Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC3 monolayer: A Promising Candidate for Metal-free Photocatalysts.

By employing first-principles computations and particle-swarm optimization calculations, we theoretically confirmed the honeycomb geometry of experimentally realized BC3 sheet, which is constructed by the hexagonal carbon-ring fragments surrounded by six boron atoms and has pronounced thermodynamic stabilities. Remarkably, the computations also demonstrate the visible-light absorption, high carrier mobilities, and promising reduction and oxidation capacities of the BC3 monolayer, indicating its efficient absorption of solar radiation, fast migration of electron and holes, and excellent capabilities of photoinduced carriers in a photocatalytic process, respectively. Additionally, its indirect band gap, spatially separated charge distributions, and great difference in mobilities of electrons and holes should lead to the restricted recombination of photoactivated e--h+ pairs within BC3 monolayer. All above-mentioned characteristics suggest that the honeycomb BC3 monolayer should be a recommendable candidate for metal-free photocatalysts, which is worthy of further verifications and explorations in experimental studies.

Introduction

Since the first experimental report on water splitting at a TiO2 photoelectrode,[1] great success has been achieved in high-performance photocatalysts over the past decades and accordingly, the photocatalytic technologies have been well developed as one of the most hopeful strategies for overcoming the global energy crisis and environmental problems.[2] Varieties of photoactive materials, especially nanostructured semiconductors with promising photochemical characteristics, were synthesized and proven excellent photocatalysts for the degradation of organic pollutants, hydrogen production from water splitting, reduction of the heavy metal ion and carbon dioxides, green synthesis of chemicals, etc.[3]

Sparked by the discovery of graphene, a great number of two-dimensional (2D) materials possessing sheetlike structures with only single- or few-atom thickness have been experimentally realized, which leads to the growing research interest in exploring the novel characteristics and potential applications of these unique 2D materials.[4] When the 2D materials are used as photocatalysts, they exhibit several intrinsic advantages for improving the photocatalytic efficiency, including the high specific surface area for light absorption, abundant reactive sites available for photocatalytic reaction, reduced migration distance and promoted separation of photoactivated e––h+ pairs, conveniently regulable electronic and optical properties, and so on.[5] Compared with the bulk materials, numerous experimental studies have found the superiority of 2D photocatalysts that can indeed lead to the improved activities in photocatalysis.[6] For instance, the freestanding SnS2 single layer exhibited high photocurrent density of 2.75 mA cm–2, which is over 70 times higher than that of bulk SnS2. The SnS2 nanosheet generated a much greater incident photon to converted electron (IPCE) ratio (38.7%) than that of bulk SnS2 (2.33%) at an irradiation wavelength of 420 nm.[6a] Moreover, Xie et al. also fabricated flexible and freestanding zinc chalcogenide single layers and found the ZnSe nanosheet demonstrates a 200 times greater photocurrent density (2.14 mA cm–2) and 170 times higher IPCE ratio (42.5%) compared to those of bulk ZnSe, respectively.[6b]

The probable high cost of noble metals, metal release and poor stability in electrolyte, and environmental toxicity of the metal-containing photocatalytic nanomaterials may place serious restrictions on their practical utilization.[7] For appropriate alternatives to the metal-based semiconductors, a great number of metal-free semiconducting materials have been experimentally synthesized or theoretically proposed as attractive photocatalysts, including the elemental semiconductors[8] and nonmetal compounds.[9] Remarkably, since the 2D metal-free polymeric carbon nitrides (g-C3N4) were synthesized and first utilized as efficient photocatalysts for hydrogen production from water, the g-C3N4 photocatalyst has attracted tremendous research interest and achieved great success due to its superiority in being earth-abundant, nontoxic, highly stable in both alkaline and acidic solution, and independent of noble metals.[10]

