Tuning band alignment and optical properties of 2D van der Waals heterostructure via ferroelectric polarization switching

Abstract

Favourable band alignment and excellent visible light response are vital for photochemical water splitting. In this work, we have theoretically investigated how ferroelectric polarization and its reversibility in direction can be utilized to modulate the band alignment and optical absorption properties. For this objective, 2D van der Waals heterostructures (HTSs) are constructed by interfacing monolayer MoS₂ with ferroelectric In₂Se₃. We find the switch of polarization direction has dramatically changed the band alignment, thus facilitating different type of reactions. In In₂Se₃/MoS₂/In₂Se₃ heterostructures, one polarization direction supports hydrogen evolution reaction and another polarization direction can favour oxygen evolution reaction. These can be used to create tuneable photocatalyst materials where water reduction reactions can be selectively controlled by polarization switching. The modulation of band alignment is attributed to the shift of reaction potential caused by spontaneous polarization. Additionally, the formed type-II van der Waals HTSs also significantly improve charge separation and enhance the optical absorption in the visible and infrared regions. Our results pave a way in the design of van der Waals HTSs for water splitting using ferroelectric materials.

Graphical abstract

Keywords

photocatalyst / ferroelectric / MoS₂ / In₂Se₃ / heterostructures / water splitting / 2D materials

1 Introduction

In the context of searching sustainable, economical, and efficient energy generation processes, nature provides perfect insight on what future energy generation should be like. In photosynthesis process, solar energy is converted into chemical energy by plants and such type of conversion can be achieved also through photocatalyst materials, even though the efficiency is far below the natural photosynthesis process. Photocatalyst material has gained immense popularity in research community since the first discovery of ZnO in 1911 [1]. Later, another popular photocatalyst, TiO₂, was reported in 1938, through production of active oxides by absorbing UV on the surface [2]. After several decades Fujishima et al. introduced electrochemical photolysis of water using TiO₂ and Pt electrodes [3]. Since then water splitting through solar irradiation into hydrogen and oxygen identified as the efficient reaction for solar chemical energy conversion. Later various photocatalyst such as CeO₂ [4], BiVO₄ [5], WO₃ [6], CdS [7], GaN [8], WSe₂ [9], and InGaP [10] were reported. Among them BiVO₄ based photoanode has demonstrated the highest conversion efficiencies around 5.2% [11]. With the discovery of graphene, the use of two-dimensional (2D) materials as a photocatalyst become major focus owing to their unique characteristics such as ideal band gap, and high surface area. 2D materials such as MoS₂ [12, 13], SiP [14], g-C₃N₄ [15], InSe [16], PdSeO₃ [17], and In₂Se₃ [18] have shown good photocatalytic properties for water splitting.

Theoretically, a perfect efficient photocatalyst should possess a bandgap higher than 1.23 eV but not greater than 3.0 eV. Here 1.23 eV is the free energy of water splitting and band gap should be less than 3.0 eV for efficient absorbance of the solar energy [19, 20]. In addition, band edges of photocatalyst material should be aligned with potentials of hydrogen evolution reaction (HER) and oxygen evolution reaction (OER). Conduction Band Minimum (CBM) should be higher than -4.44 eV (pH = 0 for efficient HER reaction) [21]. Valence Band Minimum (VBM) should be lower than -5.67 eV (pH = 0), which is the potential for OER reaction [21]. Eqs. (1) and (2) illustrate the OER and HER where electrons and holes are denoted by e and h, respectively. Additionally, e and h must be well separated. In general, it is extremely challenging to find an efficient photocatalytic material which satisfies all these requirements

\begin{array}{}\text{(1)}& {\text{H}}_2\text{O}+2{\text{h}}^{+}\to 2{\text{H}}^{+}+\frac{1}{2}{\text{O}}_2\mathrm{,}\end{array}

\begin{array}{}\text{(2)}& 2{\text{H}}^{+}+2{\text{e}}^{-}\to {\text{H}}_2\mathrm{.}\end{array}

