The Influence of Viscosity on the Truncation of Accretion Disks Around Black Holes

Nanotechnology

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Versatile mechanical properties of novel g-SiCx monolayers from graphene to silicene: a first-principles study To cite this article before publication: Y X Wang et al 2018 Nanotechnology in press https://doi.org/10.1088/1361-6528/aac337

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Versatile mechanical properties of novel g-SiCx monolayers from

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graphene to silicene: a first-principles study

X. K. Lu1, T. Y. Xin1, Q. Zhang1, Q. Xu2, T. H. Wei1, Y. X. Wang1*

1. Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Department of Nuclear Science and Technology,Fudan University, Shanghai, China 200433

Abstract

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2 Research Reactor Institute, Kyoto University, Kumatori-cho, Sennan-gun, Osaka 590-0494, Japan

Recently, a series of graphene-like binary monolayers (g-SiCx), where Si partly

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substitutes the C positions in graphene, have been obtained by tailoring the bandgaps of graphene and silicene that has made them a promising material for application in opto-

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electronic devices. Subsequently, evaluating the mechanical properties of g-SiCx has assumed great importance for engineering applications. In this study, we quantified the in-plane mechanical properties of g-SiCx (x = 7, 5, 3, 2 and 1) monolayers (also including graphene and silicene) based on density function theory (DFT). It was found that the mechanical parameters of g-SiCx, such as the ideal strength, Young’s modulus, shear

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modulus, Poisson’s ratio, as well as fracture toughness, are overall related to the ratio of Si-C to C-C bonds, which varies with Si concentration. However, for g-SiC7 and g-SiC3, the mechanical properties seem to depend on the structure because in g-SiC7, the C-C

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bond strength is severely weakened by abnormal stretching, and in g-SiC3, conjugation structure is formed. The microscopic failure of g-SiCx exhibits diverse styles depending on the more complex structural deformation modes introduced by Si substitution. We

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*

Corresponding author. Tel: +86-21-55664147. E-mail: [email protected].

1

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elaborated the structure-properties relationship of g-SiCx during the failure process, and

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particularly, found the structural transformation of g-SiC3 and g-SiC due to the singular symmetry of their structure. Due to the homogeneous phase, all the g-SiCx investigated

in this study preserve rigorous isotropic Young’s moduli and Poisson’s ratios. With versatile mechanical performances, the family of g-SiCx may facilitate the design of

advanced two-dimensional (2D) materials to meet the needs for practical mechanical

mechanical behaviors of g-SiCx monolayers.

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1. Introduction

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engineering applications. The results offer a fundamental understanding of the

Nowadays, with monolayer networks of sp2 bonded carbon atoms being realized

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for the first time [1,2], graphene [3] sets off a tidal wave of researches for twodimensional (2D) atomic crystals by virtue of its superior electronic and mechanical properties, concomitant novel promising fabrication of ultrahigh-speed electronics [4], spintronics [5] as well as nanoelectronics [6]. Si is immediately below C in the periodic table, and belongs to the same family. Moreover, Si also possesses a 2D allotrope with a

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honeycomb structure and exhibits weakly buckled local geometry viz. silicene. As an analogue of graphene, silicene has been perceived as an excellent 2D material in experiments [7-9] and theoretical studies [10,11] due to its exceptional compatibility

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with current Si-based electronics. The superior properties of the zero band gap with linear and symmetric dispersion close to the Dirac points [12,13] for both graphene and silicene arise from the massless Dirac fermion-like behavior of the charge carriers. This

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drawback limits the applications of graphene and silicene for opto-electronic devices 2

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such as field effect transistors and solar cells since these devices require a suitable band

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gap. Therefore, it is not surprising that numerous researchers are making tremendous efforts on tailoring the band gap of these materials to meet the requirement of band gap

engineering, including patterning into nanoribbons [14,15], electrostatic field tuning [16], chemical modification [17,18], and hetero atom doping and substitution [19-21].

At present, doped, substituted, or alloyed graphenes have been attractive strategies

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to tune their band gaps in use of future nano-mechanical and opto-electronic devices. A recently reported binary 2D alloy of graphitic carbon and nitrogen (C1-xNx) [22,23]

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exhibits a surprising electronic structure: C3N and C12N with mediate band gaps of 0.96 and 0.98 eV, respectively, well meet the high requirements of field effect transistors and

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solar cells. However, due to the tendency of N atoms to form molecular gas, the stability range of the largest achievable N doping concentration in graphene was naturally limited within the 33.3-37.5% range, which is less than half the range of graphene atoms. This severely limits the diversity of the structures and leads to low-level applications in optoelectronic devices. Dai et al. [24] predicted the most stable structures of 2D

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stoichiometric AlxC (x=1/3, 1, 2, and 3) monolayer sheets, where the Al2C monolayer possesses a band gap of 1.05 eV, a value suitable for photovoltaic applications. Given the tetracoordinated C atoms, the atomic structures of AlxC no longer maintain the

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hexagonal symmetry of graphene. In particular, AlC3 adopts a buckled geometry structure with an energy higher than 0.05 eV than planar ones. In addition, a relative low melting temperature of 1000-2500 K may be an adverse factor for applications requiring

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high temperature environments. Boron, an active element, has been extensively 3

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investigated. Since the first synthesized BxC1-x compounds with x = 0.17 [22], where the

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existence of a BC5 compound was proposed [25], Luo [26] has predicted the presence of B- and C-rich BxC and BCx planar 2D sheets by performing particle-swarm optimization (PSO) algorithms [27]. Unlike AlCx and C1-xNx ， almost all 2D B-C

compounds are metallic except for BC3 which has a trivial indirect gap of 0.52 eV. In

addition to alloying hetero atoms into graphene, Dai [28] analyzed B-doped silicene. The

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stable structures of 2D in plane B-Si compounds, where Si favors sp2 hybridization, are predicted for a wide range of B-Si stoichiometry. However, the lowest-energy 2D B-Si

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compounds are all metals, regardless of the B-Si stoichiometry. In this regard, the binary compounds mentioned above hold unfavorable prospects for optoelectronic applications.

