Study of formaldehyde adsorption on silicene with point defects by DFT method

Abstract

To explore the chemical activity and sorption capacity of silicene with point defects for formaldehyde (HCHO), interactions between HCHO and silicene were investigated using density functional theory (DFT) calculations. As compared to the weak adsorption on perfect silicene, HCHO molecules tend to be chemisorbed onto the Si–Si bonds of defective silicene with appreciable adsorption energy. The electronic conductance changes markedly with the adsorption of HCHO molecules on silicene containing Stone–Wales (SW) defects, whereas silicene with double vacancies still exhibits indirect semiconductor characteristics after HCHO adsorption. Moreover, the adsorption energy of HCHO on silicene with SW defects (SW-Si) undergoes a continuous increase under a tensile strain up to 10%, suggesting that the chemical reactivity of an SW-Si sheet increases by applying an external strain.

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1. Introduction

High mechanical strength, chemical stability, and unique electronic properties have made graphene a material of interest in diverse fields ranging from biotechnology to electronics in recent years.^1–3 As a counterpart of graphene, a monolayer silicon, called silicene, has also attracted considerable attention due to its unusual electronic properties similar to those of graphene and promising applications in Si nanoelectronics.⁴ Theoretical calculations have revealed that in the silicene structure, a hexagonal mesh of silicon atoms is formed and the electronic structure of silicene is equivalent to that of graphene with linear electronic dispersion, resembling that of relativistic Dirac fermions.^5–8 Experimentally, many groups have reported the successful preparation of silicene on Ag (110),^9,10 Ag (111)^11,12 and Ir (111)¹³ substrates. Structural parameters and electronic structures of these sheets are in good agreement with theoretical predictions. In addition, Si atoms in silicene prefer to undergo sp²/sp³ hybridization, which is essentially different from the sp² hybridization of graphene. Thus, silicene displays considerably higher chemical activity than graphene towards foreign adsorbates such as NO and NH₃.¹⁴

Several studies have shown that graphene usually suffers from various types of topological defects during its growth. The presence of defects such as vacancies and Stone–Wales (SW) defects can induce magnetism, tailor the electronic properties and alter the chemical activity of graphene-based structures.^15–17 Most recently, Sahin et al. studied silicene with SW defects using first-principles calculations.¹⁸ It was found that the energy barrier for the formation of SW defects in silicene is significantly lower than in graphene, and the buckled nature of silicene provides a large energy barrier for the removal of SW defects. The presence of SW defects induces a small energy gap of 0.02 eV, and this value depends on the concentration of defects. Furthermore, the investigation of other types of defects, namely, single and double vacancies (SVs and DVs), in silicene has shown that SVs are unstable and two SVs are likely to coalesce into one DV to lower the energy.¹⁹ Because SW defects and DVs exist stably in silicene and affect its electronic characteristics, it is highly intriguing to determine how point defects influence its chemical activity. In addition, according to previous studies,^20,21 mechanical strain can increase the activity of carbon nanomaterials such as nanotubes and graphene. Therefore, silicene requires further investigation.

Because silicene has a higher chemical activity, it has a potential application in the field of sensors. For nanomaterials, gas sensors are known to be based on monitoring the change in electronic conductance that results from adsorbed molecules, which act as charge acceptors or donors.^22,23 Silicene used as a gas sensor has been reported by Feng et al.¹⁴ In their study, silicene interacted with NO and NH₃ accompanied by a charge transfer between these molecules and silicene and a band gap was created upon gas adsorption. Moreover, Wu et al.²⁴ reported that the reactivity of silicene can also be increased by applying an external strain. Thus, both defects and external strain may affect the properties of silicene when it is used as a sensor.

Formaldehyde (HCHO) is the most common and well-known indoor air pollutant because of its wide applications in many construction and decorative materials. It can cause headache, nausea, coryza, childhood asthma and even lung cancer.^25,26 Therefore, monitoring and controlling exposure to it in both residential and industrial environments is very important. Many methods, such as polarography, gas chromatography and fluorometry,^27,28 have been developed to detect HCHO. However, these methods require complicated and expensive instruments and are not sensitive enough because they either have high detection limits or require long sampling intervals. Thus, many studies have investigated the interaction of HCHO with nanomaterials such as TiO₂,²⁹ CNT,³⁰ and graphene.³¹ However, very few investigations have been carried out to study the interaction of HCHO with silicene.