Ferroelectric material exhibits spontaneous polarization in the absence of external electric field. Some research works have demonstrated influence of changing polarization direction on catalytic surface absorption and desorption [22], Schottky tunnel barrier [23], e–h charge separation [24], and bandstructure engineering [25]. Recently few 2D ferroelectric materials with out-of-plane polarization have been discovered and synthesized [26-28]. One interesting question is to investigate possibility of improving the photocatalysis performance by forming van der Waals Heterostructures (HTSs) with ferroelectric materials. It is highly expected that ferroelectric materials with spontaneous polarization and reversibility of the polarization direction can significantly modulate the band alignment. Additionally, by forming Type-II van der Waals HTSs with ferroelectric materials, it can also facilitate charge separation and enhance optical absorption in photocatalyst [29]. 2D α-In₂Se₃ with a band gap of 1.3 eV is an experimentally realized ferroelectric material that have demonstrated potential for water splitting applications [18, 30]. 2D In₂Se₃ possesses considerable out-of-plane ferroelectric polarization and the band gap can be reduced when it forms a vertical HTS with WSe₂ monolayers [31]. In₂Se₃/MoS₂ HTS has been experimentally synthesized and used as a photoanode in water splitting [32]. Optical absorption properties of In₂Se₃/MoS₂ HTS has been improved compared to intrinsic In₂Se₃ and MoS₂ [33].

2D MoS₂ has a bandgap around 1.8 eV [34] which is larger than 1.23 eV and its band alignment support both HER and OER [12]. MoS₂ has good carrier mobility and high surface to volume ratio butcharge recombination is much higher considering the amount of charge generated. It also has the leverage to reduce from its original band gap of 1.8 eV to 1.23 eV, which automatically increases the light absorption at infrared wavelength. Studies have shown, constructing HTS with certain materials can reduce both band gap and charge recombination in MoS₂ [35-37]. In HTSs, charge recombination is reduced greatly when CBM and VBM appears in two different layers.

In this work, we have constructed HTSs using 2D MoS₂ and ferroelectric In₂Se₃ and found that band position can only support HER or OER depending on the ferroelectric polarization direction. When MoS₂ layer is sandwiched between top and bottom In₂Se₃ to form In₂Se₃/MoS₂/In₂Se₃ HTS, three distinct polarization states can be produced and each exhibit different band alignment with water reduction potential. One polarization state facilitates both HER and OER and the other polarizations states only facilitate either OER or HER. This allows us to shift the water reduction potentials in photocatalyst material simply by switching ferroelectric polarization in ferroelectric materials, potentially enhance the photocatalysis efficiency in water splitting. Additionally, the formed type-II van der Waals HTSs also significantly improve charge separation and enhance the optical absorption in the visible and infrared regions.

2 Methodology

All calculations were conducted by using density functional theory (DFT) with the assistance of Vienna Ab Initio Simulation Package (VASP) [38]. Core and valence electrons were interpreted using projected-wave (PAW) argument method [39, 40]. Both Perdew, Burke, and Ernzerh (PBE) method [41] and Heyd–Scuseria–Ernzerhof (HSE) method [42] were used in the calculation of bandstructure. DFT-D3 methods was used to incorporate long range van der Waals interactions in HTSs [43]. Vacuum region of 20 Å was maintained to prevent interaction between periodic images. Cut-off energy of 500 eV was used and the convergence criteria of forces on the atoms were set to -0.001 eV/Å and energy convergence criteria was set to 1E-06. 5×5×1 and 7×7×1 gamma centred k-point meshes were used for the geometry relaxations and electron structure calculations, respectively.

