered with the polarization reversal [46]. Here, β-In₂Se₃/α-In₂S₃ heterostructure is proposed as an example to explain the band edge alignments modulated mechanism. The band edge alignments of β-In₂Se₃/α-In₂S₃ heterostructure before and after connecting are plotted in figures 4(b) and (c), respectively. One can see that the CBM of β-In₂Se₃ monolayer is 0.17 eV lower than the α-In₂S₃↓ monolayer, while the corresponding VBM is 0.80 eV higher. Therefore, a type-I band edge alignment is obtained after connecting β-In₂Se₃ and α-In₂S₃ monolayers to form β-In₂Se₃/α-In₂S₃↓ heterostructure. On the other hand, both the CBM and VBM energy levels of β-In₂Se₃ monolayer are lower than those of α-In₂S₃↑ monolayer, therefore, the forming of β-In₂Se₃/α-In₂S₃↑ heterostructure reaches a type-II band edge alignment. This phenomenon can be understood as when β-In₂X₃ monolayers are connected to α-In₂X₃ monolayers to form a β-In₂X₃/α-In₂X₃ vdW heterostructure, the energy bands of the two components can essentially be considered by aligning with respect to the vacuum level, owing to the nature of weak vdW interlayer interactions. Therefore, it is possible to reverse the polarization direction of the α-In₂X₃ layers by applying an external electric field. Hence, the attached β-In₂X₃ layers can contact different surfaces of the α-In₂X₃ layers, leading to very different band alignments of β-In₂X₃/α-In₂X₃ vdW heterostructures and even altering the type of heterostructure. Consequently, the electronic structures and optical properties of β-In₂X₃/α-In₂X₃ vdW heterostructures are changed.

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The optical absorption coefficient α (ω) of β-In₂X₃/α-In₂X₃ vdW heterostructures is one of the most critical parameters for their use in photovoltaic and optoelectronic devices, which can be calculated by [77]:

$$\begin{align}\alpha (\omega ) = \sqrt 2 \omega {\left[\frac{{\sqrt {\varepsilon _1^2(\omega ) + \varepsilon _2^2(\omega )} - {\varepsilon _1}(\omega )}}{2}\right]^{{1 \mathord{\left/ \right. } 2}}}\end{align} \tag{ 3 }$$

where ${\varepsilon _{\text{1}}}$ and ${\varepsilon _{\text{2}}}$ are the real part and imaginary part of the dielectric functions, respectively. Except for the optical absorption coefficients, the other optical properties of β-In₂X₃/α-In₂X₃ vdW heterostructures were also investigated, such as refraction indexes n (ω), extinction coefficients k (ω) reflectivities R (ω) and energy loss functions L (ω), which can be calculated as follows [77]:

$$\begin{align}n(\omega ) = {\left[\frac{{\sqrt {\varepsilon _1^2(\omega ) + \varepsilon _2^2(\omega )} + {\varepsilon _1}(\omega )}}{2}\right]^{{1 \mathord{\left/ \right. } 2}}}\end{align} \tag{ 4 }$$

$$\begin{align}k(\omega ) = {\left[\frac{{\sqrt {\varepsilon _1^2(\omega ) + \varepsilon _2^2(\omega )} - {\varepsilon _1}(\omega )}}{2}\right]^{{1 \mathord{\left/ \right. } 2}}}\end{align} \tag{ 5 }$$

$$\begin{align}R(\omega ) = \frac{{{{(n - 1)}^2} + {k^2}}}{{{{(n + 1)}^2} + {k^2}}}\end{align} \tag{ 6 }$$

$$\begin{align}L(\omega ) = \frac{{{\varepsilon _2}}}{{\varepsilon _1^2(\omega ) + \varepsilon _2^2(\omega )}}.\end{align} \tag{ 7 }$$

Figures 5(a) and (b) show the optical absorption coefficients α (ω), refraction indexes n (ω), reflectivities R (ω), extinction coefficients k (ω) and energy loss functions L (ω) for β-In₂X₃/α-In₂X₃↓ and β-In₂X₃/α-In₂X₃↑ vdW heterostructures, respectively. Obviously, the optical absorption coefficients of the β-In₂X₃/α-In₂X₃↓ heterostructures have improved in the visible and ultraviolet light regions compared to the corresponding monolayers. Especially, the optical absorption coefficients of β-In₂Se₃/α-In₂Se₃↓ and β-In₂S₃/α-In₂Se₃↓ heterostructures are higher than the corresponding monolayers in nearly the entire visible and ultraviolet light regions. However, except for β-In₂S₃/α-In₂S₃↑ heterostructure, the increase of the optical absorption coefficients of the β-In₂X₃/α-In₂X₃↑ heterostructures is not as significant as the β-In₂X₃/α-In₂X₃↓ heterostructures, which is only enhanced in very narrow photon energy ranges. Furthermore, the refraction indexes of the β-In₂X₃/α-In₂X₃ vdW heterostructures were also evaluated. From figure 5, it can be observed that the refraction indexes of the β-In₂X₃/α-In₂X₃ vdW heterostructures are higher than the corresponding monolayers in all the spectral regions we investigated, and the values of refraction indexes following n (β-In₂Se₃/α-In₂Se₃) > n (β-In₂Se₃/α-In₂S₃) > n (β-In₂S₃/α-In₂Se₃) > n (β-In₂S₃/α-In₂S₃). Interestingly, the same trend as the refractive index can also be found in the reflectivities of β-In₂X₃/α-In₂X₃ vdW heterostructures and slightly increase with the increasing incident photon intensity. The extinction coefficient describes the degree of a given material absorbs or reflects light at a specific wavelength. Therefore, it exhibits a similar tendency to the optical absorption coefficients. In addition, it can also be found that the energy loss functions of β-In₂X₃/α-In₂X₃ vdW heterostructures are lower than the corresponding monolayer, which are beneficial to their solar cell related applications as well. Overall, the enhancement of the optical properties of the β-In₂X₃/α-In₂X₃ vdW heterostructures is conducive to its application in photovoltaic and optoelectronic devices.