Therefore, to understand the influence of defects on the chemical activity of silicene and to explore the potential applications of this promising material for the detection of HCHO, in this paper, we present a detailed investigation on the adsorption behavior of HCHO on pristine and two types of defective silicene (SW and DVs) based on density functional theory (DFT) calculations. It was found that different from the binding of HCHO with pristine silicene, the molecule could be chemisorbed on sheets of both types of defective silicene with high adsorption energy and obvious structural distortion. In particular, the electronic structures changed obviously with the adsorption of HCHO molecules on silicene with SW defects. Furthermore, the chemical activity of silicene with SW defects was enhanced by mechanical strain. The results may afford new insight into applications of silicene in gas sensors and provide a simple approach to control the chemical activity of silicene.

2. Computational method

All the calculations were performed within the framework of unrestricted spin-polarized DFT, implemented in the Dmol3 code.^32–34 Structure optimizations and the corresponding total energy calculations for the most stable geometries were based on the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) correction;³⁵ moreover, all-electron calculations and a double numerical basis set plus polarization functional (DNP) were adopted.³³ The DNP basis set corresponds to a double-f quality basis set with a p-type polarization function added to hydrogen and d-type polarization functions added to heavier atoms, which is comparable with the Gaussian 6-31G(d,p) basis set and exhibits higher accuracy.

In this study, a 5 × 5 × 1 supercell with a periodic boundary condition was used to model an infinite silicene sheet. The silicene sheet included 50 atoms with lattice parameters of a = b = 19.33 Å. A vacuum space of 20 Å was set in the direction normal to the sheets to avoid interactions between periodic images. The Brillouin zone was represented by a set of 10 × 10 × 1 k-points³⁶ for geometry optimizations, and 15 × 15 × 1 k-points were used to obtain the density of states (DOS). A global orbital cutoff of 4.6 Å was set in spin-unrestricted calculations for all systems, and the effect of periodic boundaries was negligible. All atoms were allowed to relax. Convergence in energy, force, and displacement was set at 10⁻⁶ Ha, 0.001 Ha/Å, and 0.001 Å, respectively. The adsorption energy E_ads of a HCHO molecule on pristine or defective silicene is defined as follows:

E_ads = E(total) − E(sheet) − E(HCHO) (1)

where E(total), E(sheet), and E(HCHO) are the total energies of relaxed silicene with an adsorbed HCHO molecule, an isolated silicene sheet and an isolated HCHO molecule, respectively. In addition, the stability of silicene is measured via its cohesive energy, which is defined as the energy required for separating a crystal into isolated free atoms. The cohesive energy of a silicene structure is given by the formula.

E_coh = (E_tot/N) − E_Si (2)

where E_tot is the total energy of the system, N is the number of atoms in the supercell and E_Si is the total energy of an isolated silicon atom.

3. Results and discussion

3.1. Pristine and defective silicene with point defects

The computed lattice parameter of relaxed silicene is 3.866 Å and Si–Si bond length is calculated to be 2.28 Å, which is consistent with previous studies,^37,38 as shown in Fig. 1(a). The larger Si–Si interatomic distance weakens the π–π overlaps and results in a low-buckled structure with h = 0.45 Å (Fig. 1(b)). Silicene was also found to be a gapless semiconductor like graphene^39,40 with the bonding π and antibonding π* bands crossing only at k-points in the hexagonal Brillouin zone (Fig. 2).

Fig. 1 Optimized geometric structures of silicene: (a) top view and (b) side view. Silicon atoms are shown in yellow.

Fig. 2 Band structure for a 5 × 5 supercell of perfect silicene. The Fermi level is set to 0 eV and is represented by dots.

As shown in Fig. 3, an SW defect in silicene can be created by rotation of a silicon dimer by 90° around the center of the Si–Si bond. After the formation of the SW defect, four neighboring hexagons of silicene are transformed into pairs of pentagons and heptagons. Moreover, through 90° rotation of a dimer, the Si–Si bond becomes stronger than that in perfect silicene and its length decreases from 2.28 to 2.189 Å. The cohesive energies of silicene and silicene with SW defects were calculated to be −3.96 and −3.92 eV, respectively. The negative cohesive energies of both structures indicate their stability.