2D In₂Se₃ and MoS₂ crystal structures were illustrated in Figs. 1(a)–(d). Table 1 shows the calculated lattice parameters for In₂Se₃ and MoS₂ and those are in good agreement with the experimental values. First, 2D In₂Se₃/MoS₂ HTSs were constructed using 3×3 In₂Se₃ and 4×4 MoS₂ unit cells to achieve minimum lattice mismatch (3%). Two different polarization states (type-A and type-B) for 2D In₂Se₃/MoS₂ HTSs were illustrated in Figs. 1(e) and (f). Next, In₂Se₃/MoS₂/In₂Se₃ HTSs were constructed as shown in Figs. 1(g)–(i) by sandwiching MoS₂ between In₂Se₃ layers. Based on polarization direction, three different polarizations states can be found and they were denoted as type-BB, type-AA, and type-AB. Reversing of the polarization occurs through rearrangement of the contacted surfaces (A and B). Naming of the HTSs are based on which side of In₂Se₃ is in contact with MoS₂. For example, HTS is named as type-AA, when the side A of In₂Se₃ layers are in contact with MoS₂ from both top and bottom sides.

3 Results and discussion

3.1 Methodology

Bandstructure of the intrinsic In₂Se₃ and MoS₂ was calculated using both HSE and PBE methods. As given in Table 2, our calculated values have good agreement to both experimental and previous theoretical calculations. According to the results, MoS₂ bandgap was accurate when calculated by PBE method, while bandgap of In₂Se₃ was accurate when calculated using HSE method.

Binding energies (E_b) of all HTSs were calculated using Eq. (3) to check the stability of HTSs. Here E_tot, E_{{\text{In}}_2\text{S}{\text{e}}_3}, and E_{{\text{MoS}}_2} are the total energy of the system, energy of In₂Se₃ layer and energy of MoS₂ layer, respectively. For the HTSs with single layer of In₂Se₃ one E_{{\text{In}}_2\text{S}{\text{e}}_3} term can be neglected. Binding energies of all HTSs having negative values as indicated in Table 3, suggesting, HTSs are stable and energy wise favourable to form

\begin{array}{}\text{(3)}& E_b=E_{\text{tot}}-E_{{\text{In}}_{\text{2}}{\text{Se}}_{\text{3}}(\text{top})}-E_{{\text{MoS}}_{\text{2}}}-E_{{\text{In}}_{\text{2}}{\text{Se}}_{\text{3}}\text{(bottom)}}\mathrm{.}\end{array}

3.2 In₂Se₃/MoS₂ heterostructures

Bandstructure and contribution of each layer to its bands have been investigated in In₂Se₃/MoS₂ HTS as illustrated in Fig. 2. Type-A HTS shows band gap of 0.66 eV (PBE method) which is a reduced bandgap compared to intrinsic In₂Se₃ and MoS₂. On the other hand, type-B HTS has band gap of 0.84 eV (PBE method) which is similar to the band gap of In₂Se₃. According to the bandstructure of type-A HTS, valence band is contributed by MoS₂ layer and conduction band is contributed by In₂Se₃ layer. Hence charge recombination is greatly reduced owing to transferring of holes and electrons into two different layers. But in type-B configuration, both conduction and valence bands are mainly contributed by the In₂Se₃ layer, thus charge separation is not effectively facilitated. According to research, intrinsic In₂Se₃ also has better charge separation compared to MoS₂ monolayer due to its inbuilt polarization [18].

Figures 3(a) and (b) illustrate the electrostatic potential (eV) of both HTSs and it shows different vacuum levels at each end owing to the inbuilt polarization. Figures 4(c) and (d) show the band alignment of type-A and type-B configurations. Type-A have CBM above the HER potential and VBM below the OER potential, thus, it supports both reactions. However, in type-B configuration, VBM is higher than the OER potential and CBM is below the HER potential. Therefore, only HER is supported in type-B configuration. Results exhibit change of polarization direction has strong impact on band alignment. Through close observation, it can be seen that spontaneous polarization also plays critical role in the context of band aligning. In type-A HTS, CBM is -5.42 eV and yet it is above the HER potential. This happens as a result of different vacuum levels at both ends. Only In₂Se₃ have contributed to the CBM and vacuum level is lower at the end of In₂Se₃. Therefore, when considering HER potential respect to the vacuum level at the side of In₂Se₃, it becomes lower than the CBM. On the other hand, in type-B HTS, both CBM and VBM occurs in In₂Se₃ layer, which has the greater vacuum level. Therefore, the both potential values taken respect to same vacuum level. This feature is unique to polarized materials and it can assist better band alignment in photocatalyst materials.

