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C: Physical Processes in Nanomaterials and Nanostructures
Adsorption Induced Indirect-to-Direct Band Gap Transition in Monolayer Blue Phosphorus Wangping Xu, Jinzhu Zhao, and Hu Xu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05125 • Publication Date (Web): 11 Jun 2018 Downloaded from http://pubs.acs.org on June 12, 2018
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Adsorption Induced Indirect-to-Direct Band Gap Transition in Monolayer Blue Phosphorus
Wangping Xu, Jinzhu Zhao,* and Hu Xu* Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
ABSTRACT: In this work, we systematically studied adsorption induced indirect-to-direct band gap transition in monolayer blue phosphorus from first-principles calculations by combining one-shot GW approximation and the Bethe-Salpeter equation. Our results revealed that surface adsorption (i.e., O2, -OH, -COOH, -CN) strongly modifies the conduction and valence band edges, resulting in an indirect-to-direct band gap transition. More importantly, the direct band gap can be dramatically tuned by either the in-plane strain or the coverage ratio of adsorbates, which enables monolayer blue phosphorus to efficiently adsorb visible light. The mechanism of strain effect and surface adsorption on band gap tuning was deeply discussed. Moreover, our results clearly showed that the adsorbates have an important influence on the exciton binding energies (EBE), while the coverage of adsorbates play a crucial role in the linear scaling behavior between EBE and quasi-particle band gap. Our findings suggest that monolayer blue phosphorus has potential applications in electro-optical devices.
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1. INTRODUCTION Recently, layered black phosphorus (BP) has received considerable attention due to its unique properties including the tunable band gap with high carrier mobility, the on/off current ratio up to 105 and a negative Poisson’s ratio, which make it to be a promising material in electronic and optical application.1-9 It is interesting to note that the ‘brother’ of BP, monolayer blue phosphorus (MBP) in hexagonal phase, is predicted to have the similar structural stability as BP from first-principles calculations.10-12 Experimentally, the MBP-like islands by epitaxial growth have been widely reported.13-16 Similar to BP, MBP is also proposed to have high carrier mobility17 that makes it become a nice candidate for future electronic devices, especially in optical-electronic industry. However，there are mainly two problems need to be well solved. On the one hand, its band gap of 3.23 eV18-21 is beyond the visible light region. On the other hand, the indirect band gap feature of MBP suppresses the direct optical absorption and strongly limits its potential optical-electronic applications as well. Therefore, it is desired to decrease the band gap width of MBP and tune the indirect feature to a direct one. We noticed that previous works18, 20 have already paid attention to these two issues. Although the in-plane strain approach is able to modify the band gap width of MBP, it remains an indirect band gap.22-23 To engineer a direct band gap, several approaches have been tried but never perfectly solve this problem. For instance, an extreme high external electric field of 0.2 V/Å might lead to a direct band gap, however, which is not easy to be achieved experimentally.20 An alternative option is to substitute P atoms by other elements, e.g. B, Al, or Sb.24-26 Unfortunately, neither the doping sites nor the band gap width can be well manipulated.24 Surface adsorption is another promising option. Similar to the case of BP, oxygen adsorption also significantly affects the electronic properties of MBP.27-28 For example, Yang et al. proposed that MBP with full coverage of oxygen experiences a strain-induced quantum phase transition.27 However, the mechanism of indirect-to-direct band gap transition caused by oxygen adsorption is still lacking. Furthermore, it is interesting 2
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and important to reveal the interplay between the indirect-to-direct band gap transition and surface adsorption. In this work, by performing first-principles calculations combined with one-shot GW approximation (G0W0)29 and Bethe-Salpeter equation (BSE),30-31 we investigated surface adsorption induced indirect-to-direct band gap transition of MBP. The coverage of adsorbates and in-plane strain effect on the electronic and optical properties were discussed. Our results showed that the electronic behaviors of MBP can be effectively modified to form a direct band gap by surface adsorption. The band gap is able to be tuned to the visible light region by means of in-plane strain and the coverage of adsorbates, while the corresponding mechanisms were analyzed in detail. In addition, the influence of surface adsorption on the exciton binding energy (EBE) was also studied. Our work not only provides a feasible method to realize indirect-to-direct band gap transition but also offers a more efficient way to manipulate the band gap width of MBP.