In view of the great achievements of 2D metal-free g-C3N4 photocatalytic materials, a question arises naturally; is there any other 2D metal-free semiconductor that can be utilized as a highly efficient, nontoxic, and environmentally friendly photocatalyst? With this problem, we prescreened some metal-free graphenelike materials and found that the BC3 nanosheet may be a good candidate for metal-free photocatalysis. The metallic BC3 crystal with graphitelike structure was first synthesized through interaction of boron trichloride with benzene at 800 °C: 2BCl3 + C6H6 → 2BC3 + 6HCl. The electron micrographs showed that the BC3 is a homogeneous product with a sheetlike character and the sheets are seen to be 3–4 Å apart.[11] Similar to the fabrication of graphene, the BC3 nanosheets may be easily obtained through physical/chemical exfoliation from bulk crystal, chemical vapor deposition method, or epitaxial growth on the substrate lattice. Yanagisawa et al. found that a microscopic uniform sheet of BC3 with excellent crystalline quality can be epitaxially grown on the NbB2(001) surface through carbon-substituted technique in a boron honeycomb.[12] Moreover, by employing the ab initio pseudopotential local-orbital method, this graphenelike BC3 monolayer was theoretically predicted to be a semiconductor with an energy gap of 0.66 eV, which could have visible-light absorptions. Therefore, the experimentally feasible BC3 nanosheets may be one of the unexplored photocatalysts with earth-abundant elements and promising photoactivities, implying that it is essential to further investigate the intrinsic characteristics, especially the photochemical properties, of the BC3 nanosheet for exploring the potential metal-free photocatalysts.

Through first-principles computations and global minimum search using particle-swarm optimization (PSO) method, we herein theoretically analyzed possible geometries and photoactivities of BC3 monolayer. Among the three geometries from PSO computations, the global minimum structure of BC3 (GM-BC3) monolayer, which is composed of the fragments of hexagonal carbon ring surrounded by six boron atoms, has the best kinetic and thermal stabilities, and accordingly, the GM-BC3 has the greatest possibility to be the experimentally realized BC3 sheet. Moreover, we also computed the electronic and optical properties of experimentally obtained BC3 monolayer, namely, the GM-BC3 monolayer, and evaluated its photocatalytic activities. The computational results suggest that the GM-BC3 sheet has not only the broad visible-light absorption but also efficient migration and separation of photoactivated carriers, as well as the promising reduction and oxidation capacities of electrons and holes, indicating that the BC3 monolayer is a potential candidate for 2D metal-free photocatalysts.

Results and Discussion

Possible Geometries of the BC3 Monolayer

A comprehensive PSO search of ground states was performed to explore the global minimum structure of BC3 monolayer, which led to three possibly stable structures. All theoretically predicted BC3 monolayers have honeycomb geometries (Figure ). In the BC3-I monolayer, there is C6 ring with regular hexagon geometry, which are surrounded by six boron atoms. Both the B and C atoms within BC3-I are equivalent to other homogeneous atoms, in which all boron atoms are bonded with three carbon atoms, whereas each carbon atom is connected to one boron and two carbon atoms. The BC3-II is composed of BC5 or B2C4 hexagonal rings, in which all B atoms are connected with three C atoms and C-2 atoms are bonded with one C-1 and two C-3 atoms and each C-1 (C-3) is connected with two B (C-2) and one C-2 (B) atoms. Dissimilarly, there are B–B bonds parallel to one side of C6 hexagonal rings within the BC3-III monolayer. The B atoms are equivalent to each other, and C atoms can be classified into C-4 and C-5, which also have different coordinations. Notably, the computed cohesive energies of BC3-I, BC3-II, and BC3-III defined as Ecoh = (xEB + 3xEC – xEBC)/4x were also computed, where EB, EC, and EBC are total energies of a single B, a single C, and one unit cell of the BC3 monolayer. The BC3-I, BC3-II, and BC3-III monolayer have cohesive energies of 6.86, 6.54, and 6.43 eV per atom, respectively, indicating the strongest connection between the boron and carbon atoms in the three predicted BC3 monolayers. Relatively, the BC3-I has the lowest energy and thus may be the global minimum structure of BC3 sheets.

Figure 1

Relaxed geometries of three BC3 monolayers predicted by CALYPSO. Blue dashed lines present the crystal lattices, whereas the red-line frameworks are guidelines for the B2C6 unit cell.

To assess the kinetic stabilities of these BC3 monolayers, the phonon dispersions were computed along the high-symmetry lines in the first Brillouin zone (Figure ). There is no appreciable imaginary frequency in the phonon spectrum of BC3-I monolayer, implying the kinetic stability of this geometry (Figure a). The highest frequency in the phonon spectrum can reach up to 1518 cm–1, which indicates the robust B–C interaction in the BC3-I monolayer. However, the obvious negative frequencies in the phonon spectra of BC3-II and BC3-III monolayer suggest their kinetic instability. According to the results of cohesive energies and phonon dispersion, the BC3-I geometry with the lowest total energy and greatest cohesive energy is the global minimum and the most kinetically stable structure and thus has the greatest feasibility in experiments, which agree well with the experimental and theoretical predictions.[12] Therefore, only the global minimum structure, namely the BC3-I monolayer, was further computed to uncover its structural, electronic, and optical properties and to evaluate the feasibility of its photocatalytic application. All “BC3” mentioned in the following discussion represent the BC3-I structure.