3.3 In₂Se₃/MoS₂/In₂Se₃ heterostructures

Here, HTSs were constructed by sandwiching the MoS₂ layer between In₂Se₃ layers. In this arrangement, polarization becomes stronger and its controllability can be improved. Based on polarization direction, three states, namely, AA, BB, and AB can be identified. In type-AA and type-BB HTSs, In₂Se₃ layers have polarization directions opposing to each other, hence HTSs have zero net spontaneous polarization. In AB HTS, polarization direction of the two layers are in same direction, thus, it is subjected to very high spontaneous polarization. Figure 4 shows the band structure and contribution to bands from each layer. In type-AA, CBM is contributed by both top and bottom In₂Se₃ layers and VBM is by the MoS₂. In type-BB, both CBM and VBM occur in both In₂Se₃ layers. In type-AB CBM occurs in top In₂Se₃ layer and VBM occurs in bottom In₂Se₃ layer. Therefore, type-AA and type-AB types have good charge separation while charge separation has not improved in type-BB.

As illustrated in Fig. 5, type-AA configuration supports only HER and type-BB configuration supports only OER. But Type-AB configuration supports both reactions owing to assistance from existing inbuilt spontaneous polarization. This dramatic change of band alignment respect to the direction of polarization can lead to tuneable photocatalyst materials which its selectivity and efficiency of the chemical reactions can be controlled by changing the polarization direction.

Finally, optical absorption coefficient of all HTSs, MoS₂ and In₂Se₃ have been studied as shown in Fig. 6. All HTSs except type-B HTS have shown increased optical absorption coefficient compared to In₂Se₃ and MoS₂ around the ultraviolet (UV) region. In visual light region, all HTSs, MoS₂, and In₂Se₃ have shown similar absorption coefficient values. As highlighted in Fig. 6(b), optical absorption at infrared region also has shown considerable increase in all HTSs compared to MoS₂ The increase of infrared light absorption occurs due to reduction of bandgap in HTSs.

4 Conclusion

In this work, In₂Se₃/MoS₂ heterostructures with two polarization states, namely, type-A and type-B and In₂Se₃/MoS₂/In₂Se₃ heterostructures with three polarization states, namely, type-AA, type-BB, and type-AB have been studied. Type-A and Type-AB HTSs support both OER and HER. Type-B and type-AA HTSs only support HER while Type-BB HTS only supports OER. Through HTSs formation, charge separation can be enhanced in photocatalyst materials as reflected by type-A, type-AA, and type-AB configurations. All HTSs have shown increased optical absorption coefficient at UV and infrared region of the solar spectrum compared to monolayer MoS₂ and In₂Se₃.

The ability to change band alignment through altering the polarization direction is one of the important findings of this work. This feature is evident in type-AA, type-BB, and type-AB HTSs which their band alignments were drastically changed respective to the polarization direction. Applying this principle, external electric field can be used to reverse the polarization and change the material properties from photoanode to photocathode or vice versa. This also allows to change reaction paths according to the concentration of the products generated in the solution. If the products of one reaction is greater than the other, then the polarization direction can be changed to facilitate the other reaction using the same photocatalyst material. Interestingly, fast switching between In₂Se₃/MoS₂/In₂Se₃ three polarization states (AA, AB, BB) perhaps can provide an interesting result in water splitting applications. Another important finding is how the spontaneous polarization have shifted OER or HER potential values as shown in type-A and type-AB HTS configurations to facilitate the band alignment. Overall results of this work provide valuable insight on how ferroelectric materials can be used in photocatalyst application to improve their performance and may lead to development of next generation tuneable photocatalyst for water splitting.

