Rolling the WSSe Bilayer into Double-Walled Nanotube for the Enhanced Photocatalytic Water-Splitting Performance

For the emerging Janus transition metal dichalcogenides (TMD) layered water-splitting photocatalysts, stacking the monolayers to form bilayers has been predicted to be an effective way to improve their photocatalytic performances. To achieve this, the stacking pattern plays an important role. In this work, by means of the density functional theory calculations, we comprehensively estimate energetical stability, light absorption and redox capacity of Janus WSSe bilayer with different stacking patterns. Unfortunately, the Janus WSSe bilayer with the most stable configuration recover the out-of-plane symmetry, which is not in favor of the photocatalytic reactions. However, rolling the Janus WSSe bilayer into double-walled nanotube could stabilize the appropriate stacking pattern with an enhanced instinct dipole moment. Moreover, the suitable band edge positions, high visible light absorbance, outstanding solar-to-hydrogen efficiency (up to 28.48%), and superior carrier separation promise the Janus WSSe double-walled nanotube the potential for the photocatalytic water-splitting application. Our studies not only predict an ideal water-splitting photocatalyst, but also propose an effective way to improve the photocatalytic performances of Janus layered materials.


Introduction
Since the discovery of "Honda-Fujishima effect" in 1972 [1], overall water-splitting for hydrogen production with semiconductor-based photocatalysts has attracted extensive attentions [2][3][4]. The solar hydrogen generation is considered as a green technology to solve the growing energy crisis and environmental pollution problems [5][6][7][8][9]. The mechanism of photocatalytic water-splitting on semiconductor based photocatalysts could be elaborated as follows [6]. Under solar illumination, the photon absorption in a semiconductor gives rise to an electronic transition between the conduction and valence bands, which brings in the photo-excited carriers. After the interior and exterior recombination, the residual photo-generated electrons at the surface reduce protons in the water to form hydrogen gas (H + /H 2 ), while the residual photo-generated holes oxidize water molecules to produce oxygen gas (O 2 /H 2 O). Normally, a high-performance water-splitting photocatalyst needs to fulfill three requirements, namely a high visible photons utilization efficiency, the excellent capability for carriers separate and transfer, and suitable band edge potentials for sufficient redox ability of photo-generated carriers. Unfortunately, an irreconcilable contradiction arises that, the high light utilization rate demands a adequately narrow band gap, however, the competent redox capability usually calls for a large band gap (≥1.23 eV). The new fast-developing two dimensional (2D) polar materials bring a new dawn for solving this problem.
verified to be unremarkably [17], here we do not apply the SOC correction in our calculations for saving the computing resource. More computational details of theoretical STH efficiency and DWNT surface ΔΦ can be found in the Supplementary Materials. In this study, we choose the AB stacking mode (translation symmetry) for the Janus WSSe bilayers, which has been widely demonstrated to be more stable than the other stacking modes for transition metal dichalcogenides layered materials [11,12,15,24,25]. In the AB stacking mode, the S/Se (W) atom of the first layer locate above the W (S/Se) atom of the second layer. As depicted as in Figure 1, based on the atomic species at the interface, three kinds of WSSe bilayer stacking configurations are considered, namely S-Se (denoted as AB1), Se-Se (denoted as AB2), and S-S (denoted as AB3) with the point groups of C3v,

Computational Methods
In this work, we employ the Vienna Ab initio Simulation Package (VASP) (version 5.3) software to carry out the DFT calculations for both geometry relaxations and electronic structures [18,19]. We use the projector augmented wave (PAW) pseudo potentials to describe the electron-ion interaction [20,21]. We choose generalized gradient approximations of Perdew-Burke-Ernzerhof (GGA-PBE) as the exchange-correlation functional [22]. In order to avoiding the interactions with neighboring mirror images, we set a 20 Å vacuum space vertical to each Janus WSSe bilayer, and more than 15 Å vacuum spaces along x and y directions for each Janus WSSe DWNT, whose periodic boundary condition is along the z axis. We apply the DFT-D3 approach of Grimme to address the van der Waals (vdW) force [17,23]. The Brillouin zone is regulated with a 9 × 9 × 1 gamma-pack k-mesh for the Janus WSSe bilayers, and a 1 × 1 × 7 one for the Janus WSSe double-walled nanotube. The cutoff energy is set to 500 eV, and the convergence criteria for the force and energy is 10 −2 eV/Å and 10 −5 eV, respectively. Although tungsten is a heavy element, since the effect of spin-orbital coupling (SOC) on the band gap of WSSe monolayer has been verified to be unremarkably [17], here we do not apply the SOC correction in our calculations for saving the computing resource. More computational details of theoretical STH efficiency and DWNT surface ∆Φ can be found in the Supplementary Materials.