Figure 2

Phonon dispersions of (a) BC3-I, (b) BC3-II, and (c) BC3-III monolayer.

The thermal stability of BC3 monolayer was further tested by performing the ab initio molecular dynamics (AIMD) simulations on a relatively large 4 × 4 supercell (32 boron and 96 carbon atoms) at the temperatures of 500, 800, 1200, and 1500 K. Snapshots of the geometries at the end of 10 ps simulation are represented in Figure . The framework of BC3 is well kept in its original configuration with tightly connected B–C bonds at the temperature of 1500 K (Figure d), demonstrating the promising thermal stability of BC3 monolayer.

Figure 3

Snapshots of BC3 equilibrium structures at the temperatures of (a) 500 K, (b) 800 K, (c) 1200 K, and (c) 1500 K at the end of 10 ps AIMD simulations.

For deep insights into the unique bonding nature and stabilizing mechanism of BC3 monolayer, the electron localization function (ELF) calculation[13] was performed. The ELF is useful for the classification of chemical bonds and can effectively describe the electron localization in molecules and solids. Generally, the ELF values of 1.0 and 0.5 designate the completely localized and delocalized electrons, respectively, whereas a value approaching 0 denotes very low charge density.[14] The isosurface of ELF for BC3 monolayer was plotted at the isosurface value of 0.30 au, and the ELF values at different regions are visualized by colors in ELF maps (Figure S1). The ELF results suggest that the electrons within BC3 are well delocalized and homogeneous electron gas is evenly distributed in the whole B–C framework. The well-delocalized electrons in this BC3 monolayer should lead to a robust connection between the boron and carbon atoms. Additionally, the blue region in the ELF map suggests the electron deficiency around the boron atoms and at the center of the hexagonal C6 rings, implying the electron transfer from B to C atoms and thus the strong covalent B–C and C–C bonds. The electrons accumulated at the center of B–C and C–C bonds, indicated by the red region of ELF map, demonstrate the possible σ bonds in this BC3 system. Therefore, due to the completely delocalized electrons and strongly connected boron and carbon atoms, this 2D BC3 framework has promising thermal and dynamic stabilities and is a reasonable structure of the experimentally obtained BC3 sheet.[12]

Electronic Structure and Optical Properties

To verify the possibility of photocatalytic application, band structure of BC3 monolayer was computed to investigate its semiconducting characteristics. According to the Perdew–Burke–Ernzerhof (PBE) computational results, the BC3 monolayer is a semiconductor with an indirect gap of 0.64 eV (Figure S2), which agrees well with the results of previous theoretical studies (0.62–0.66 eV).[15] In view of the underestimated band gaps by PBE functional, the hybrid density functional theory (DFT) method of Heyd–Scuseria–Ernzerhof (HSE06) functional was used to calculate the electronic structures of the BC3 monolayer (Figure a). The HSE06 computational results also demonstrate the indirect gap of this boron carbide system, which indicates the different positions of electrons and holes in momentum space and accordingly the restricted recombination of photoactivated electrons and holes with different momentums.[16] The conduction-band minimum (CBM) and valence-band maximum (VBM) are located at M and Γ point, respectively. The CBM is composed of B-p and C-p states, and the VBM mainly consists of p orbital of the C carbons (Figure a).

Figure 4

(a) Band structures (left) and projected density of states (PDOS) (right), as well as (b) the imaginary parts of dielectric constants for the BC3 monolayer, computed by HSE06 functionals. The green dashed line in the band structure represents the Fermi level at 0 eV, whereas the red dashed line tangent to ε2 curves is used to determine the optical gap of BC3 monolayer.

The band gap of 1.83 eV suggests the visible-light absorption of the BC3 monolayer, which is a great advantage when it was used as photocatalyst. To confirm its visible-light absorption, we computed the dielectric constants of BC3 monolayer and found that this 2D boron carbide has an optical gap of 2.01 eV and thus pronounced visible-light absorption. The high peak of absorption spectrum at 2.53 eV and large area under the ε2 curve at the visible-light region (<3.0 eV) demonstrate high absorption coefficients at these frequencies and accordingly high-efficiency visible-light absorption (<414 nm) of BC3 monolayer. Therefore, the BC3 monolayer could be a potential visible-light responsive semiconductor for high-performance nonmetal photocatalysis, solar harvesting process, or other light-emitting devices.