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What will happen if Si atoms would replace some of the C atoms in the graphene hexagons? The existing literature, including theoretical simulations [29-35] and experimental synthesis [36, 37] focused on single atomic graphene-like Si-C compounds (g-SiCx). Previous studies have demonstrated that SiC with a 1:1 stoichiometric ratio

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energetically stable and attainable as one-dimensional (1D) Si-C nanotubes, 2D sheets

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and three-dimensional (3D) bulk materials [37-39]. Two-dimensional Si1-xCx structures (𝑥 ≠1), i.e., structures with stoichiometry deviating from the 1:1 stoichiometry are also considerably stable since their formation energies are just above 0.057--0.329 eV when

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compared to SiC. It is also worth mentioning that the Si-C binary graphene-like monolayers possess robust thermal stability at 3000-3500 K [29-32], except the planar tetracoordinate pt-SiC2 with a relatively low melting temperature of 800-1000K [33].

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The Si-C binary monolayers are even superior to C1-xNx, AlxC, and BxC. The 2D Si-C 4

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compounds present two different structural phases, i.e., homogeneous and in-plane

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hybrid phase, both possessing distinct electronic structures and mechanical properties. The electronic properties for the two phases have been successfully predicted according to their lattice structures using the 3N rule [40]. The delocalized π electrons for the hybrid phase become localized by introducing Si dopants due to the potential

perturbation, and this structure shows uniform semiconducting properties with a widely

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tunable band gap from 0 to 2.87 eV. By contrast, the homogeneous phase can be either

semiconducting or semimetallic depending on the superlattice vector. These findings

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suggest that 2D single atomic Si-C compounds may present a new ‘family’ of 2D materials with diverse properties for applications in electronics and optoelectronics.

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Considering the chemical stability due to strong in-plane covalent bonds and the robust thermal stability due to low formation energy, synthesizing 2D Si-C compound ‘families’ in laboratories is to be expected. However, it is difficult to obtain a perfect 2D Si-C monolayer in experiments using chemical vapor deposition (CVD) because of mismatching between monolayer and substrate and substrate surface corrugation [41,42]

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Which always results in the distortion of the monolayer. This will significantly influence the properties of the Si-C monolayer. Therefore, in-depth understanding of the mechanical stability of 2D Si-C monolayers is crucial for practical applications,

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especially for electronic devices, and for revealing the profound physics beneath the response relationship between the electronic properties and the strain. The ideal tensile strength of a 2D material [43] is a crucial mechanical parameter

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fundamentally characterizing the nature of chemical bonding and elastic limit of single5

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or a-few-layer thin films. Until now, the utmost limit of elastics, characterized by the

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ideal strength and critical strain (the value corresponding to the ideal strength), has been studied for many pure 2D materials, such as graphene [44-46], borophene [47,48] and silicene [49]. However, studies on 2D binary materials are still missing. Mechanical

responses under strain are even more complex in 2D binary compounds than in their

pure counterparts with the same honeycomb network since the homogeneous bonding

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characteristics are broken by mixing heterogeneous bonds of different species. In this

study, we used the first-principle density functional theory (DFT) to evaluate the

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mechanical properties of 2D Si-C monolayers thoroughly with different chemical stoichiometry by calculating their ideal tensile strength, critical strain, Young’s moduli,

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Poisson ratios, and toughness. A systematic analysis of the relationship between structure and mechanical properties was carried out. It was found that the mechanical parameters of g-SiCx monolayers are associated with the concentration of Si-dopants, while the failure mechanism is influenced by the atomic steric distribution related to the Si replacing C position. The 2D SiCx family exhibits versatile mechanical traits that provide

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flexible choices in mechanical applications of 2D functional materials. In addition, the isotropic features Young’s modulus and Poisson’s ratio, can facilitate the design of advanced composites with direction-independent properties. We hope that our

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investigation can propel SiCx-based nanostructures toward novel nano-mechanical applications.

2. Computational methods

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We performed DFT calculations using the Vienna Ab initio Simulation Package 6

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(VASP) [50] based on the projector augmented wave (PAW) method [51] employed to

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describe the electron-ion interaction. We used the generalized gradient approximation (GGA) [52] exchange-correlation energy designed by Perdew, Burke, and Ernzerhof

(PBE) [53]. The plane wave cut off energy was set to 500eV. The convergence criterion

of self-consistent calculations for ionic relaxations was 10-5eV between two consecutive steps. The atomic positions and unit cells were optimized until the atomic forces were

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less than 0.01eV/Å without any symmetry constraints. The vacuum layer thickness was

20 Å to avoid image layer interactions. The size of the primitive cell was set according

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to the stoichiometry of g-SiCx from graphene to silicene. Correspondingly, the different mesh grids of k-points were adopted for the different primitive cell of these monolayers.