Geometric and Electronic Structures of Janus WSSe Bilayer
In this study, we choose the AB stacking mode (translation symmetry) for the Janus WSSe bilayers, which has been widely demonstrated to be more stable than the other stacking modes for transition metal dichalcogenides layered materials [11,12,15,24,25]. In the AB stacking mode, the S/Se (W) atom of the first layer locate above the W (S/Se) atom of the second layer. As depicted as in Figure 1, based on the atomic species at the interface, three kinds of WSSe bilayer stacking configurations are considered, namely S-Se (denoted as AB1), Se-Se (denoted as AB2), and S-S (denoted as AB3) with the point groups of C 3v , D 3d , and D 3d , correspondingly. After the full optimization, the lattice constants (a = b) of AB1, AB2, and AB3 stacking configurations nearly are the same (about 3.25 Å) (listed in Table 1), which well agrees with the result (3.228 Å) in the previous report [24]. Table 1. Detailed information of Janus WSSe bilayers with different stacking patterns. Calculated lattice constants (a and b), binding energy (E b ), surface potential energy difference (∆Φ), and band gap (E g ). In the beginning, we check the stacking-dependent stability by calculating the binding energy E b , which could be obtained with the following equation [13,24]:

Stacking Patterns
where E ML and E BL are separately the total energies of Janus WSSe monolayer and bilayer. Under this definition, a more negative E b value indicates a higher energetical stability. According to the calculated binding energy listed in the Table 1, the AB2 stacking configuration has the most negative E b among all the three cases, forecasting the most stable stacking approach for the Janus WSSe monolayer to form its bilayer.
Next, we study the electronic properties of Janus WSSe bilayer by investigating their electronic band structures with various stacking models. Here, we choose the results achieved by PBE functional instead of Heyd−Scuseria−Ernzerhof (HSE06) hybrid functional or G 0 W 0 calculations. This is because that, as shown in Figure 2a, the calculated band gap of WSSe monolayer at PBE level (1.68 eV) is closer to the experimental optical gap (1.83 eV) [26] than the one at the HSE06 level (2.13 eV) or at the G 0 W 0 level (2.68 eV) [17]. Our treatment agrees with the one applied in the previous studies on Janus TMD studies [27][28][29]. As plotted in Figure 2, all these Janus WSSe bilayers present indirect band gap, different from the case of its monolayer. Specifically, for the Janus WSSe monolayer, the conduction band minimum (CBM) and valence band maximum (VBM) are both at the K point [17]; nevertheless, as to the bilayer cases, all the VBM moves to the Γ point, meanwhile the CBM of AB1 and AB3 stacking modes still stay at the K point, and the one of AB2 stacking mode lies on the Γ-K path. The direct−indirect transition of band gap, caused by the forming bilayer, is able to suppress the photogenerated electron-hole recombination, hopefully raising the quantity of free carriers. Furthermore, as summarized in the Table  1, all the band gaps of bilayer (0.90, 1.36, and 1.01 eV for AB1, AB2 and AB3 modes, respectively) are obviously smaller than the one of monolayer (1.68 eV). The narrowed band gap in these bilayers is conducive to improve the light absorption efficiency, which will be discussed later. Intriguingly, the spatial distribution of CBM and VBM in the Janus WSSe bilayers depends on the stacking pattern. As illustrated in the inset of Figure 2, for the AB1 stacking pattern, the CBM distributes at the first layer, while the VBM is dominantly distributed at the second layer. However, for the AB2 and AB3 stacking patterns, both VBM and CBM uniformly spread over the both layers. Under illumination, the photo-excited electron jump to