References

[1] J. M. Coronado, A Historical Introduction to Photocatalysis, in: Design of Advanced Photocatalytic Materials for Energy and Environmental Applications, J. M. Coronado, F. Fresno, M. D. Hernández-Alonso, and R. Portela (Eds.), Springer London: London, 2013, pp 1–4

[2] C. F. Goodeve and J. A. Kitchener, The mechanism of photosensitisation by solids, Trans. Faraday Soc. 34(0), 902 (1938)

[3] A. Fujishima and K. Honda, Electrochemical photolysis of water at a semiconductor electrode, Nature 238(5358), 37 (1972)

[4] D. Channei, B. Inceesungvorn, N. Wetchakun, S. Ukritnukun, A. Nattestad, J. Chen, and S. Phanichphant, Photocatalytic degradation of methyl orange by CeO₂ and Fe-doped CeO₂ films under visible light irradiation, Sci. Rep. 4(1), 5757 (2014)

[5] F. F. Abdi, L. Han, A. H. M. Smets, M. Zeman, B. Dam, and R. van de Krol, Efficient solar water splitting by enhanced charge separation in a bismuth vanadate-silicon tandem photoelectrode, Nat. Commun. 4(1), 2195 (2013)

[6] P. Dong, G. Hou, X. Xi, R. Shao, and F. Dong, WO₃-based photocatalysts: Morphology control, activity enhancement and multifunctional applications, Environ. Sci. Nano 4(3), 539 (2017)

[7] M. Luo, Y. Liu, J. Hu, H. Liu, and J. Li, One-pot synthesis of CdS and Ni-doped CdS hollow spheres with enhanced photocatalytic activity and durability, ACS Appl. Mater. Interfaces 4(3), 1813 (2012)

[8] T. Kida, Y. Minami, G. Guan, M. Nagano, M. Akiyama, and A. Yoshida, Photocatalytic activity of gallium nitride for producing hydrogen from water under light irradiation, J. Mater. Sci. 41(11), 3527 (2006)

[9] A. Eftekhari, Tungsten dichalcogenides (WS₂, WSe₂, and WTe₂): Materials chemistry and applications, J. Mater. Chem. A 5(35), 18299 (2017)

[10] P. Varadhan, H. C. Fu, Y. C. Kao, R. H. Horng, and J. H. He, An efficient and stable photoelectrochemical system with 9% solar-to-hydrogen conversion efficiency via InGaP/GaAs double junction, Nat. Commun. 10(1), 5282 (2019)

[11] L. Han, F. F. Abdi, R. van de Krol, R. Liu, Z. Huang, H. J. Lewerenz, B. Dam, M. Zeman, and A. H. M. Smets, Efficient water-splitting device based on a bismuth vanadate photoanode and thin-film silicon solar cells, ChemSusChem 7(10), 2832 (2014)

[12] Y. Li, Y. L. Li, C. M. Araujo, W. Luo, and R. Ahuja, Single-layer MoS₂ as an efficient photocatalyst, Catal. Sci. Technol. 3(9), 2214 (2013)

[13] J. Mao, Y. Wang, Z. Zheng, and D. Deng, The rise of two-dimensional MoS₂ for catalysis, Front. Phys. 13(4), 138118 (2018)

[14] Z. Ma, J. Zhuang, X. Zhang, and Z. Zhou, SiP monolayers: New 2D structures of group IV-V compounds for visible-light photohydrolytic catalysts, Front. Phys. 13(3), 138104 (2018)