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In order to conveniently and reliably calculate the ideal strength of g-SiCx from graphene to silicene, the rectangular, while not rhombic unit cell, shown by the black line in Fig. 1, was adopted where the two basis vectors were set along armchair (A) and zigzag (Z) directions, respectively. Since the two basic vectors are perpendicular to each other, it is convenient to impose uniaxial strains directly along the vector direction.

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Tension simulation was carried out by imposing in-plane tensile strain at a rate of 0.01 per step in the uniaxial direction (A or Z). The tensile strain is defined as ε =(L-L0)/L0, where L0 and L are the equilibrium and strained lattice constant, respectively. In order to

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mimic the tensile process, geometry relaxation was performed for both lattice basis vectors, while the atoms were fixed in the in-plane strain direction and simultaneously relaxed in the perpendicular direction. This process is accomplished by slightly

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modifying the VASP code with specific constraints for the strain components. At large 7

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strain, the crystal symmetry may be changed and the Brillouin zone significantly

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deformed. Thus, we increased the cutoff energy by 30% and verified the convergence of the stress-strain calculations with different k-point grids. This method was initially introduced by Round et.al and Luo et.al [54,55] for the 3D bulk crystal.

For a 2D material, the elastic constants and modulus can be determined from the Hook’s law (1) under the plane-stress condition [56-58]:

where 𝐸𝑖 ,

0

𝜀𝑥𝑥 0 0 ] [ 𝜀𝑦𝑦 ] 𝐶66 2𝜀𝑥𝑦

𝐶12 𝐶22 0

0 0

𝜀𝑥𝑥 𝜀 ] [ 𝑦𝑦 ] 𝐺𝑥𝑦 (1 − 𝜈𝑥𝑦 𝜈𝑦𝑥 ) 2𝜀𝑥𝑦

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𝐶11 = [𝐶21 0

𝜈𝑦𝑥 𝐸𝑥 𝐸𝑦

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𝐸𝑥 𝜎𝑥𝑥 1 [𝜎𝑦𝑦 ] = [𝜈𝑥𝑦 𝐸𝑦 1 − 𝜈𝑥𝑦 𝜈𝑦𝑥 𝜎𝑥𝑦 0

and 𝜈𝑖𝑗 defined as 𝐸𝑖 =

𝜎𝑖 𝜀𝑖

(1)

Δ𝜀

, and 𝜈𝑖𝑗 = − Δ𝜀𝑗 respectively, represent 𝑖

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Young’s modulus and Poisson’s ratio under uniaxial tensile along the 𝑖 direction. Therefore, we can derive several principal elastic mechanical parameters for a 2D material, including Young’s modulus (E), shear modulus (G), and Poisson’s ratio(ν) based on Eq. (1) as follows:

𝐶11 𝐶22 − 𝐶12 𝐶21 , 𝐶22

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𝐸𝑥 =

𝐶

𝐸𝑦 =

𝐶11 𝐶22 − 𝐶12 𝐶21 𝐶11

𝐶

𝜈𝑥𝑦 = 𝐶21 , 𝜈𝑦𝑥 = 𝐶12 , 𝐺𝑥𝑦 = 𝐶66 22

11

(2)

However, there is a controversial definition of the elastic modulus for the 2D system.

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Wei [57] described the mechanical properties of phosphorene using the 𝐶𝑖𝑗 = 1

(

𝜕𝐸𝑠

𝐴0 𝑑0 𝜕ℇ𝑖 𝜕ℇ𝑗

) formula expressed in GPa by rescaling the dimensional length Z/𝑑0 , while 1

𝜕𝐸

𝑠 Fu [59] gave an in-plane one formula 𝐶𝑖𝑗 = 𝐴 (𝜕ℇ 𝜕ℇ ) expressed in N/m by rescaling

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0

𝑖

𝑗

the dimensional length Z. In the above formulas, 𝐸𝑠 represents the strain energy, and 𝐴0 , Z and 𝑑0 are the area of unit cell, vacuum thickness, and effective thickness, 8

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respectively. As the material system experiences lengths varying from the macroscopic

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scale to multi and single atomic 2D crystals, the effective thickness is a vaguely defined parameter. In a 3D bulk system, 𝑑0 is the interlayer distance [60,61]. However, for a 2D system, 𝑑0 usually equals the buckling distance, which is the out of plane direction

distance between the topmost and bottommost atoms [62]. In some studies[63,64], 𝑑0 was replaced by the summation of the buckling distance and the van der Waals (vdW)

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radii of the outer-most surface atoms. Since vdW radii are different for different element types, this will lead to different effective thickness values, thus giving ambiguous elastic

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properties depending on the different definitions used for 𝑑0 . What thickness should we adopt? The ambiguity for the thicknesses of mono-atomic layers such as graphyne,

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graphene, and carbon nanotubes has been previously discussed [65], suggesting that the stress and elastic moduli of the 2D monolayer should be expressed as a unit of force N/m rather than GPa. Here, we opted for the formula proposed by Fu [59] to calculate the elastic modulus and express it in N/m. Moreover, the calculated modulus using this formula was marked using a ‘2D’superscript. In addition, the corresponding in-plane

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bulk values expressed in GPa were calculated using Wei’s formula, and were labeled using the ‘3D’ superscript.