the CBM, at the same time, the photo-excited hole stay at the VBM. Hence, the locational separation of CBM and VBM normally could lower the carrier recombination. Consequently, from the perspective of increasing the free photo-generated carrier number, the AB1 stacking pattern is a better choice. It is also better than the monolayer, where both the CBM and VBM are located partially at the W atomic layer (see the inset of Figure 2a). Nanomaterials 2021, 11, x FOR PEER REVIEW 4 of 10 achieved by PBE functional instead of Heyd−Scuseria−Ernzerhof (HSE06) hybrid functional or G0W0 calculations. This is because that, as shown in Figure 2a, the calculated band gap of WSSe monolayer at PBE level (1.68 eV) is closer to the experimental optical gap (1.83 eV) [26] than the one at the HSE06 level (2.13 eV) or at the G0W0 level (2.68 eV) [17]. Our treatment agrees with the one applied in the previous studies on Janus TMD studies [27][28][29]. As plotted in Figure 2, all these Janus WSSe bilayers present indirect band gap, different from the case of its monolayer. Specifically, for the Janus WSSe monolayer, the conduction band minimum (CBM) and valence band maximum (VBM) are both at the K point [17]; nevertheless, as to the bilayer cases, all the VBM moves to the Γ point, meanwhile the CBM of AB1 and AB3 stacking modes still stay at the K point, and the one of AB2 stacking mode lies on the Γ-K path. The direct−indirect transition of band gap, caused by the forming bilayer, is able to suppress the photogenerated electron-hole recombination, hopefully raising the quantity of free carriers. Furthermore, as summarized in the Table 1, all the band gaps of bilayer (0.90, 1.36, and 1.01 eV for AB1, AB2 and AB3 modes, respectively) are obviously smaller than the one of monolayer (1.68 eV). The narrowed band gap in these bilayers is conducive to improve the light absorption efficiency, which will be discussed later. Intriguingly, the spatial distribution of CBM and VBM in the Janus WSSe bilayers depends on the stacking pattern. As illustrated in the inset of Figure 2, for the AB1 stacking pattern, the CBM distributes at the first layer, while the VBM is dominantly distributed at the second layer. However, for the AB2 and AB3 stacking patterns, both VBM and CBM uniformly spread over the both layers. Under illumination, the photo-excited electron jump to the CBM, at the same time, the photo-excited hole stay at the VBM. Hence, the locational separation of CBM and VBM normally could lower the carrier recombination. Consequently, from the perspective of increasing the free photogenerated carrier number, the AB1 stacking pattern is a better choice. It is also better than the monolayer, where both the CBM and VBM are located partially at the W atomic layer (see the inset of Figure 2a).

Redox Potential of the Photoexcited Carriers in Janus WSSe Bilayer
In order to overall split water, the adequate redox capacities of photo-generated carriers require that, in photocatalysts, the CBM position should be higher than the H + /H 2 reduction potential (−4.44 eV at pH = 0), while the VBM one lower than the H 2 O/O 2 oxidation potential (−5.67 eV at pH = 0). For the polar materials, such as Janus TMD and ferroelectrics, the intrinsic dipole has a strong impact on the CBM and VBM positions [17,[30][31][32]. Usually, for a given 2D polar material, the intrinsic dipole can be evaluated by the ∆Φ [31,32]. By this means, we study the stacking-dependent dipole moment of the Janus WSSe bilayers. As illustrated in Figure S1, due to the recovery of out-of-plane symmetry, the ∆Φ is 0 eV in the AB2 and AB3 stacking configurations. However, as to the case of AB1 stacking configuration, ∆Φ (1.37 eV) is nearly twice the one (0.73 eV) of Janus WSSe monolayer [17], which indicates an enhanced intrinsic dipole moment.