Fig. 3 Optimized geometric structures of silicene with SW defect: (a) top view and (b) side view.

The electronic band dispersion of silicene with SW defects (SW-Si) is presented in Fig. 4. Because the formation of the SW defect breaks the six-fold symmetry of the silicene lattice, a small band gap of 0.028 eV occurs at the crossing point, which agrees with previous studies on silicene and graphene.^19,41 It is noted that silicene with SW defects almost retains a linear dispersion relationship near the Fermi level. Hence, silicene with SW defects would create a small band gap without affecting high-velocity carriers. Nevertheless, the SW defect induces a defective π state, which is located about 0.3 eV above E_F. The defective π state is significantly localized at the Si1 and Si1′ sites, as seen in the PDOS presented in Fig. 4(b). In addition, because no dangling bonds are introduced into the silicene lattice with the creation of an SW defect, all the atomic orbitals of Si atoms in the vicinity of the defect are paired and thus there is no defect-originated magnetism.

Fig. 4 (a) Band structure of SW-Si and (b) PDOS for atoms around its defect region. The Fermi energy is set to 0 eV and is indicated by dots.

The case of divacancy is different from that of the SW defect. Here, we could not observe obvious increased buckling behavior of atoms from the mean plane (see Fig. 5) after optimization. The Si atoms around the vacancies have moved inward forming two new bonds and have been rearranged to form a 585 defect, as shown in Fig. 6. The rearranged structure encloses a central octagon and two opposing pentagons. The Si–Si length between 2 unit cells is reduced to approximately 6.55 Å compared with the pristine Si–Si bond, which has a length of about 7.73 Å. It is found that the length of the newly formed Si–Si bond increases to 2.43 Å, and the other Si–Si bonds that form the octagon also stretch to share the tension induced by the two missing atoms. The cohesive energy of this silicene with double-vacancy defects (DV-Si) is −3.90 eV, which is a little higher compared with that of silicene with SW defects.

Fig. 5 Top (a) and side (b) views of reconstructed atomic structures of silicene in the presence of 585 defects. The lengths of Si–Si bonds are denoted by red numbers in Å.

Fig. 6 Deformation electron density of defected region of DV-Si.

The calculated band structure for this 585 DV-Si sheet is shown in Fig. 7(a). The effect of the presence of the divacancy is apparent; it induces a dispersionless band state of π-character above the Fermi level and an indirect semiconductor with a band gap of 0.14 eV is obtained. From the PDOS for Si atoms around the defect region depicted in Fig. 7(b), it is found that the main contributions to such a band are dominated by the p_z states of Si1 atoms and Si3 and Si4 atoms next to the vacancy site. In a previous study on graphene with 585 defects,⁴² it was predicted that a single impurity in sublattice A of graphene induces an impurity state that is mostly localized in sublattice B and vice versa due to the presence of two non-equivalent Dirac points. Because Si2 atoms belong to a sublattice different to that of Si1 and Si3 atoms, there is almost no contribution from the former atoms to the defect states above the Fermi level.

Fig. 7 (a) Band structure of DV-Si and (b) its PDOS for atoms around its defect region. The Fermi energy is set to 0 eV and is indicated by dots.

3.2. Formaldehyde adsorption on pristine silicene

For the adsorption of HCHO on pristine silicene, different possible adsorption configurations were considered, including the approach of a C, O or H atom of HCHO molecule towards the Si atom, a C=O bond formed parallel to the Si–Si bond and the HCHO molecule located parallel to the hexagonal ring of silicene. All the five stable configurations were observed after full optimization, and the adsorption energies calculated for these cases are given in Table 1. The obtained adsorption energies and the large interaction distances for all the five configurations imply weak physisorption of HCHO molecule on the pristine silicene sheet.