The toughness (Γ) of g-SiCx was explored in order to quantify the fracture resistance

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capability when replacing part of C atoms with Si in graphene. Γ was defined as the integration of the stress-strain curves, as follow [66]:

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𝜀

Γ = ∫0 𝑚𝑎𝑥 𝜎𝑑𝜀

where 𝜀𝑚𝑎𝑥 is the critical strain corresponding to the ideal strength upon failure. 9

(3)

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3. Results and discussion

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3.1 Structures of 2D Si-C monolayers Carbon-rich SiCx (g-SiCx, x = 7, 5, 3, 2, and 1) obtained from the family of the

graphene-like 2D Si-C monolayers were the focus of this study. We chose C-rich monolayers for the reasons listed below. (1) C-rich homogeneous phases avoid the

formation of Si-Si bonds, which substantially destabilize the hexagonal system [34]. (2)

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g-SiC2 and g-SiC7 possess surprising band gaps of 1.13 and 1.09 eV, respectively, that fairly satisfy the demand of highly energetic conversion efficiency for donor materials

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in solar cells through superior sunlight optical absorbance in infrared and visible photon ranges [67], thus holding great potential applications in photovoltaics [29,32]. (3) g-SiC3

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inherited the electronic structure of graphene and silicene, and more importantly, is a novel 2D topological insulator with enhanced spin-orbital coupling (SOC) superior to that of graphene [31]. (4) g-SiC5, benefitting from its semimetallic nature, is superior to other siligraphenes in terms of gas sensing, which ensures fast charge transport [30]. Further, the g-SiCx studied here possesses relatively low formation energy and

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comparable cohesion energy that rival those of graphene, whose parameters are listed in Table 1. Their relatively high thermal stability makes their synthesis feasible. For instance, quasi-2D SiC2 has been successfully synthesized in experiments [36]. For

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convenience and systematic investigation, a series of g-SiCx, in order of increasing Si concentration (beginning with graphene and ending with silicene), was the focus of this study.

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Table 1 Lattice constants, bond lengths, formation energy, cohesion energy and extracted band gap as reported in previous studies [29].

Graphene g-SiC7 g-SiC5 g-SiC3 g-SiC2 SL-SiC Silicene

a = b (Å)

C-C

C-Si

Si-Si

Ef (eV/atom)

2.468 5.273 4.639 5.633 5.002 3.096 3.867

1.424 1.437/1.550 1.454 1.438 1.444 -

1.691 1.766 1.813 1.797 1.786 -

2.277

0.00 0.52 0.43 0.44 0.43 0.31 0.00

Ec (eV/atom)

Band gap (eV) [29]

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Bonds length (Å)

-8.14 -6.95 -6.91 -6.62 -6.35 -5.91 -3.66

0 1.13 0 0 1.09 2.90 0

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We started by defining the formation energy (Ef) and cohesive energy (Ec) of the gSiCx (x = 7, 5, 3, 2, and 1) monolayers, as follows:

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𝐸𝑓 = 𝐸𝑠𝑖𝑥 𝐶𝑦 − 𝑥𝜇𝑆𝑖 − 𝑦𝜇𝐶 𝐸𝑐 = 𝐸𝑠𝑖𝑥 𝐶𝑦 − 𝑥𝐸𝑆𝑖 − 𝑦𝐸𝐶

(5) (6)

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where 𝐸𝑠𝑖𝑥 𝐶𝑦 , 𝐸𝑆𝑖 and 𝐸𝐶 are the energies of the g-SiCx monolayer with a single Si atom and a single C atom in supercell. In addition, 𝜇𝑆𝑖 𝑎𝑛𝑑 𝜇𝐶 are the chemical potentials of silicene and graphene, while x and y correspond to the concentrations of Si

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and C in g-SiCx.

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Fig. 1 Structures of (a) graphene/silicene; (b) g-SiC7; (c) g-SiC5; (d) g-SiC3; (e) g-SiC2; and (f) gSiC. The rectangular and rhombic unit cells are highlighted using black and red solid lines, respectively. Specifically, the C ring in g-SiCx (x = 7, 3) and the Si-C ring in g-SiC3 are circled with red dotted lines. The bond lengths are expressed in angstroms. It should be noted that silicene has a

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low-buckled structure, but its top view has a lattice plane similar to graphene.

For the different concentrations of Si substituting C, the most energetically favorable atomic structures are depicted in Fig. 1. These have been already analyzed in previous studies [10, 29-32]. For these structures, the primitive and habit cells

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corresponding to the rhombic and rectangular unit cells are framed by red and black lines, respectively. First we firstly optimized graphene, the g-SiCx (x = 7, 5, 3, 2, and 1) and silicene monolayers. The calculated lattice parameters listed in Table 1 are in good

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agreement with previous results [10, 29-32]. Similar to graphene, Si-doped g-SiCx still maintained a honeycomb structure. However, the honeycomb six-membered rings of gSiCx are severely distorted when x =7, 5, and 2 (Fig. 1(b), (c) and (e)). The most severe

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distortion happens in g-SiC7, which has the scarcest concentration of Si doping. In g12

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SiC7, the lengths of the C-C bonds are uniform, some still maintaining the 1.424 Å length

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as in graphene, while others are elongated to 1.550 Å, as shown in Table 1. On the other hand, Si-C bonds in the six-membered rings are shrunk compared with those in the SiC crystal. Inconsistency in bonding length leads to the heavy deformation of the sixmembered rings in g-SiC7. Furthermore, g-SiC3 preserved the synergetic conjugation between C and Si rings, which guarantees its Dirac cones and topologically nontrivial

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electronic structure.