After taking the ∆Φ in consideration, we evaluate the redox potential of the photoexcited carriers in Janus WSSe bilayer through Yang's method [7,10]. As shown in Figure  S2, since both the oxidation potential of O 2 /H 2 O and reduction potential of H + /H 2 lie in the gap, the samples with AB1 and AB2 stacking patterns meet the redox potentials requirements for overall water-splitting reactions at pH = 0. In order to make the practical photocatalytic applications low-cost and eco-friendly, these overall water-splitting reactions would better take place in neutral environment (pH = 7). Because the pH dependence of band edge positions is exactly consistent with that of water redox potentials (0.059 × pH) [33,34] AB1 and AB2 configurations in theory are capable to catalyze watersplitting at pH = 7 (see Figure 3) Whereas, for the AB3 configuration, the VBM potential is higher than the O 2 /H 2 O oxidation potential, making it incapable for the oxygen evolution reaction (OER). For the hydrogen evolution reaction (HER), the CBM location of AB2 and AB3 configurations is too much close to the H + /H 2 reduction level, so that their hydrogen evolution performances probably cannot be high. However, with the help of enhanced ∆Φ, the CBM location of AB1 mode is greatly lifted, which is obviously higher than the H + /H 2 reduction level, promising a outstanding hydrogen evolution ability.

Redox Potential of the Photoexcited Carriers in Janus WSSe Bilayer
In order to overall split water, the adequate redox capacities of photo-generated carriers require that, in photocatalysts, the CBM position should be higher than the H + /H2 reduction potential (−4.44 eV at pH = 0), while the VBM one lower than the H2O/O2 oxidation potential (−5.67 eV at pH = 0). For the polar materials, such as Janus TMD and ferroelectrics, the intrinsic dipole has a strong impact on the CBM and VBM positions [17,[30][31][32]. Usually, for a given 2D polar material, the intrinsic dipole can be evaluated by the ΔΦ [31,32]. By this means, we study the stacking-dependent dipole moment of the Janus WSSe bilayers. As illustrated in Figure S1, due to the recovery of out-of-plane symmetry, the ΔΦ is 0 eV in the AB2 and AB3 stacking configurations. However, as to the case of AB1 stacking configuration, ΔΦ (1.37 eV) is nearly twice the one (0.73 eV) of Janus WSSe monolayer [17], which indicates an enhanced intrinsic dipole moment.
After taking the ΔΦ in consideration, we evaluate the redox potential of the photoexcited carriers in Janus WSSe bilayer through Yang's method [7,10]. As shown in Figure  S2, since both the oxidation potential of O2/H2O and reduction potential of H + /H2 lie in the gap, the samples with AB1 and AB2 stacking patterns meet the redox potentials requirements for overall water-splitting reactions at pH = 0. In order to make the practical photocatalytic applications low-cost and eco-friendly, these overall water-splitting reactions would better take place in neutral environment (pH = 7). Because the pH dependence of band edge positions is exactly consistent with that of water redox potentials (0.059 × pH) [33,34] AB1 and AB2 configurations in theory are capable to catalyze water-splitting at pH = 7 (see Figure 3) Whereas, for the AB3 configuration, the VBM potential is higher than the O2/H2O oxidation potential, making it incapable for the oxygen evolution reaction (OER). For the hydrogen evolution reaction (HER), the CBM location of AB2 and AB3 configurations is too much close to the H + /H2 reduction level, so that their hydrogen evolution performances probably cannot be high. However, with the help of enhanced ΔΦ, the CBM location of AB1 mode is greatly lifted, which is obviously higher than the H + /H2 reduction level, promising a outstanding hydrogen evolution ability.

Optical Absorption and STH Efficiency of Janus WSSe Bilayer
With regard to an ideal photocatalyst, the solar absorptivity, which could be assessed by the absorption coefficient ( ), should be good. Hereon, we study the stacking-dependent ( ) of Janus WSSe bilayer based on the formula below [32,35]: The pH is set to 7.