[15] Y. Wang, M. Miao, J. Lv, L. Zhu, K. Yin, H. Liu, and Y. Ma, An effective structure prediction method for layered materials based on 2D particle swarm optimization algorithm, J. Chem. Phys. 137(22), 224108 (2012)

[16] H. L. Zhuang and R. G. Hennig, Single-layer group-III monochalcogenide photocatalysts for water splitting, Chem. Mater. 25(15), 3232 (2013)

[17] M. Qiao, J. Liu, Y. Wang, Y. Li, and Z. Chen, PdSeO₃ monolayer: Promising inorganic 2D photocatalyst for direct overall water splitting without using sacrificial reagents and cocatalysts, J. Am. Chem. Soc. 140(38), 12256 (2018)

[18] P. Zhao, Y. Ma, X. Lv, M. Li, B. Huang, and Y. Dai, Two-dimensional III₂-VI₃ materials: Promising photocatalysts for overall water splitting under infrared light spectrum, Nano Energy 51, 533 (2018)

[19] R. M. Navarro Yerga, M. C. Álvarez Galván, F. del Valle, J. A. Villoria de la Mano, and J. L. G. Fierro, Water splitting on semiconductor catalysts under visible-light irradiation, ChemSusChem 2(6), 471 (2009)

[20] A. Kudo and Y. Miseki, Heterogeneous photocatalyst materials for water splitting, Chem. Soc. Rev. 38(1), 253 (2009)

[21] M. Ni, M. K. H. Leung, D. Y. C. Leung, and K. Sumathy, A review and recent developments in photocatalytic water-splitting using TiO₂ for hydrogen production, Renew. Sustain. Energy Rev. 11(3), 401 (2007)

[22] A. Kakekhani and S. Ismail-Beigi, Ferroelectric-based catalysis: Switchable surface chemistry, ACS Catal. 5(8), 4537 (2015)

[23] X. Liu, Y. Wang, J. D. Burton, and E. Y. Tsymbal, Polarization-controlled Ohmic to Schottky transition at a metal/ferroelectric interface, Phys. Rev. B 88(16), 165139 (2013)

[24] D. Kim, H. Han, J. H. Lee, J. W. Choi, J. C. Grossman, H. M. Jang, and D. Kim, Electron hole separation in ferroelectric oxides for efficient photovoltaic responses, Proc. Natl. Acad. Sci. USA 115(26), 6566 (2018)

[25] D. Wijethunge, C. Tang, C. Zhang, L. Zhang, X. Mao, and A. Du, Bandstructure engineering in 2D materials using ferroelectric materials, Appl. Surf. Sci. 513, 145817 (2020)

[26] F. Liu, L. You, K. L. Seyler, X. Li, P. Yu, J. Lin, X. Wang, J. Zhou, H. Wang, H. He, S. T. Pantelides, W. Zhou, P. Sharma, X. Xu, P. M. Ajayan, J. Wang, and Z. Liu, Room-temperature ferroelectricity in CuInP₂S₆ ultrathin flakes, Nat. Commun. 7(1), 12357 (2016)

[27] S. Yuan, X. Luo, H. L. Chan, C. Xiao, Y. Dai, M. Xie, and J. Hao, Room-temperature ferroelectricity in MoTe₂ down to the atomic monolayer limit, Nat. Commun. 10(1), 1775 (2019)

[28] A. Chandrasekaran, A. Mishra, and A. K. Singh, Ferroelectricity, antiferroelectricity, and ultrathin 2D electron/hole gas in multifunctional monolayer MXene, Nano Lett. 17(5), 3290 (2017)

[29] J. Low, J. Yu, M. Jaroniec, S. Wageh, and A. A. Al-Ghamdi, Heterojunction Photocatalysts 29(20), 1601694 (2017)