Table 1 Adsorption energy (E_ads) and interaction distance between HCHO and pristine silicene surface (d)

Model	Eads (eV)	d^a (Å)
a For the models where O, C, or H atom approaches a Si atom, d means the distance between the O, C, or H and the Si atom it approaches. When C=O bond is parallel to the Si–Si bond, d is the distance between O and Si atoms.
O approaches Si atom	−0.076	3.622
C approaches Si atom	−0.087	4.000
H approaches Si atom	−0.086	2.878
C=O bond parallel to Si–Si bond	−0.116	3.177
HCHO located parallel to hexagonal ring	−0.113	3.489

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3.3. Formaldehyde adsorption on silicene with SW defects

Similarly, to find the most stable adsorption configuration, a HCHO molecule was initially placed in various positions above SW-Si with different orientations. The most stable configuration of HCHO on SW-Si is shown in Fig. 8, in which the C=O bond of HCHO attacks an Si–Si bond of the pentagon ring. Adsorption occurs through the bonding of O atom of the HCHO molecule with the Si atom of the silicene with SW defects and the bonding of C atom with another Si atom, forming a four-membered ring. E_ads is −1.389 eV, which is 1.173 eV lower than that on pristine silicene in the most stable configuration, and it is also much lower than that for the adsorption of HCHO on SW-defective graphene,⁴³ indicating that the interaction between of HCHO with SW-Si is considerably stronger than that with pure silicene or SW-defective graphene. The interaction distances of C–Si and O–Si bonds are 1.963 and 1.704 Å, respectively. Moreover, HCHO adsorption induces a local structural deformation in both HCHO molecule and SW-Si sheet. The bond angles of H–C–H and two H–C–O in HCHO are significantly decreased from 116°, 122°, and 122° in free HCHO to 109.7°, 109.4°, and 110.1°, respectively, in the adsorbed form. The HCHO-adsorbed Si–Si bond is pulled outward from the silicene with the bond length increasing from 2.345 Å in SW-Si to 2.405 Å. This structural deformation is attributed to the change in the hybridization of Si atoms from sp²/sp³ to sp³. The deformation of the system structure and the appreciable binding energy suggest that the interaction in this configuration relates to chemisorption.

Fig. 8 Optimized configuration of HCHO adsorbed on SW-Si.

To investigate charge transfer between the SW-Si sheet and HCHO molecule, electron density difference, Δρ, for this configuration was determined, which illustrates how the charge density changes during the adsorption process and is defined as follows:

Δρ = ρ_total − (ρ_sheet + ρ_HCHO)

where ρ_total, ρ_sheet and ρ_HCHO denote the electron densities for an SW-Si sheet with an adsorbed formaldehyde molecule, for an SW-Si sheet and for a HCHO molecule in the adsorbed form, respectively. Fig. 9 shows the electron density difference isosurfaces for this configuration; loss of electrons is indicated in yellow, whereas enrichment of electrons is indicated in blue. It is clear that charge is transferred from silicene to HCHO, in which some charge accumulates around the C and O atoms, especially around the O atom. Therefore, both reconstruction and charge transfer confirm the strong binding between HCHO and SW-Si sheet induced by adsorption. The abovementioned results are supported by data from Mulliken charge analysis, where electrons around O atom increase from 0.312 to 0.557 e, whereas those around C atom increase to 0.161 e. About 0.436 electrons are transferred from silicene to HCHO molecule, which is much greater than the charge transfer that occurs in the adsorption of NO or NH₃ on silicene.¹⁴

Fig. 9 Charge density of HCHO adsorbed SW-Si sheet.

3.4. Formaldehyde adsorption on silicene with DV defects

To investigate the effect of divacancy defects on the adsorption of HCHO, except for those configurations with binding energies comparable to that of pristine silicene, two stable configurations were studied (panels 1 and 2) after geometry optimization, as shown in Fig. 10. In both configurations, similar to the adsorption of HCHO on SW-Si, the C=O bond of the molecule interacts with the Si–Si bond, either in the pentagonal ring or the octagonal ring. Adsorption of HCHO molecule pulls the HCHO-adsorbed Si–Si bond out slightly from the silicene and induces a local structural deformation in the HCHO molecule. For example, as shown in panel 2, the HCHO-adsorbed Si–Si bond is pulled outward from the silicene and the interaction distances of C–Si and O–Si bonds are 1.950 and 1.708 Å, respectively. The adsorption energies of a HCHO molecule on divacancy silicene are −1.577 and −1.644 eV, which are in contrast to the values using pristine silicene substrates; this shows that the introduction of a vacancy significantly enhanced the interaction between the silicene and HCHO molecule. Charge transfer between the HCHO molecule and the divacancy silicene substrate was also calculated by Mulliken population analysis. The results suggested that the HCHO molecule behaved as an acceptor, capturing charge from silicene, which is the same as HCHO adsorption on SW-Si sheet. The charge in panel 2 is 0.431 e, which is a little higher than that in panel 1, and this agrees with its larger adsorption energy. Charge transfer analysis again indicated that the defect affected the adsorption process of HCHO molecule.