Optical Absorption and STH Efficiency of Janus WSSe Bilayer
With regard to an ideal photocatalyst, the solar absorptivity, which could be assessed by the absorption coefficient a(ω), should be good. Hereon, we study the stackingdependent a(ω) of Janus WSSe bilayer based on the formula below [32,35]: where ε 1 and ε 2 separately stand for the real and imaginary part of dielectric function. As plotted in Figure 4, among the visible range (380-780 nm), these three kinds of Janus WSSe bilayers each has several significant absorption peaks (>10 5 cm −1 ), indicating that they are hopeful visible-light-response candidates. Here, we draw the a(ω) of Janus WSSe monolayer as a contrast (black line). It could be found that, nearly all the visible light absorption peaks of the bilayers exhibit red-shift, comparing with the ones of monolayer, which could be explained by the visibly narrower band gaps as noted earlier. where ε1 and ε2 separately stand for the real and imaginary part of dielectric function. As plotted in Figure 4, among the visible range (380-780 nm), these three kinds of Janus WSSe bilayers each has several significant absorption peaks (>10 5 cm −1 ), indicating that they are hopeful visible-light-response candidates. Here, we draw the ( ) of Janus WSSe monolayer as a contrast (black line). It could be found that, nearly all the visible light absorption peaks of the bilayers exhibit red-shift, comparing with the ones of monolayer, which could be explained by the visibly narrower band gaps as noted earlier. Normally, for a water-splitting photocatalyst, the strong light absorption capacity in the visible-light area heralds a high STH efficiency. According to the method proposed by Yang's group [36], we estimate the energy conversion efficiency of these Janus WSSe bilayers with the data of band alignments (see Table S1) discussed above. As shown in Table  2, because of the ultrahigh light absorption efficiency (ηabs) and carrier utilization efficiency (ηcu), the STH efficiency (ηSTH) of AB1 mode arrives 31.22%. Even considering the instinct dipole contribution into the total energy, the corrected solar-to-hydrogen efficiency (η'STH) of AB1 mode is still up to 19.46%, far over the standard conversion efficiency of commercial applications for hydrogen production through photocatalytic water splitting (10%) [37]. It also markedly outstrips the one of Janus WSSe monolayer (11.68%) [17] and other reported photocatalysts, i.e., AgBiP2Se6 (10.04%) [30], and most M2X3 (M = Al, Ga, In; X = S, Se, Te) monolayers [36]. Nevertheless, the ηSTH of AB2 and AB3 configurations are not as high as the one of AB1 configuration, due to the badly low ηcu. As stated before, in the AB1 mode, the large ΔΦ raises the redox potentials, causing a high ηcu. Whereas, the out-of-plane symmetry recovery makes the ΔΦ disappear in AB2 and AB3 configurations. Their redox potentials are therefore much lower than the ones in AB1 configuration, so are their ηcu. To sum up, based on the staking-dependent STH efficiency of Janus WSSe bilayers considered in our study, the AB1 configuration is most suitable for the photocatalytic water-splitting application.  Normally, for a water-splitting photocatalyst, the strong light absorption capacity in the visible-light area heralds a high STH efficiency. According to the method proposed by Yang's group [36], we estimate the energy conversion efficiency of these Janus WSSe bilayers with the data of band alignments (see Table S1) discussed above. As shown in Table 2, because of the ultrahigh light absorption efficiency (η abs ) and carrier utilization efficiency (η cu ), the STH efficiency (η STH ) of AB1 mode arrives 31.22%. Even considering the instinct dipole contribution into the total energy, the corrected solar-to-hydrogen efficiency (η' STH ) of AB1 mode is still up to 19.46%, far over the standard conversion efficiency of commercial applications for hydrogen production through photocatalytic water splitting (10%) [37]. It also markedly outstrips the one of Janus WSSe monolayer (11.68%) [17] and other reported photocatalysts, i.e., AgBiP 2 Se 6 (10.04%) [30], and most M 2 X 3 (M = Al, Ga, In; X = S, Se, Te) monolayers [36]. Nevertheless, the η STH of AB2 and AB3 configurations are not as high as the one of AB1 configuration, due to the badly low η cu . As stated before, in the AB1 mode, the large ∆Φ raises the redox potentials, causing a high η cu . Whereas, the out-of-plane symmetry recovery makes the ∆Φ disappear in AB2 and AB3 configurations. Their redox potentials are therefore much lower than the ones in AB1 configuration, so are their η cu . To sum up, based on the staking-dependent STH efficiency of Janus WSSe bilayers considered in our study, the AB1 configuration is most suitable for the photocatalytic water-splitting application.