Albeit the distortion exiting in the six-membered rings, Si or C atoms are still

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constrained in the plane of g-SiCx (x =7, 5, 2 and 1) monolayers. This is mainly attributed to the in-plane sp2 hybridization of both Si and C atoms, which avoids the formation of

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the dangling bonds. Actually, Si atoms tend to adopt sp3 hybridization, such as in silicene where the buckled pattern forms, but sp3 hybridization is responsible for the higher 2.56 eV/atom formation energy in Si (Table 1). The hybridization transformation of Si from sp3 to sp2 favors the chemical stability of 2D g-SiCx monolayers, as compared with buckled silicene, and facilitates the exfoliation from substrates in experiments. This

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strong in-plane covalent bonding feature is reflected by the calculated cohesion energy in Table 1, where the cohesive energies in the new g-SiCx ( x = 7, 5, 3 ,2, and 1) structures

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are all lower than that of pt-SiC2 :6.05 eV/atom [33]. 3.2 Ideal strength and failure behavior Based on the study of stable structures, the mechanical response under uniaxial

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strain along either the armchair (A) or zigzag (Z) direction can be evaluated. Ideal tensile strength i.e., the ultimate strength, the maximum strength a material can withstand when 13

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being stretched to critical strain, is an important parameter. Beyond its critical value, the

cri pt

material will be meta-stable and easily destroyed by long wave-length perturbations, vacancy defects, and high temperatures [68]. The calculated stress-strain relationships are presented in Fig. 2, and the corresponding ideal strength and critical strain for each

stoichiometric Si-C monolayer are summarized in Table 2. Our calculations for graphene, g-SiC, and silicene are consistent with those existing in the literature, indicating the

us

reliability of the present computational scheme [69-71]. Similar to previous studies on

graphene and silicene [71-73], all mono-sheets with different stoichiometry also obey

an

linear relationships with ε within a small strain range, and deviate from linear elasticity under large deformation. However, the yielding point, corresponding to the critical strain

dM

in this study, and failure mechanisms are different. In graphene and g-SiCx (x = 7, 5), the yielding point is always higher in the Z direction than in the A direction. On the contrary, the yield value in this direction is slightly higher for g-SiC3 and silicene, while the yielding points in the two directions are almost equal for g-SiC2 and g-SiC. In addition, graphene and g-SiCx (x = 5, 2) sheets undergo more ductile failure processes at maximum

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load along the Z direction than along the A direction. On the contrary, silicene, g-SiC, and g-SiC7 exhibit more ductile properties along the A direction. In particular, g-SiC7 exhibits a robust brittleness along the Z direction and a great ductility along the A

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direction, forming a strong anisotropy of in-plane strength. This discrepancy between the A and Z uniaxial tensions remarkably reaches 57.59% for the ideal strength, and 60% for the critical strain. This can be attributed to the nonequivalent feature of the in-plane

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Page 14 of 42

bond strength and will be discussed infra. It is interesting to note that after yielding under 14

Page 15 of 42

Z-direction loading, g-SiC3, g-SiC and silicene all undergo structural transitions through

an

us

cri pt

forming new bonds.

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Fig. 2 Stress-Strain relationships for (a) graphene and silicene; (b) g-SiC7; (c) g-SiC5; (d) g-SiC3; (e) g-SiC2 and (f) g-SiC under strains along the A and Z directions.

Table 2 shows that graphene exhibits outstanding ideal strength to resist the external deformation (34.38 N/m at 0.18 strain for the A direction and 38.12 N/m at 0.21 strain for Z direction), whereas, silicene presents the least strength (7.17 and 5.83 N/m along

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the A and Z directions). It is no wonder that silicene is responsible for this lowest value since its buckled structure stems from sp3-hybridized bonds, which are more apt to be ruptured than the in-plane sp2 bond weaving networks. Our observation is in agreement

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with the fact that sp3 hybridization deteriorates the mechanical properties in graphene allotropes [73]. With increasing content of Si (starting from graphene and ending with silicene), the ideal strength generally declines with increasing concentration of Si atoms

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along the two directions except for g-SiC7 and g-SiC3. Since the ideal strength may be limited by the phonon instabilities, we calculated the phonon dispersion for each case 15

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we studied here. It was found that along the A direction, the critical strain corresponding

cri pt

to phonon instability was very close to the critical value of elastic instability for each monolayers. For the Z axial tension, cases were very similar to those of the A direction except for g-SiC7 and g-SiC3. For g-SiC7 and g-SiC3, an imaginary frequency can be

observed clearly ahead of the critical strain of elastic instability. This further lowers the

values of the critical strain, the ideal strength and the toughness of g-SiC7 and g-SiC3,

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and enhances the peculiarity of the two cases. The resulting reasons will be discussed

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later.