Besides the light absorption, the separation efficiency of photoactivated electron and holes is also a significant factor in the photocatalytic activities of BC3 monolayer. Because most photogenerated electrons and holes are locating at the CBM and VBM of the semiconductor, the spatial distribution of photoactivated electron and holes depends on the real-space locations of CBM and VBM, respectively. Accordingly, we computed the spatial distributions of CBM and VBM for BC3 monolayer to further analyze the separation efficiency of the photoinduced carriers (Figure ). For the investigated BC3 monolayer, the VBM at Γ point mainly consist of p states of carbon atoms whereas the CBM at M point is predominantly comprised of p orbitals of boron atoms, which is in good consistence with the PDOS results (Figure a). Additionally, the spatial charge distributions of CBM and VBM mainly locating around boron and carbon atoms suggest that the photoactivated electrons and holes should be predominantly distributed around the B and C atoms, respectively, implying the spatially separated photoinduced e––h+ pairs within the BC3 sheet.

Figure 5

Top (upper) and side (lower) view of (a) CBM and (b) VBM spatial distribution at Γ and M point within BC3 monolayer, respectively. The isosurfaces value is set at 0.016 au.

Redox Capacities and Carrier Mobilities of Electrons and Holes

The redox potential of photogenerated electrons and holes is another predominant factor that has a remarkable impact on the activities of photocatalysts. Consequently, the band edge position of BC3 monolayer was also computed to further analyze its reduction and oxidation capacities. The band edge alignments were determined by the CBM/VBM energy levels relative to the vacuum levels set at 0 eV. For a hypothetical example, if the vacuum level is calculated to be 1.5 eV, the CBM level at −3.6 eV should be determined at −(3.6 + 1.5) = −5.1 eV relative to the vacuum level at 0 eV.

The redox potential of a normal hydrogen electrode (ENHE), which equals to −4.5 V with respect to the absolute vacuum scale (EAVS = 0 V), is often utilized as a reference to evaluate the redox capacities of carriers. Generally, the valence-band holes with chemical potential of +1.0 to +3.5 V (vs ENHE) are good oxidants, whereas the conduction-band electrons with chemical potential varying from +0.5 to −1.5 V (vs ENHE) should have impressive reduction capabilities.[17] Our computations suggest that the chemical potentials of conduction-band electrons and valence-band holes are +0.15 and +1.98 V (vs ENHE), respectively, implying the promising reduction and oxidation capacities of photogenerated electrons and holes within BC3 monolayer (Figure ). Especially, the holes within BC3 have redox potential (+1.98 V) comparable to those of other high-performance photocatalysts, such as the g-C3N4 (+1.97 V), holey C2N (+2.05 V), etc. Therefore, the activated BC3 photocatalysts could directly capture the electrons and thus oxidize the organic pollutants or other reactants. The chemical potential of electrons at +0.15 V (vs ENHE) belonging to the numerical interval of +0.5 to −1.5 V (vs ENHE) also suggests the possible reduction capability of activated electrons, which could reduce the heavy metal ions or other reactants.

Figure 6

Band edge positions of BC3 monolayer, TiO2, g-C3N4, and holey C2N crystals. The redox potentials of H+/H2 (Φ(H+/H2) = 0 vs ENHE) and H2O/O2 (Φ(H2O/O2) = 1.23 vs ENHE) at pH = 0 are also given as a reference.

Compared with the excellent oxidation capability of holes, performance of electrons in reduction process may be not that impressive. However, the band alignment could be engineered through the introduction of impurity dopants, crystal defects, heterostructure nanosheets, biaxial strain, and so on, which is worthy to attempt.[18] For instance, the uniaxial strains not only result in varying band structures and band gaps of BC3 but also lead to the variation of band alignments (Figure S3). The BC3 monolayer with uniaxial strain of 3, 4, and 5% at x direction has chemical potentials of electrons at −0.02, −0.07, and −0.12 V (vs ENHE), respectively, implying the improved reduction capacities of photoinduced electrons within these strained modified BC3 systems. Meanwhile, the chemical potentials of holes within BC3 with x direction strain of 3, 4, and 5% are 2.02, 2.03, and 2.01 V (vs ENHE), respectively, also indicating the tiny enhancement of their oxidation capability (Figure S3a). The uniaxial strain of −3, −4, and −5% at y direction leads to the enhanced reduction capability of electrons but a little poorer oxidation capacity of holes within the BC3 monolayer. Notably, the BC3 monolayers with strain of 3, 4, and 5% at x direction, as well as the strain of −3, −4, and −5% at y direction, can simultaneously reduce H+ and oxidize OH– to generate H2 and O2.