Geometric and Electronic Structures, and Photocatalytic Properties of Janus WSSe Double-Walled Nanotube
As the previous lines already suggested, due to the most negative bonding energy, the AB2 configuration is the most stable stacking pattern among these three cases, however, its photocatalytic performance is far less excellent than the one of AB1 mode. Therefore, making the stacking pattern of WSSe bilayer uniformly follow the AB1 style is a hopeful way to improve its photocatalytic performance. Rolling the WSSe bilayer into double-walled nanotube may realize this. It has been reported that, for a given diameter, the Janus MoSSe nanotubes with outer shell of selenium atoms and inner shell of sulfur atoms (Se-W-S, outside→inside) have a lower strain energy than the corresponding ones with the opposite structures (S-W-Se), which can be explained that, the selenium atom has a larger radius than the sulfur atom, and the inner atoms with larger radius likely have stronger repulsive force [28]. Therefore, it can be expected that, for the Janus WSSe DWNT, the stacking pattern predictably follows the AB1 style, which could be checked through comparing the strain energy of Janus WSSe DWNTs with different stacking patterns [38][39][40]. Since the armchair Janus MoSSe nanotube has been reported to be more energetically stable than the zigzag one [27,28], here we choose (15,15) and (8,8) Janus WSSe armchair nanotubes to build the Janus WSSe DWNT, where the distance between the inner and outer layers is close to the one of Janus WSSe bilayer (about 3 Å, see Figure 1). As shown in the Figure 5, four stacking configurations for the Janus WSSe DWNT are considered, namely Se-W-S-Se-W-S, Se-W-S-S-W-Se, S-W-Se-Se-W-S, and S-W-Se-S-W-Se (outer layer→inner layer), which are separately labelled as DWNT1, DWNT2, DWNT3, and DWNT4. In this work, the strain energy E str is defined as follows: where E NT and N NT are the total energy and the number of unit cells in the Janus WSSe DWNTs, meanwhile, E M and N M are corresponding value for the WSSe monolayer, respectively. As displayed in Figure 1, the DWNT1 configuration has the lowest strain energy among these four cases, which is in line with our expectation. Then we explore the electronic properties of Janus WSSe DWNT1 by investigating its electronic band structure and density of state (DOS). As shown in Figure 6a, it exhibits an indirect band gap of 0.65 eV, which is even smaller than the one of WSSe bilayer, ensuring the intensive light absorption in the visible areas (see Figure 4, red line). Moreover, as illustrated in Figure 6b-c, similar to the case of AB1 mode, in the Janus WSSe DWNT1, the CBM mainly scatter on the outer layer, while the VBM generally is located at the inner layer, which is coincident with the results of DOS (see Figure 6a). To get the ∆Φ between the inner and outer layers of Janus WSSe DWNT1, we calculate the ∆Φ of its component parts ((8, 8) and (15,15) nanotubes) with corresponding building block models first. Then, based on the fitting line of the relationship between the ∆Φ for Janus MXY (M = Mo, W; X, Y = S, Se, Te) monolayers and the ones for their own bilayers, we estimate the ∆Φ of Janus WSSe DWNT1. More calculation details could be found in the Supplementary Materials (see Figures S3-S5).