2. COMPUTATIONAL METHODS All first-principles calculations were performed in the framework of density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP).32-33 Potentials based on the projector augmented wave (PAW)34 method were used to describe ion-electron interactions. The energy cutoff for plane-wave basis set was 450 eV. All atoms were fully relaxed until the forces acting on each atom are less than 0.01 eV/Å. The Perdew-Burke-Ernzerhof35 functional combined with the G0W0 approximation (PBE-G0W0) was used while the excitonic effects were considered by employing BSE method.30-31 To obtain quasi-particle (QP) band structures, the wavefunction was expanded using maximally localized Wannier functions basis36. For G0W0-BSE calculations, the energy cutoff for the response functions was set to 200 eV. For QP band structures and G0W0-BSE calculations, a p(2×2) supercell and (16×16×1) k-point meshes were used. The QP band structures and optical absorption spectra were obtained using a spatial separation of Lz=25 Å. The van der Waals (vdW) 3
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interactions were considered using the semi-empirical DFT-D3 approach.37 The minimum energy pathways were investigated using the climbing image nudged elastic band (CI-NEB) method.38
3. RESULTS AND DISCUSSION The MBP has a buckled hexagonal structure with an indirect band gap, as shown in Figure 1(a), which is in excellent agreement with prior results.18-21 Our calculated G0W0 band gap is 3.23 eV, while the PBE result ( i.e., 1.96 eV)13-15 underestimates the band gap of MBP by ~39.3% in comparison with the G0W0 one. The valence band maximum (VBM) locates at the Γ point (denoted by V1), which is mainly composed by in-plane px and py orbitals. It is worth mentioning that the highest occupied state of sp orbitals deviated from the Γ point (denoted by V2) is only 0.15 eV lower in energy than the VBM. The conduction band minimum (CBM) which is mostly contributed by the pz orbitals lies along the Γ-M direction.
Figure 1． ．(a) The G0W0 band structure and the corresponding density of states. The charge distribution of the VBM and CBM are shown as insets, where the isosurface values are set to 0.02 e/Å3. (b) The value of G1 and G2 as functions of in-plane biaxial strain.
As mentioned before, the band gap of MBP can be gradually tuned by varying in-plane strain. The essential mechanism of strain approach is to modify the lattice crystal field, and eventually changes the corresponding energy states. To systematically analyze this, we further investigated the evolution of band structures with respect to in-plane strain. As shown in Figure 1(b), the band gaps between V1 and V2 to CBM are referred as G1 and G2, respectively, and the smaller value of which 4
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denotes the corresponding global band gaps. The calculated results clearly show that the in-plane strain remarkably affects the relative position of V1 and V2. We find that G1 increases from 2.58 eV to 3.44 eV while G2 decreases from 3.49 eV to 3.01 eV when the biaxial in-plane strain is changed from -4% to +4%. The value of G1 becomes larger with the increasing in-plane tensile strain, which mainly relates to the repulsion effect between px and py states. By contrast, G2 is dominated by lattice crystal field induced splitting between bonding and anti-bonding states of pz electrons. Therefore, G2 becomes smaller with increasing lattice constant as the lattice crystal field is weakened. Thus, G1 and G2 follow different evolution trends under in-plane strain.
Figure 2. Band structures of p(4×4) MBP at (a) 9.3% and (b) 12.5% oxygen coverages, where the charge distribution of the VBM and CBM are plotted as insets. The corresponding isosurface values are set to 0.005 e/Å3 and 0.04 e/Å3 in (a) and (b), respectively. The corresponding DOS is plotted at the right panel.