Thanks to the highly developed experimental methods and techniques, tensile strains at an appropriate strength could be easily realized in 2D materials. For instance, the graphene film to a circular mold with a strain of 12% has been successfully fabricated by utilizing the laser-induced shock pressure.[19] By adhering graphene to a bending PMDS substrate, a tensile strain of 18.7% was obtained in stretched graphene.[20] By indenting the center of the graphene film with an atomic force microscopy tip, Lee et al. observed strains up to 12% when measuring the intrinsic strength of monolayer graphene.[21] However, the experimental realization of compressing strain may be not that easy, which should be further explored in future experimental studies.

As another prerequisite for an excellent photocatalyst, the carrier mobilities of BC3 monolayer were computed to further evaluate the efficiency of charge migration in this 2D material. By employing the deformation potential (DP) theory, the carrier mobilities of BC3 monolayer were computed along x and y directions. The computational details of the stretching modulus (C) and DP constant (E1) along armchair (x) and zigzag (y) directions are represented in Figure S4. The calculated values of C, E1, effective mass (m*), and carrier mobilities are listed in Table . The electrons and holes within BC3 monolayer have carrier mobilities of 2.09 × 102 and 5.13 × 104 cm2 V–1 S–1 along the x direction, as well as 1.06 × 102 and 4.22 × 102 cm2 V–1 S–1 along the y direction, respectively, which are comparable to those of MoS2 single-layer transistors (ca. 200 cm2 V–1 S–1).[22] Remarkably, the carrier mobility of holes along x direction (5.13 × 104 cm2 V–1 S–1) reaches up to a value approaching those of phosphorene (1.0–2.6 × 104 cm2 V–1 S–1),[23] which is a novel 2D material that can be used in high-performance electronic devices.

Table 1

Values of the Stretching Modulus C, DP Constants |E1|, Effective Mass |m*| and Carrier Mobilities μ along x and y Directions for Global Minimum Structure of BC3 Monolayer

	C (N m–1)	|E1| (eV)	|m*| (me)	μ (cm2 V–1 S–1)
electrons (x)	295.98	3.80	1.18	2.09 × 102
holes (x)	295.98	9.54	0.03	5.13 × 104
electrons (y)	303.01	5.04	1.26	1.06 × 102
holes (y)	303.01	4.84	0.66	4.22 × 102

These high values of carrier mobility should result in fast movements of photoinduced electrons and holes within BC3 monolayer, and accordingly, the photoactivated e––h+ pairs can fast migrate to the surface and participate in the electron-exchange reactions. It is noteworthy that the mobility of holes can reach up to 2 orders of magnitude higher than the electron mobility in the armchair (x) direction, whereas the holes have mobility about four times greater than that of electrons along zigzag (y) direction. The dramatic difference between mobilities of electrons and holes should result in their quite different velocities and thus restrained recombination of photoinduced e––h+ pairs along both the armchair and zigzag directions.[28]

Conclusions

Through DFT computations and global minimum search, we computed the structural, electronic, and optical characteristics of BC3 monolayers, which were experimentally realized in 2004. The computational results suggest that the BC3 monolayer containing the fragments of hexagonal six-carbon rings surrounded by six boron atoms is the global minimum structure, which has promising kinetic and thermal stabilities and thus the greatest possibility to be the experimentally obtained BC3 sheet.

The global minimum structure of BC3 monolayer has an electronic gap of 1.83 eV and an optical gap of 2.01 eV, indicating its visible-light absorption. Moreover, the carrier mobility of holes can reach up to 5.13 × 104 cm2 V–1 S–1 along x direction whereas other carrier mobilities at x or y direction are comparable to those of MoS2 single-layer transistors, which suggest the fast migration of photoinduced carriers within the BC3 monolayer. The chemical potentials of CBM and VBM at +0.15 and 1.98 eV (vs ENHE), respectively, which can be further improved by uniaxial-strain modification, demonstrate the promising reduction and oxidation capabilities of the photogenerated electrons and holes in this BC3 sheet. The BC3 monolayer even has the spatially separated charge distribution of CBM and VBM, indirect band gap, and great difference in the mobilities of electrons and holes, implying the spatially separated distribution and effectively restricted recombination of photoinduced e––h+ pairs. According to the above-mentioned intrinsic properties, the BC3 monolayer is a promising metal-free candidate for the green photocatalyst, which is worthy of experimental confirmation and exploration.