Fig. 10 Optimized stable configurations of HCHO adsorbed on DV-Si sheet: (a) panel 1 and (b) panel (2).

3.5. Electronic structures

For detecting a molecule using a given sheet, one of the properties of the sheet (such as electrical conductivity) should be changed by chemisorption of the molecule. The obtained results have shown that HCHO molecule is chemisorbed on SW-Si or DV-Si sheets. Thus, to determine the sensitivity of the sheets for the adsorption of HCHO molecules, the electronic structures of the considered configurations were investigated by analyzing their band structures, spin-polarized DOS and partial density of states (PDOS) spectra.

In the case of a DV-Si sheet, the presence of the divacancy induces an almost dispersionless band state above the Fermi level and the system becomes an indirect semiconductor. It is clear from Fig. 11 that upon chemisorption of a HCHO molecule on a DV-Si sheet, no distinct change occurs in its electronic properties; therefore, this sheet is not suitable for detecting HCHO molecules.

Fig. 11 Band structure for HCHO adsorption on DV-Si with panel 2 configuration. The dotted line indicates the Fermi level.

In the case of a silicene sheet with SW defects, the distortion of Si–Si bonds creates a band gap of 0.028 eV and does not affect the linear dispersion of bands near the Fermi level E_F, which implies the possibility of still having a high Fermi velocity. Although the presence of SW defects changes the characteristics of the material, the conductivity of the charge carriers will not be very different from the case of pristine silicene. For the system involving HCHO adsorption, the band structure and PDOS spectrum are depicted in Fig. 12. The band gap of the HCHO/SW-Si system is distinctly increased to a value of 0.190 eV and a new state near the Fermi level is introduced. On inspection of its PDOS, it can be seen that the O 2p states strongly hybridize with Si 3p in the upper region of the valence band, giving the main contribution to HCHO/SW-Si sheet interaction. In this process, the sheet donates electrons to the n′_o orbitals of HCHO and strengthens the interaction to make it chemisorption. On the other hand, a small amount of empty orbitals at the bottom of the conduction band indicates that some π_CO* orbitals accept electrons and fall into the region of the valence band. This process weakens the intramolecular interaction between C and O. As a result, the C=O bond of HCHO is elongated to 1.452 Å, which is ∼20% longer than that in the free gas phase. The electron donation of the sheet and the downshift in the frontier orbitals of HCHO increase the band gap and make the sheet a semiconductor with a larger band gap than the sheet without HCHO. The changes that originate from the adsorption of HCHO make the sheet a promising candidate for its use in detecting HCHO.

Fig. 12 (a) Band structure and (b) PDOS for HCHO adsorption on silicene with SW defects. The dotted line indicates the Fermi level.

3.6. Effect of strain on formaldehyde adsorption on SW-Si sheet

As is well known, an external stress applied to graphene affects the orbital hybridization of carbon atoms and leads to the formation of an extended π orbital with a localized electron (p_z orbital). As a result, the reactivity of graphene increases.⁴⁴ Motivated by this knowledge, we suspected that nanomechanical modulation of strain may influence the chemical activity of silicene with SW defects. Thus, we investigated the effects of strain applied to silicene with SW defects upon the adsorption of HCHO by varying the isotropic strain from 0% to 10% without any change in the crystal symmetries and honeycomb-like structures of the adsorbent. In-plane tensile strain was uniformly applied along lattice directions. Isotropic strain is defined as ε = Δa/a₀, where the lattice constants of the unstrained and strained supercell are equal a₀ and a = Δa + a₀, respectively. SW-Si sheet was stretched by first elongating the optimized lattice constant from a₀ to a, and then the supercell was re-optimized. While undergoing strain, we find that the energy of an SW-Si sheet increases with tensile strain (shown in Fig. 13(a)), which is in agreement with previous reports of graphene under tensile strain. The Si atoms in silicene prefer sp³ hybridization, which leads to the formation of a low-buckled sheet. When tensile strain increases, the buckling of the sheet decreases, which results in destabilization of silicene. As a result, the total energy of the sheet increases. The variations in HCHO adsorption energies with lattice strain are shown in Fig. 13(b). As the strain increases, the adsorption energy of HCHO decreases. Under strain, a HCHO molecule is still chemisorbed on the SW-Si sheet with the configuration same as that on a sheet without strain. Fig. 14 shows the PDOS of HCHO adsorption on SW-Si at ε = 7%, in which the p orbitals of O and C atoms hybridize with the Si 3p orbital of SW-Si. Hybridization broadens and downshifts the Si 3p band. The obvious hybridization among the p-orbitals of Si and HCHO implies that the HCHO molecule is still chemisorbed on the defective silicene under tensile strain. It is also noted that at ε = 10%, the defective silicene after HCHO adsorption begins to exhibit an obvious distortion in the defect area with an Si–Si bond moving away from its original position.