Despite all these efforts, neither the tensile nor compressive strain is able to change the indirect band gap feature of MBP. To accomplish the indirect-to-direct band gap 5
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transition of MBP, it requires both VBM and CBM to locate at the same point in momentum space. Surface adsorption, especially the dissociative adsorption, strongly affects electronic properties of MBP, which may change the type of band gap. Indeed, according to our calculations, the adsorption of O2, -OH, -COOH and -CN can effectively give rise to the indirect-to-direct band gap transition of MBP. In the following, we mainly focus on the adsorption of O2 on MBP as a typical example, while the adsorption configurations and band structures of other adsorbates (i.e., -OH, -COOH and -CN) are shown in Figure S1. The adsorption energy per O2 in dissociative form is about 4.10 eV, which is lower than the one in molecular form on MBP. The dissociated O2 are chemically bonded to P atoms with a bond length of 1.49 Å, which is perpendicular to the MBP plane (see Figure 2(a)). The possible adsorption sites for O atom are shown in Fig. S2 in Supplementary Information. The calculated results show that O oxygen does not prefer to locate at hollow sites. The adsorption energies per O atom are -2.82 eV and -2.12 eV for top and bridge configurations, respectively. In other word, O atom prefers to adsorb on top of the upper P atoms. The O coverage is defined as the ratio between the number of surface sites occupied by O atoms and the total number of adsorption sites. The oxygen adsorption dramatically modifies the electronic band edges, and the amplitude of modification is significantly enhanced by increasing oxygen coverage. As shown in Figure 2(a), the indirect-to-direct band gap transition occurs at the critical oxygen coverage of 9.3%. As the mechanism of band gap transition at other coverages are essentially the same, we first focus our study on the case of 12.5% oxygen coverage (see Figure 2(b)), while the results of 100% oxygen coverage will be briefly analyzed later. Upon oxygen adsorption, the location of VBM, which is mainly originated from px and py states of the adsorbed oxygen, remains at the Γ point. Importantly, the location of CBM, mainly composed by the s and pz orbitals of P atoms bonding to the adsorbed oxygen, moves to the Γ point. Our results show that the O-P bond strongly modifies the lattice crystal field, leading to a significant reduction of the bonding-antibonding splitting of sp orbitals. As a result, 6
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oxygen adsorption nicely achieves the indirect-to-direct band gap transition of MBP with a direct band gap of 2.76 eV at the 12.5% oxygen coverage. It is worth mentioning that electronic properties of the rest P atoms without O-P bond are essentially the same with the ones in the pristine MBP.
Figure 3. (a) Percentage elongation of optimized lattice constants (denoted by ∆L) compared to the pristine MBP as a function of oxygen coverage. (b) Adsorption energy per oxygen (∆E in red rhombuses) and QP band gap (blue circles) as functions of oxygen coverage are represented. (c) QP band gaps of the pristine MBP, MBP with 12.5% oxygen coverage and 100% oxygen coverage as functions of lattice constant are represented by purple rhombuses, red rhombuses, and blue triangles, respectively. The pink dotted lines indicate the optimized lattice constants at different oxygen coverages.
The adsorbed O atoms tend to repel each other, which gradually enlarges the lattice constants of MBP by increasing oxygen coverage. As shown in Figure 3(a), the lattice 7
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constant is elongated by 11.9% at full coverage of oxygen. Consequently, the adsorption energy per oxygen decreases from 2.05 eV to 1.76 eV with increasing oxygen coverage due to repulsion among O atoms (see Figure 3(b)). Meanwhile, the direct QP band gap of MBP decreases from 2.76 eV to 0.81 eV when the oxygen coverage increases from 12.5% to 100% (see Figure 3(b)), suggesting a tunable band gap engineering of MBP through surface adsorption.
Figure 4. Reaction pathways and energy barriers of O2 dissociation on the MBP surface in the (a) absence and (b) presence of one water molecule. (c) Reaction pathway and energy barrier of H2O2 dissociation on the MBP surface. The atomic structures of each marked state are shown as insets.