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It is noted that β-In₂S₃/α-In₂Se₃↓ heterostructure possesses type-II band alignment with a suitable band gap, favorable optical absorption, and relatively small band offset, indicating its great potential application in the heterostructure solar cells. Therefore, to evaluate the performance of β-In₂S₃/α-In₂Se₃↓ heterostructure in solar cells, the power conversion efficiencies (PCE) η was calculated based on the following equation [78, 79]:

$$\begin{align}\eta {\text{ = }}\frac{{{\beta _{{\text{FF}}}}{V_{{\text{OC}}}}{J_{{\text{SC}}}}}}{{{P_{{\text{solar}}}}}} = \frac{{{\text{0}}{\text{.65(}}{E_{\text{g}}}{\text{ - }}\Delta E{}_{\text{c}}{\text{ - 0}}{\text{.3)}}\int_{{E_{\text{g}}}}^\infty {\frac{{P(\hbar \omega )}}{{\hbar \omega }}d(\hbar \omega )} }}{{\int_0^\infty {P(\hbar \omega )d(\hbar \omega )} }}\end{align} \tag{ 8 }$$

where the fill factor (β_FF) is set to 0.65, V_OC, J_SC, P_solar, E_g and ΔE_c represent the open-circuit voltage, short circuit current, total incident solar energy, band gap of donor and conduction band offset, respectively. Figure 6(a) shows the band structures of β-In₂S₃ and α-In₂Se₃ monolayers with respect to vacuum levels, which are calculated by fixing β-In₂S₃ and α-In₂Se₃ monolayers in the lattice of β-In₂S₃/α-In₂Se₃↓ heterostructure. The schematic diagram of the band edge alignment of β-In₂S₃/α-In₂Se₃↓ heterostructure is described in figure 6(b). It can be observed that the β-In₂S₃ and α-In₂Se₃ layers play as the acceptor and donor in heterostructure solar cell, respectively. Moreover, the band gap of the α-In₂Se₃ layer and band offset between the two layers are 1.49 eV and 0.35 eV, respectively. Based on the given parameters, a contour map depicting the relationship between E_g, ΔE_c and PCE is presented in figure 6(c). The calculated PCE of the β-In₂S₃/α-In₂Se₃↓ heterostructure can be achieved at 17.9%, which is at the same level as other promising heterostructures solar cells, such as ZrS₃/HfS₃ (16%–18%) [80], In₂SSe₂/In₂SeS₂ (15.5%) [81], and SnS/GaSe (18%) [82], and is higher than ZrS₃/MoS₂ (14.2%) [83], GaTe/InS (11%) [21], and BP/SnSe (11.96%) [84].

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In summary, four types of the β-In₂X₃/α-In₂X₃ (X = S and Se) vdW heterostructures have been constructed, and each type of heterostructure has two different stacking configurations due to connecting of β-In₂X₃ monolayer to the different polarization direction of the α-In₂X₃ monolayer, namely the β-In₂X₃/α-In₂X₃↓, and β-In₂X₃/α-In₂X₃↑ vdW heterostructures. Their electronic structures, band edge alignments, and optical absorption coefficients have been systematically studied using DFT calculations. All the β-In₂X₃/α-In₂X₃ vdW heterostructures belong to indirect band gap semiconductors. The band alignment of β-In₂S₃/α-In₂S₃, β-In₂Se₃/α-In₂Se₃, and β-In₂Se₃/α-In₂S₃ heterostructures can be transferred from type-I to type-II by switching the out-of-plane polarization direction of α-In₂X₃ layers. Moreover, the optical absorption coefficients can be significantly enhanced by constructing the β-In₂X₃/α-In₂X₃ vdW heterostructures. Meanwhile, the PCE of β-In₂S₃/α-In₂Se₃↓ heterostructure can reach up to 17.9%. These results not only demonstrate the out-of-plane polarization α-In₂X₃ monolayers can efficiently modulate the electronic structures of the α-In₂X₃-based vdW heterostructures, but also provide the theoretical insight for the application of the β-In₂X₃/α-In₂X₃ vdW heterostructures in the field of optoelectronic devices and heterostructure solar cells.

This work was supported by the National Key Research and Development Program of China (No. 2022YFB3807200), the National Natural Science Foundation of China (No. 21973012), the Natural Science Foundation of Fujian Province (Nos. 2021J06011, 2020J01351, 2020J01474, 2021J01590 and 2021H6011), the Open Foundation of Key Laboratory of Green Perovskites Application of Fujian Province Universities (No. JXKFB202203), and the 'Qishan Scholar' Scientific Research Project of Fuzhou University.

All data that support the findings of this study are included within the article (and any supplementary files).

The authors declare that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.