Table 2 Ideal strength (N/m), critical strain, and the differences in these values between the A and Z directions. Ideal Strength

g-SiC7 g-SiC5 g-SiC3 g-SiC2 g-SiC

Z

34.38 30.84 14.55 24.88 19.11 21.52 19.75 22.8 7.17 7.07

38.12 34.13 22.93 27.01 18.04 21.9 19.95 20.5 5.83 5.66

Critical Strain A

10.88 10.67 57.59 8.56 5.60 1.77 1.01 1.01 18.69 19.94

0.18 0.19 0.1 0.15 0.14 0.17 0.18 0.17 0.17 0.18

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Silicene

A

Difference in ideal strength (%)

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Graphene

Z

0.21 0.23 0.16 0.18 0.11 0.17 0.18 0.17 0.16 0.14

Difference in critical strain (%) 16.67 21.05 60 20 21.42 0 0 0 5.88 22.22

Source

PBE [69]

MD [70] [71]

ce

To further understand the failure mechanism, we monitored the variation of bond lengths in g-SiCx (x = 7, 5, 3, 2, and 1) monolayers, in conjunction with the snapshots based on the valence charge density differences (VCDDs) before and after bonds are broken (M1 and M2, respectively in Fig. 3). Figure. 3 reveals the atomistic tensile

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deformation. 16

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Fig. 3 VCDDs snapshots at the M1 and M2 points for g-SiCx (x = 7, 5, 3, 2, and 1). It should be noted that M1 and M2 represent the before and after failure points for A and Z direction loadings (marked by the arrows), respectively. The isosurface values of g-SiC3 along the A direction and g-SiCx(x=7, 5, and 2) correspond to ±0.0125 /Bohr3; The isosurface values of g-SiC3 along the Z direction and gSiC correspond to ±0.0085 /Bohr3.

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Failure occurs at ε=0.1 for g-SiC7 under the strain applied in the A direction. Subsequently, after experiencing a long plastic process, a collapse occurs at ε=0.27 (M2 point), where among the bonds parallel to the A direction, Si1-C1 and C5-C6

ce

dramatically elongate and break completely, while the others, C3-C4 and Si2-C2 are stretched at first and then contract close to their initial lengths of 1.55 and 1.69 Å, respectively. It may be surprising that at the yielding point, the C-C bonds along the A

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cri pt

Page 17 of 42

direction (C3-C4 and C5-C6) broke before the Si-C bonds (Si1-C1), as seen in the VCDD at M1. This should be ascribed to the extremely unbalanced bonds strength 17

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introduced by the scarce Si atoms, which distorts the perfect hexagon rings of the

cri pt

honeycomb. As shown in Fig. 1(b), the C- and Si-C-rings generate the g-SiC7 network together. The C-ring is severely deformed and the C-C bond length (1.437 Å) is still close to 1.424 Å (the inherent length of C-C sp2 hybridization in graphene). However, in

the Si-C-ring, larger distortion in bonds has to occur in order to seal the ring since the two C-C bonds significantly elongated to 1.550 Å compared to 1.424Å. In addition, the

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two Si-C bonds contract to 1.69 Å compared to 1.768 Å (the inherent length of Si-C sp2

hybridization). Under uniaxial tension along the A direction, the pre-elongated C-C

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bonds (C3-C4, C5-C6) and pre-contracted Si-C bonds (Si1-C1, Si2-C2) are all simultaneously subjected to the stretch. In Si-C bonding, the charge distribution of the

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Si atom is more easily reshaped than that of the C atom since the former has much low electronegativity, where the difference in (Pauling) electronegativity between C and Si atoms is quite large, ΔΧ=0.7 [66]. This makes Si-C bonds much more tolerant of the stretched process, thus enabling them to bear a large loading. As a result, C-C bonds seem somewhat rigid, while Si-C bonds are more flexible. Under tensile strain along the

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A direction, the pre-elongated C-C bonds (C3-C4, C5-C6) follow a continuing stretch, resulting in first breaking at a 0.1 critical strain and a low peak strength of 14.55 N/m, while the contracted Si-C bonds (Si1-C1 and Si2-C2) release the inner stress through

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stretching themselves, and preserve up to the catastrophic collapse at M2. The asynchronous break of the C-C and Si-C bonds results in a unique behavior during plastic deformation, where the stress-strain curve experiences a huge elongation of strain in the

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0.087- 0.26 range before fracture. When loading occurs along the Z direction, the pre18

Page 19 of 42

elongated and rigid C-C bonds (C4-C7 and C6-C8 in Fig. 3(a)) with a bond length 1.550

cri pt

Å are also first broken after the 0.16 critical strain point. This requires a relatively large strength of 22.93N/m. However, those pre-contracted Si-C bonds (such as Si3-C9) only undergo a slight stretch to 1.92 Å, as shown by the M1 and M2 points in VCDDs (Fig.

3(a)). A large polygonal ring is thus created under Z-directional loading. One can see that the ideal strength along the A direction is much lower than that along the Z direction.