Fig. 13 (a) Calculated total energy of silicene containing an SW defect and (b) HCHO adsorption energy, E_ads, on an SW-Si sheet as a function of lattice strain.

Fig. 14 PDOS of HCHO adsorption on SW-Si at ε = 7%. The dotted line indicates the Fermi level.

Fig. 15 shows that the center of the Si 3p band in SW-Si without HCHO molecule increases monotonically with tensile strain. The reduction in E_ads with increasing tensile strain is due to the shift in the energies of the 3p orbital of Si atom and reduction in coordination. After HCHO adsorption, the center of the Si 3p band downshifts with an increase in strain, as shown in Fig. 15(b). The downshift in the energy of the 3p band from the pristine state to the state with the adsorption of HCHO is a measure of relative relaxation in the system energy upon HCHO adsorption. The shift becomes more negative for ε ≤ 10%. The relative increase in the magnitude of this downshift for ε ≤ 10% means that the system energy is reduced by a larger amount upon chemisorption of HCHO when strained, which favors stronger chemisorption with lower E_ads, leading to increased reactivity. The same tendency has also been reported for the dissociative adsorption of hydrogen on silicene under strain.²⁴

Fig. 15 (a) Center of bonded Si 3p band before HCHO adsorption. (b) Shift in center of bonded Si 3p band from the pristine to the HCHO-adsorbed state, relative to the zero-strain condition.

Furthermore, at ε = 3%, 5% and 7%, the silicene with SW defects after HCHO adsorption still exhibits semiconductor characteristics with band gaps of 0.163, 0.136 and 0.109 eV, respectively (shown in Table 2). Although silicene with SW defects becomes a metal when the strain increases to 9% and 10%, the electronic properties of silicene with SW defects still display an apparent variation before and after HCHO adsorption under a tensile strain of 0–7%. This indicates that even when external stress is present in the adsorption system, SW-Si still exhibits the characteristics of a gas sensor.

Table 2 Mulliken charge transfers from silicene with SW defects under strain to a HCHO molecule and band gaps of silicene with SW defects before (E_gb) and after (E_ga) HCHO adsorption

ε	Q (e)	Egb (eV)	Ega (eV)
3%	−0.418	0	0.163
5%	−0.411	0.028	0.136
7%	−0.409	0.028	0.109
9%	−0.408	Metal	Metal
10%	−0.412	Metal	Metal

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4. Conclusions

To summarize, the adsorption of HCHO molecules on the pristine, Stone–Wales defective and double-vacancy defective silicene was investigated using the first-principles density functional theory. Detailed analyses of the geometrical structures and electronic properties of optimized configurations were performed. It was found that the adsorption of a HCHO molecule on pristine silicene does not induce any structural distortion in either the molecule or silicene. Moreover, a HCHO molecule can be chemisorbed on silicene with SW defects and double vacancies. Both defective silicene sheets exhibit large binding energies and short bond lengths. However, the electronic conductance changed markedly when a HCHO molecule was adsorbed on SW-Si but not on DV-Si. Interestingly, the chemical reactivity of SW-Si can be tuned by applying an external strain. The application of strain provides an increase in the rate of reactivity by a factor of up to 10. This suggests that strain engineering of silicene provides a feasible way to modify the chemical activity of silicene.

Acknowledgements

The authors gratefully acknowledge financial support from NSFC (No. 21303054), China Postdoctoral Science Foundation (2013M540332) and the Fundamental Research Funds for the Central Universities under projects No. 222201414040 and 222201314049. All the computation simulation was undertaken with the resources provided from the High Performance Computing Center of East China University of Science and Technology.

Notes and references

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