To better achieve a fine-tuning of the band gap, we also combined the strain approach with the surface adsorption. Compared with the pristine MBP, the band gap of MBP with oxygen adsorption follows a very similar trend in terms of in-plane strain, i.e., it hosts the maximum band gap size at around the optimized lattice 8
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constant of the pristine MBP. As the oxygen coverage has a remarkable influence on the lattice constant of MBP, it seems that the band gap widths with different surface adsorption vary differently with respect to in-plane strain, which is confirmed by our calculated QP band gaps (see Figure 3(c)). In the case of 12.5% oxygen coverage, the optimized lattice constant of 3.29 Å is slightly elongated by 0.3%. The band gaps change in the range of between 2.13 eV and 2.79 eV when -5%~2% strains are applied. For the 100% oxygen coverage, the lattice constant (i.e., 3.67 Å) of MBP dramatically increases by 11.9% comparing to the pristine one. In such case, the QP band gap decreases monotonously from 1.43 eV to 0 eV in the strain range from -3% to 1.7%. Interestingly, when the band gap closes due to the tensile strain, MBP undergoes a topological phase transition.27 To achieve the dissociation of O2 on MBP, it needs to overcome an appropriate energy barrier. It is well known that O2 promptly dissociates on the BP surface with a reaction barrier lower than 0.2 eV.39-40 For O2 dissociation on the MBP surface, Zhou et al.28 showed that the corresponding energy barrier is 0.52 eV. We looked back to this reaction pathway, however, we are not able to reproduce their energy barrier. Our results clearly showed the corresponding reaction barrier is 1.56 eV (see Figure S3(a)). In other words, the dissociated O2 would be likely trapped to form a metastable adsorption structure (see Figure S3(b)) along this reaction pathway. To further study the dissociation barrier of O2, we carried out calculations using a p(4×4) supercell. Our PBE results reveal that the reaction barrier of O2 dissociation on MBP is 0.72 eV shown in Figure 4(a), implying a weak reactivity of O2 in the oxidative degradation of MBP. Interestingly, we notice that the presence of H2O reduces the reaction barrier of O2 dissociation to 0.60 eV (see Figure 4(b)) that makes the dissociation of O2 more easily. Water molecule prefers to physically absorb on the MBP surface by forming two hydrogen bonds pointing to the dissociated O atoms. The interaction between H2O and O2 weakens the O-O bond, resulting in a reduction of the dissociation energy barrier. Therefore, it can be expected to oxidize MBP more easily in a wet environment. Besides, the adsorption of H2O2 on the MBP surface is an alternative 9
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way to provide adsorbed oxygen. Unexpectedly, the reaction barrier is only 0.36 eV in the splitting process of H2O2 that forms a H2O molecule and an O atom (see Figure 4(c)). These results indicate a more feasible pathway for oxygen adsorption on the MBP surface.
Figure 5. Calculated QP band structures and BSE spectra. The QP band structures of (a) the pristine MBP, the MBP with the dissociated O2 in the (c) absence and (e) presence of one H2O molecule, and the corresponding BSE spectra are shown in (b), (d), and (f), respectively.
As excitons in semiconductors play a crucial role in optical properties, in the following we investigated the influence of surface adsorption on the optical band gap and the EBE of MBP. Since BSE calculations are extremely time consuming, we performed calculations using a relatively small p(2×2) supercell. The value of EBE is calculated as the energy difference between the QP band gap and optical band gap, where the QP band gap is the lowest value of the direct band gap. As shown in Figure 5(a), the lowest direct band gap is 3.61 eV for the pristine MBP, while the optical band gap is 2.70 eV (see Figure 5(b)). As a result, it presents a large EBE of 0.91 eV, which is consistent with the literature.41 It is worth pointing out that our calculated EBE for the pristine MBP is also in excellent agreement with the linear scaling law 10
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proposed by W. H. Duan et al.,42 namely, the value of EBE is approximately one quarter of the QP band gap (EEBE=Eg/4). As discussed above, oxygen adsorption introduces an indirect-to-direct band gap transition and markedly reduces the band gap width. As shown in Figure 5(c) and 5(d), the QP and optical band gaps of MBP with oxygen adsorption are 2.46 eV and 2.17 eV, respectively. In this case, the calculated EBE is only 0.29 eV, which severely deviates from the linear scaling law of EEBE=Eg/4.42 Distinguished from oxygen adsorption, water adsorption on MBP almost does not change the electronic properties due to its weak physical interaction between H2O and MBP. The calculated QP and optical band gaps for MBP with the co-adsorption of O2 and H2O are respective 2.46 eV (see Figure 5(e)) and 2.23 eV (see Figure 5(f)), while the corresponding EBE is 0.23 eV. Our calculations clearly show that the ratio of EEBE/Eg significantly depends on the oxygen coverage. Thus, the value of EEBE/Eg dramatically deviates from the 1/4 law at the oxygen coverage of 12.5% (see Figure 6(a) in which the QP band gap width are tuned by the in-plane biaxial strain). Our results strongly suggest that the linear scaling law of EEBE=Eg/4 is not compatible with non-pristine two-dimensional semiconductors (with defects or adsorbates). At full coverage of oxygen, the oxidized MBP can be considered as a new pristine compound where the 1/4 law remains valid (see Figure 6(b)).