Notably, the ∆Φ (1.82 eV) of Janus WSSe DWNT1 is larger than the one (1.37 eV) of AB1 mode, indicating that rolling into DWNT could strengthen the ∆Φ of WSSe bilayer, which may be related to the strains appeared during the coiling processes [17]. The ∆Φ in the Janus WSSe DWNT1 will generate a built-in electric field pointing from the outer shell to the inner one. As depicted in the Figure 6d, this built-in electric field effectively pushes the photoexcited electrons to run to the outer layer, meanwhile it also forces the photoexcited holes to stay at the inner layer. Besides, as illustrated in Figure 3, the band edge positions of DWNT1 broadly straddle the standard water-splitting redox potential, which separately ensures the competent reduction ability for the HER at the outer surface, and the sufficient oxidation capacity the OER at the inner surface. What's more, the η' STH of DWNT1 reach up to 28.48%, which greatly transcends the ones of corresponding bilayers. Additionally, as a complement, we also perform a computational study on the energy conversion efficiency of WSSe (15,15) single-walled nanotube (SWNT), which is summarized in Table 2. Due to the narrower band gap (1.56 eV) and larger ∆Φ, the WSSe (15, 15) SWNT has an obviously higher η' STH (22.18%) than its monolayer (η' STH = 11.68%) [17]. This is consistent with the situation of DWNT. However, the η' STH of the SWNT is still inferior to the one of DWNT, indicating that, similar to the case of bilayer, the photocatalytic performance of SWNT could be improved by forming DWNT as well.
As the previous lines already suggested, due to the most negative bonding energy, the AB2 configuration is the most stable stacking pattern among these three cases, however, its photocatalytic performance is far less excellent than the one of AB1 mode. Therefore, making the stacking pattern of WSSe bilayer uniformly follow the AB1 style is a hopeful way to improve its photocatalytic performance. Rolling the WSSe bilayer into double-walled nanotube may realize this. It has been reported that, for a given diameter, the Janus MoSSe nanotubes with outer shell of selenium atoms and inner shell of sulfur atoms (Se-W-S, outside→inside) have a lower strain energy than the corresponding ones with the opposite structures (S-W-Se), which can be explained that, the selenium atom has a larger radius than the sulfur atom, and the inner atoms with larger radius likely have stronger repulsive force [28]. Therefore, it can be expected that, for the Janus WSSe DWNT, the stacking pattern predictably follows the AB1 style, which could be checked through comparing the strain energy of Janus WSSe DWNTs with different stacking patterns [38][39][40]. Since the armchair Janus MoSSe nanotube has been reported to be more energetically stable than the zigzag one [27,28], here we choose (15,15) and (8,8) Janus WSSe armchair nanotubes to build the Janus WSSe DWNT, where the distance between the inner and outer layers is close to the one of Janus WSSe bilayer (about 3 Å, see Figure  1). As shown in the Figure 5, four stacking configurations for the Janus WSSe DWNT are considered, namely Se-W-S-Se-W-S, Se-W-S-S-W-Se, S-W-Se-Se-W-S, and S-W-Se-S-W-Se (outer layer→inner layer), which are separately labelled as DWNT1, DWNT2, DWNT3, and DWNT4. In this work, the strain energy Estr is defined as follows: where and are the total energy and the number of unit cells in the Janus WSSe DWNTs, meanwhile, and are corresponding value for the WSSe monolayer, respectively. As displayed in Figure 1, the DWNT1 configuration has the lowest strain energy among these four cases, which is in line with our expectation.  