[30] C. Cui, W. J. Hu, X. Yan, C. Addiego, W. Gao, Y. Wang, Z. Wang, L. Li, Y. Cheng, P. Li, X. Zhang, H. N. Alshareef, T. Wu, W. Zhu, X. Pan, and L. J. Li, Intercorrelated in-plane and out-of-plane ferroelectricity in ultrathin two-dimensional layered semiconductor In₂Se₃, Nano Lett. 18(2), 1253 (2018)

[31] W. Ding, J. Zhu, Z. Wang, Y. Gao, D. Xiao, Y. Gu, Z. Zhang, and W. Zhu, Prediction of intrinsic two-dimensional ferroelectrics in In₂Se₃ and other III₂-VI₃ van der Waals materials, Nat. Commun. 8(1), 14956 (2017)

[32] Y. Jiang, Q. Wang, L. Han, X. Zhang, L. Jiang, Z. Wu, Y. Lai, D. Wang, and F. Liu, Construction of In₂Se₃/MoS₂ heterojunction as photoanode toward efficient photoelectrochemical water splitting, Chem. Eng. J. 358, 752 (2019)

[33] J. R. Zhang, X. Z. Deng, B. Gao, L. Chen, C. T. Au, K. Li, S. F. Yin, and M. Q. Cai, Theoretical study on the intrinsic properties of In₂Se₃/MoS₂ as a photocatalyst driven by near-infrared, visible and ultraviolet light, Catal. Sci. Technol. 9(17), 4659 (2019)

[34] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically thin MoS₂: A new direct-gap semiconductor, Phys. Rev. Lett. 105(13), 136805 (2010)

[35] H. Li, K. Yu, Z. Tang, H. Fu, and Z. Zhu, High photocatalytic performance of a type-II α-MoO₃@MoS₂ heterojunction: From theory to experiment, Phys. Chem. Chem. Phys. 18(20), 14074 (2016)

[36] Q. Li, N. Zhang, Y. Yang, G. Wang, and D. H. L. Ng, High efficiency photocatalysis for pollutant degradation with MoS₂/C₃N₄ heterostructures, Langmuir 30(29), 8965 (2014)

[37] F. Li, C. Shi, D. Wang, G. Cui, P. Zhang, L. Lv, and L. Chen, Improved visible-light absorbance of monolayer MoS₂ on AlN substrate and its angle-dependent electronic structures, Phys. Chem. Chem. Phys. 20(46), 29131 (2018)

[38] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54(16), 11169 (1996)

[39] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59(3), 1758 (1999)

[40] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50(24), 17953 (1994)

[41] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77(18), 3865 (1996)

[42] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118(18), 8207 (2003)

[43] S. Grimme, J. Antony, S. Ehrlich, and, H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys. 132(15), 154104 (2010)

[44] C. Ataca, M. Topsakal, E. Aktürk, and S. Ciraci, A comparative study of lattice dynamics of three- and two-dimensional MoS₂, J. Phys. Chem. C 115(33), 16354 (2011)

[45] P. Joensen, R. F. Frindt, and S. R. Morrison, Single-layer MoS₂, Mater. Res. Bull. 21(4), 457 (1986)

[46] Z. Xie, F. Yang, X. Xu, R. Lin, and L. M. Chen, Functionalization of α-In₂Se₃ monolayer via adsorption of small molecule for gas sensing, Front. Chem. 6, 430 (2018)

[47] A. Kuc, N. Zibouche, and T. Heine, Influence of quantum confinement on the electronic structure of the transition metal sulfide TS₂, Phys. Rev. B 83(24), 245213 (2011)

[48] S. Lebègue and O. Eriksson, Electronic structure of two-dimensional crystals from ab initio theory, Phys. Rev. B 79(11), 115409 (2009)

[49] C. Ataca and S. Ciraci, Functionalization of single-layer MoS₂ honeycomb structures, J. Phys. Chem. C 115(27), 13303 (2011)

[50] K. Kośmider and J. Fernández-Rossier, Electronic properties of the MoS₂–WS₂ heterojunction, Phys. Rev. B 87(7), 075451 (2013)

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