Figure 6. The values (denoted by blue rectangles) of the QP band gap and EBE at oxygen coverages of (a) 12.5% and (b) 100% where the values of QP band gap are tuned by in-plane biaxial strain. The corresponding values of EBE according to ideal 1/4 law are represented by red triangles.
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4. CONCLUSION In this work, we studied adsorption induced indirect-to-direct band gap transition of MBP by combining first-principles calculations and GW-BSE method. The CBM of MBP can be effectively modified by the adsorption of O2, -OH, -COOH, or -CN to form a direct band gap. As an example, the case of oxygen adsorption was investigated in detail. The band gap of MBP can be reduced to the visible spectrum region by either applying in-plane strain or tuning the coverage of adsorbates. Surprisingly, the dissociation of H2O2 on MBP offers an effective way to provide O atoms with a lower energy barrier of 0.36 eV. Moreover, we proposed a different linear scaling behavior between EBE and QP band gap depending on the coverage of the adsorbates. Our findings not only contribute to the indirect-to-direct band gap transition of MBP but also shed light on the potential electro-optical applications.
■ ASSOCIATED CONTENT Supporting Information See Supporting Information for band structures of more adsorbates (i.e., -OH, -COOH and -CN), adsorption configurations of oxygen, and reaction barrier of O2 dissociation on MBP.
■ AUTHOR INFORMATION Corresponding Authors *E-mail:
[email protected] *E-mail:
[email protected]
Notes The authors declare no competing financial interest.
■ ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China 12
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(NSFC, Grant Nos. 11674148, 11334004 and 11404159), the Guangdong Natural Science Funds for Distinguished Young Scholars (No. 2017B030306008), and the Basic Research Program of Science, Technology, and Innovation Commission of Shenzhen
Municipality
(Grant
Nos.
JCYJ20160531190054083,
JCYJ20170412154426330)
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Mater. 2017, 1, 061002. (17) Xiao, J.; Long, M.; Zhang, X.; Ouyang, J.; Xu, H.; Gao, Y., Theoretical Predictions on the Electronic Structure and Charge Carrier Mobility in 2d Phosphorus Sheets. Sci. Rep. 2015, 5, 9961. (18) Chowdhury, C.; Jahiruddin, S.; Datta, A., Pseudo-Jahn–Teller Distortion in Two-Dimensional Phosphorus: Origin of Black and Blue Phases of Phosphorene and Band Gap Modulation by Molecular Charge Transfer. J. Phys. Chem. Lett. 2016, 7, 1288-1297. (19) Villegas, C. E. P.; Rodin, A. S.; Carvalho, A.; Rocha, A. R., Two-Dimensional Exciton Properties in Monolayer Semiconducting Phosphorus Allotropes. Phys. Chem. Chem. Phys. 2016, 18, 27829-27836. (20) Ghosh, B.; Nahas, S.; Bhowmick, S.; Agarwal, A., Electric Field Induced Gap Modification in Ultrathin Blue Phosphorus. Phys. Rev. B 2015, 91, 115433. (21) Ding, Y.; Wang, Y., Structural, Electronic, and Magnetic Properties of Adatom Adsorptions on Black and Blue Phosphorene: A First-Principles Study. J. Phys. Chem. C 2015, 119, 10610-10622. (22) Li, Y.; Ma, F., Size and Strain Tunable Band Alignment of Black-Blue Phosphorene Lateral Heterostructures. Phys. Chem. Chem. Phys. 2017, 19, 12466-12472. (23) Kaur, S.; Kumar, A.; Srivastava, S.; Tankeshwar, K., Electronic Structure Engineering of Various Structural Phases of Phosphorene. Phys. Chem. Chem. Phys. 2016, 18, 18312-18322. (24) Sun, M.; Hao, Y.; Ren, Q.; Zhao, Y.; Du, Y.; Tang, W., Tuning Electronic and Magnetic Properties of Blue Phosphorene by Doping Al, Si, as and Sb Atom: A Dft Calculation. Solid State Commun. 