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This is because in A-directional loading, the main pre-elongated C-C bonds (C3-C4, C5C6) are directly subjected to loading in the tensile process, whereas for Z-directional

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loading, all the pre-elongated C-C bonds (C4-C7 and C6-C8) and pre-contracted Si-C bonds (such as Si3-C9) in zigzag chains present a minimum deviation angle of 30º from

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the Z direction. Thus, the external loads can be dissipated, concomitant with bond elongating and angle enlarging. Therefore, lower ideal strength and critical strain were expected for A-directional tension. The presence of Si substitution in graphene leads to a reduction of 4.2% and 6% in graphene tensile strength along the A and Z directions, respectively. It should be mentioned that a more seriously distorted honeycomb lattice

pte

usually results in poorer strength. It was also found that g-SiC7 possesses the highest anisotropy of orientation, where the discrepancies between A and Z are 57.59% in ideal strength and 60% in critical strain. This is scarcely observed in other g-SiCx monolayers.

ce

The concentration of Si in g-SiC5 is slightly higher compared to g-SiC7. With the ratio of Si-C bonds to C-C bonds increasing, the degrees of stretching for the C-C bonds and contracting for the Si-C bonds are alleviated. At the same time, the degree of unbalanced

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bonds between Si-C and C-C diminishes. However, distortion in bonds is still present. 19

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Unlike g-SiC7, all C-C bonds in g-SiC5 have the same 1.454 Å length, slightly longer

cri pt

than the inherent 1.424 Å length, while Si-C bonds, 1.766 Å, are slightly shorter than the 1.786 Å inherent length. The C-C bonds (C1-C2, C3-C4, C5-C6, and C7-C8),

deviating a small angle (7.4º) from the A direction, are first to be broken under A directional loading. This is still due to the pre-elongating and weakening of C-C bonds, as shown in VCDDs at the critical points (Fig. 3(b)). As a result, akin to the failure modes

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in g-SiC7, the rigid C-C (C1-C2, C3-C4, C5-C6 and C7-C8) bonds are liable to break. This behavior dominates the peak stress of 24.88 N/m at the 0.15critical strain. When

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imposing a Z directional tension, the pre-elongated and weakened C-C bonds (C6-C9 and C8-C10) in the zigzag chains are first to rupture, as shown in Fig. 3(b). Since the

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bonds in zigzag chains have a large angle deviating from the Z direction, stronger ideal tensile strength and critical strain were attainable for the Z direction than for the A direction. Our observations are in agreement with the fact that atomic bonds are straightforwardly parallel to the direction of loading, thus susceptible to failure [65,68]. In additional, the degree of unbalanced bonds between Si-C and C-C is alleviated,

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compared to g-SiC7. Hence, the increase in Si-dopant content in g-SiC5 alleviates the local distortion, which results in the ideal strength of g-SiC5 surpassing that of g-SiC7. With the ratio of Si to C further increasing to 1:3, the hexagonal symmetry seemed

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to return, as shown for g-SiC3 in Fig. 1(d). The C rings remain perfectly hexagonal. The Si-containing rings are also hexagonal with equal internal angles, but different bond lengths. This somewhat high symmetry originates from a synergetic conjugation

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between the C and Si atoms. However, the high symmetry is gained at the expense of 20

Page 21 of 42

largely stretching Si-C bonds (1.813 Å). Here，Si-C is in a tensile state, and is no longer

cri pt

as flexible as the contracted Si-C bonds mentioned above, while the C-C bond length is 1.438 Å, approaching 1.424 Å in graphene. Therefore, the bond instability primarily

exists in those pre-elongated Si-C bonds (Si1-C1, Si2-C2 and Si3-C3) under the A directional loading, as shown in VCDDs in Fig. 3(c). Since stretching the Si-C bonds occurs in both zigzag and armchair chain, the ideal strengths for the two directions

us

significantly decline to 19.11 N/m at ε=0.14 along the A direction and 18.04 N/m at ε=0.11 along the Z direction. The C ring nearly preserves its hexagonal symmetry during

an

the A/Z directional loading process, where the C-C bonds in the C ring change only from 1.438 to 1.482 Å at the critical strain when stretched in the A strained direction,

dM

suggesting the integrality of C rings as benzenoid C6 units. For the A directional loading, the Si-C bond is the first to break at the critical strain point, M1. No bond breaking occurs for the Z directional loading, as shown in Fig. 3(c), whereas the Si-C hexagon circled in Fig. 4(a) is heavily elongated along the Z direction, where its inner angles α and βsynchronously increase from 120° to 149°, unlike the benzenoid C ring where the

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inner angles are almost unchanged. The large deformation of this Si-C hexagon causes failure, reflecting in the stress-strain curve. In g-SiC2, the C-C and Si-C bond lengths are 1.444 and 1.797 Å, respectively. The

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ideal strengths for the two directions are almost equal, around 21.5 N/m. The C-C bonds (C1-C2 and C3-C4 in Fig. 3(d)) are broken during A directional loading, while the Si-C bonds (Si1-C5 and Si2-C6 in Fig. 3(d)) are broken during Z directional loading. g-SiC2

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is more ductile in the Z than in the A direction, which may be due to the flexibility of Si21

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an

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C bonds and the rigidity of C-C bonds.

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Fig. 4 Snapshots of the failure process: (a) g-SiC, (b) g-SiC3 at different strain points. The bond angles in the Si-C rings circled with red dotted lines are labeled as α and β in g-SiC3 and 𝛼 ′ and 𝛽 ′ in g-SiC, respectively.