Computational Details

Our periodic density functional theory (DFT) computations were carried out by employing the projector-augmented plane-wave[24] method as implemented in the Vienna ab inito simulation package (VASP)[25] code. The electron-exchange-correlation functional was described by the generalized gradient approximation in the form proposed by Perdew, Burke, and Ernzerhof (PBE).[26] The tolerance for energy convergence was set to 10–6 eV, whereas the ionic relaxation is converged when the force on each atom was less than 10–3 eV Å–1. The 500 eV cutoff for plane-wave basis set was adopted in all computations. The Brillouin zone was integrated by using Monkhorst–Pack generated sets of k-points. In the geometry relaxation and self-consistent computations, 15 × 15 × 1 Monkhorst–Pack k-points mesh was used. In view of the underestimated band gaps by the PBE functional, the Heyd–Scuseria–Ernzerhof (HSE06)[27] hybrid functional, which was proven a reliable method for evaluating the electronic and optical properties,[28] was employed to calculate the band structures and dielectric constants of the BC3 monolayer. In the geometric optimization process, the dipole correction has been considered and there is no spin polarization in this BC3 system.

The particle-swarm optimization (PSO) method within the evolutionary algorithm, as implemented in CALYPSO code,[29] was employed to search for global minima of the BC3 monolayer sheets. As an unbiased global optimization method, PSO algorithm has successfully predicted highly stable structures of 2D materials.[30] In our PSO calculations, the population size is set to 50 and the number of generations was maintained at 30. Unit cells containing total atoms of 4 (BC3), 8 (B2C6), and 16 (B4C12) were considered. After complete relaxations of the required structure by using PBE functional, as implemented in VASP code, three stable monolayers of BC3, namely, BC3-I, BC3-II, and BC3-III, (Figure ) were obtained. The BC3 monolayers were placed in the xy plane, with the z direction perpendicular to the layer plane, and a vacuum space of 20 Å in the z direction was adopted to avoid interactions between adjacent layers.

To evaluate the dynamic stability of three predicted BC3 monolayers, phonon dispersion analysis was performed by using the Phonopy code[31] interfaced with density functional perturbation theory,[32] as implemented in VASP. The ab initio molecular dynamics (AIMD) simulations using PBE functional were carried out to assess the thermal stabilities of the lowest-energy BC3 monolayer. In the molecular dynamics (MD) simulations, a 4 × 4 supercell of BC3 containing 32 B atoms and 96 C atoms was annealed at temperatures of 500, 800, 1200, and 1500 K, respectively. The MD simulation in the NVT ensemble lasted for 10 ps, with a time step of 2.0 fs, and the temperature was controlled by employing the Nosé–Hoover method.[33]

To analyze the optical properties of the lowest-energy BC3 monolayer, the frequency-dependent dielectric matrix was calculated by using the HSE06 functional and expanding over a 21 × 21 × 1 k-point mesh. The complex dielectric constants at a given frequency can be defined as ε(ω) = ε1(ω) +iε2(ω). The expression for the absorption coefficient I(ω) was given as[34]According to the equation, we can obtain the light absorbing information from the value of the imaginary part. Only if the imaginary part ε2(ω) > 0, the absorption coefficient I(ω) was above zero. The imaginary part is determined by a summation over empty states using the equation[35]where the indices c and v represent conduction and valence-band states, respectively, and uc is the cell periodic part of the orbitals at the k point. A large number of empty conduction-band states (almost twice more than the number of valence bands) are included for the summation of eq .

The deformation potential theory was applied to evaluate the carrier mobility of 2D materials, which was derived from the following analytical expressionwhere the temperature (T) was adopted at 300 K in this study and m* is the effective mass of charge in the transport direction and defined as m* = ℏ2(∂2E(k)/∂k2)−1. C is the stretching modulus caused by uniaxial strain, which is given by C = (∂2Etotal/∂ε2)/S0, where Etotal represents the total energy of a unit cell under different strengths of uniaxial strain (ε) and S0 represents the area of the optimized unit cell. E1 is the deformation potential (DP) constant of VBM for holes and CBM for electrons, calculated by using E1 = ∂Eedge/∂ε. Eedge is the energy level of VBM or CBM along the transport direction. e is the electron charge, ℏ is the reduced Planck constant, and kB is the Boltzmann constant.