2016, 242, 36-40. (25) Zheng, H.; Yang, H.; Wang, H.; Du, X.; Yan, Y., Electronic and Magnetic Properties of Nonmetal Atoms Doped Blue Phosphorene: First-Principles Study. J. Magn. Magn. Mater. 2016, 408, 121-126. (26) Sun, M.; Tang, W.; Ren, Q.; Wang, S.-k.; Yu, J.; Du, Y., A First-Principles Study of Light Non-Metallic Atom Substituted Blue Phosphorene. Appl. Surf. Sci. 2015, 356, 110-114. (27) Zhu, L.; Wang, S.-S.; Guan, S.; Liu, Y.; Zhang, T.; Chen, G.; Yang, S. A., Blue Phosphorene Oxide: Strain-Tunable Quantum Phase Transitions and Novel 2d Emergent Fermions. Nano Lett. 2016, 16, 6548-6554. (28) Nanshu, L.; Si, Z., Gas Adsorption on Monolayer Blue Phosphorus: Implications for Environmental Stability and Gas Sensors. Nanotechnology 2017, 28, 175708. (29) Hybertsen, M. S.; Louie, S. G., Electron Correlation in Semiconductors and Insulators: Band Gaps and Quasiparticle Energies. Phys. Rev. B 1986, 34, 5390-5413. (30) Bickers, N. E.; Scalapino, D. J.; White, S. R., Conserving Approximations for Strongly Correlated Electron Systems: Bethe-Salpeter Equation and Dynamics for the Two-Dimensional Hubbard Model. Phys. Rev. Lett. 1989, 62, 961-964. (31) Charlesworth, J. P. A.; Godby, R. W.; Needs, R. J., First-Principles Calculations of Many-Body Band-Gap Narrowing at an Al/Gaas(110) Interface. Phys. Rev. Lett. 1993, 70, 1685-1688. (32) Kresse, G.; Furthmüller, J., Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169-11186. (33) Kresse, G.; Joubert, D., From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758-1775. (34) Blöchl, P. E., Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953-17979. (35) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K., Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett. 2008, 100, 136406. 14
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(36) Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N., Wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions. Computer. Phys. Commun. 2008, 178, 685-699. (37) Marom, N.; Tkatchenko, A.; Rossi, M.; Gobre, V. V.; Hod, O.; Scheffler, M.; Kronik, L., Dispersion Interactions with Density-Functional Theory: Benchmarking Semiempirical and Interatomic Pairwise Corrected Density Functionals. J. Chem. Theory. Comput. 2011, 7, 3944-3951. (38) Henkelman, G.; Uberuaga, B. P.; Jónsson, H., A Climbing Image Nudged Elastic Band Method for Finding Saddle Points and Minimum Energy Paths. J. Chem. Phys. 2000, 113, 9901-9904. (39) Xu, W. P.; Xu, H., Role of Surface Adsorption in Tuning the Properties of Black Phosphorus. Phys. Chem. Chem. Phys. 2018, 20, 112-117. (40) Ziletti, A.; Carvalho, A.; Trevisanutto, P. E.; Campbell, D. K.; Coker, D. F.; Castro Neto, A. H., Phosphorene Oxides: Bandgap Engineering of Phosphorene by Oxidation. Phys. Rev. B 2015, 91, 085407. (41) Choi, J.-H.; Cui, P.; Lan, H.; Zhang, Z., Linear Scaling of the Exciton Binding Energy Versus the Band Gap of Two-Dimensional Materials. Phys. Rev. Lett. 2015, 115, 066403. (42) Jiang, Z.; Liu, Z.; Li, Y.; Duan, W., Scaling Universality between Band Gap and Exciton Binding Energy of Two-Dimensional Semiconductors. Phys. Rev. Lett. 2017, 118, 266401.
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