Then we explore the electronic properties of Janus WSSe DWNT1 by investigating its electronic band structure and density of state (DOS). As shown in Figure 6a, it exhibits an indirect band gap of 0.65 eV, which is even smaller than the one of WSSe bilayer, ensuring the intensive light absorption in the visible areas (see Figure 4, red line). Moreover, as illustrated in Figure 6b-c, similar to the case of AB1 mode, in the Janus WSSe DWNT1, the CBM mainly scatter on the outer layer, while the VBM generally is located at the inner layer, which is coincident with the results of DOS (see Figure 6a). To get the ΔΦ between the inner and outer layers of Janus WSSe DWNT1, we calculate the ΔΦ of its component parts ((8, 8) and (15,15) nanotubes) with corresponding building block models first. Then, based on the fitting line of the relationship between the ΔΦ for Janus MXY (M = Mo, W; X, Y = S, Se, Te) monolayers and the ones for their own bilayers, we estimate the ΔΦ of Janus WSSe DWNT1. More calculation details could be found in the Supplementary Materials (see Figures S3-S5). Notably, the ΔΦ (1.82 eV) of Janus WSSe DWNT1 is larger than the one (1.37 eV) of AB1 mode, indicating that rolling into DWNT could strengthen the ΔΦ of WSSe bilayer, which may be related to the strains appeared during the coiling processes [17]. The ΔΦ in the Janus WSSe DWNT1 will generate a built-in electric field pointing from the outer shell to the inner one. As depicted in the Figure 6d, this built-in electric field effectively pushes the photoexcited electrons to run to the outer layer, meanwhile it also forces the photoexcited holes to stay at the inner layer. Besides, as illustrated in Figure  3, the band edge positions of DWNT1 broadly straddle the standard water-splitting redox potential, which separately ensures the competent reduction ability for the HER at the outer surface, and the sufficient oxidation capacity the OER at the inner surface. What's more, the η'STH of DWNT1 reach up to 28.48%, which greatly transcends the ones of corresponding bilayers. Additionally, as a complement, we also perform a computational study on the energy conversion efficiency of WSSe (15,15) single-walled nanotube (SWNT), which is summarized in Table 2. Due to the narrower band gap (1.56 eV) and larger ΔΦ, the WSSe (15, 15) SWNT has an obviously higher η'STH (22.18%) than its monolayer (η'STH = 11.68%) [17]. This is consistent with the situation of DWNT. However, the η'STH of the SWNT is still inferior to the one of DWNT, indicating that, similar to the case of bilayer, the photocatalytic performance of SWNT could be improved by forming DWNT as well.

Conclusions
From the geometric, electronic, optical and chemical properties, we have presented comprehensive DFT calculations to investigate stacking-dependent photocatalytic performance of the Janus WSSe bilayer. Though the AB2 stacking sample is energetically most stable, due to the recovery of the out-of-plane symmetry, its redox capacity is fairly low, which seriously restricts the STH conversion efficiency. Moreover, the spatial overlap of CBM and VBM probably causes a high carrier recombination. Fortunately, changing the stacking pattern into AB1 style could effectively overcome these drawbacks above. Based on the strain energy, we find that rolling the Janus WSSe bilayer into double-walled nanotube is a promising path to stably realize the AB1 configuration. Notably, because of the enhanced intrinsic dipole, the STH conversion efficiency of the WSSe DWNT1 is even higher than that of the AB1 configuration. Therefore, constructing the double-walled nanotubes is a promising approach to improve the photocatalytic performances for the Janus WSSe layered materials.
Supplementary Materials: The following are available online at https://www.mdpi.com/2079-499 1/11/3/705/s1, Figure S1: Planar average electrostatic potential energy of the bilayer WSSe with (a) AB1, (b) AB2, and (c) AB3 stacking models, respectively, Figure S2: The potential of the CBM (blue region) and VBM (red region) of WSSe bilayer with different stacking patterns (AB1, AB2, and AB3) and WSSe DWNT1, with respect to the vacuum level (labeled as 0 eV), Figure S3: Average electrostatic potentials in the plane normal to the tube axis for the whole structure of Janus WSSe DWNT1, Figure S4: The fitting line of the relationship between the electrostatic potential difference for Janus MXY (M = Mo, W; X, Y = S, Se, Te) monolayers (∆Φ monolayer ) and the ones for their own bilayers (∆Φ bilayer ), Figure S5: Average electrostatic potentials in the plane normal to the tube axis for the whole structure (left) and building block (right) of (a) (8,8) and (b) (15, 15) Janus WSSe armchair nanotubes, Table S1: Over-Potential for Hydrogen Evolution Reaction χ(H 2 ), Over-Potential for Oxygen Evolution Reaction χ(O 2 ), Direct Band Gaps, and Difference of Electrostatic Potential (∆Φ) of Janus WSSe A-NTs.