When the Si: C reaches 1:1, the hexagonal network is only knitted by Si-C bonds and no more C-C bonds appear. Thus, g-SiC exhibits a perfectly hexagonal symmetry,

pte

where the Si-C bond length is 1.786 Å. The ideal strengths are 19.95 and 19.75 N/m along the Z and A directions, respectively. Similar to the highly symmetric g-SiC3, the Si-C bonds (Si1-C1 in Fig. 3(e)) were broken at the critical strain along the A directional

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loading since the Si-C bonds aligned in the A direction are directly subjected to the load. No breaking in the Si-C bonds cccurs along the Z direction, only the transformation of the 2D structure takes place at the critical strain (Fig. 4(b)). From Table 2 and Fig. 5 (a) and (b), looking at the results of anisotropy for ideal

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Page 22 of 42

strength and critical strain, we can see that the magnitude of the in-plane strength 22

Page 23 of 42

difference between the A and Z directions generally decreases with the increase of Si-

cri pt

dopants concentration from 57.59% for g-SiC7 to 1.01% for the g-SiC sheet to. Specifically, g-SiC2, g-SiC3, and g-SiC show nearly isotropic mechanical strength in the

an

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two loading directions. This should be associated with the increase of Si-C bond content.

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Fig. 5 Ideal strength and critical strain under both uniaxial strains for g-SiCx (x = 7, 5, 3, 2, and 1), graphene, and silicene as a function of Si doping concentration.

Fig. 6 Variation of bond lengths (C-C and Si-C) and bond angles (α and β) along the zigzag chains as a function of strains along the Z direction in g-SiC3.

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Reviewing the tensile processes of the g-SiCx monolayers, bond breaking is usually responsible for the failure in both loading directions. Surprisingly, in g-SiC3 and g-SiC, no bond breaking but structure transformation takes place at the critical strain when

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applying Z tension, as shown in Fig. 4(a) and 4(b). During the transformation of g-SiC3, the Si-C bonds in zigzag chains are not significantly stretched, but the α and β C-Si-C 23

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angles in Fig. 4(a), which are symmetric with regard to the mirror plane perpendicular

cri pt

to the A direction, are synchronously enlarged from 120 to nearly 149º, as shown in Fig. 6. This leads to the vertices of α and β, the two Si atoms, to approach each other until

they form a new Si-Si bond, thus a new structure is formed at the 0.15 strain as seen in

Fig. 4(a). Similarly, in g-SiC, 𝛼 ′ and 𝛽 ′ denoted in Fig. 4(b) also synchronously enlarged until 180º, leading to the two vertices to directly form a new Si-C bond at the

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0.27 strain. Why does the structural transformation occur just in g-SiC3 and g-SiC, while not in other g-SiCx monolayers under Z loading? If we consider these monolayers as

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networks composed of zigzag chains linked by armchair chains, the zigzag chains will bear the Z loading and are responsible for the strength of the network under Z tension.

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Meanwhile, the strength of the zigzag chain depends on both bonding strength and angle. For the benzenoid C6 ring, the π electrons conjugate perfectly as shown in Fig. 7(a) contributing to the stability of the C6 ring, in other words, the firmness of the bond angle. However, for the Si-C ring, the conjugation of π electrons severely deteriorates as shown in Fig. 7(b), therefore the bond angle structured by the Si-C bonds is more flexible. It

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can be seen that Si-C bonds of the zigzag chains in g-SiC3 and g-SiC are not significantly stretched , but the angles between the two Si-C bonds in the zigzag chain are easily opened when tensile strain is imposed in the Z direction (Figs.4 and 6). Moreover,

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because of the high symmetry of g-SiC3 and g-SiC, the synergetic movement of α and β in g-SiC3 (or 𝛼 ′ and 𝛽 ′ in g-SiC), i.e., the synchronous enlarging of the two bond angles, can be accomplished harmonically under Z direction tensile. This situation is

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absent in other g-SiCx monolayers since there are no pairs of symmetric angles 24

Page 25 of 42

constituted of Si-C bonds. Therefore, we can understand that an interesting structural

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transformation occurs in g-SiC3 and g-SiC.

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Fig. 7 Sideview of density distribution of π electrons in (a) C-C and (b) Si-C bonds in g-SiC3.

Although the distribution of Si-dopants more or less introduces irregularities to the

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honeycomb structure, the ideal strength is not predominantly dependent on the deformation degree of the structure. In principle, the overall trends of ideal strength and

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critical strain variations with the concentration of Si demonstrate an appreciable reduction from graphene, g-SiCx (x = 7, 5, 3, 2, and 1) to silicene, as seen in Fig. 5, since the strength of Si-C bonds is weaker than that of C-C bonds. For the honeycomb lattice, bonds in zigzag chains deviate from the zigzag direction by 30º, while bonds in the

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armchair chains directly parallel to the armchair direction. This enables failures to occur in the A direction with less ideal strength than in the Z direction for a given Si concentration. On the other hand, the structural deformation induced by Si doping

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usually influences the nature of the bond through stretching or contracting bonds. The detailed failure mechanism of g-SiCx lies in the comprehensive effects from the bond length and angle, and their variations caused by Si doping. We summarized the fracture

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behavior in Table 3. The relative length of stretching (due to Si doping) to their stable states (C-C bonds in graphene and Si-C bonds in g-SiC were set to be the stable states), 25

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as well as the deviated angle (Δθ) of the broken bonds from the loading direction have

cri pt

been counted. Here, the relative length of bonds is defined as ΔD% = (D-D0)/D0, where Table 3 Summary of deformation mechanism: type of broken bonds, ΔD, and Δθ. A direction

Z direction

Systems Broken Bonds

ΔD(%)

Δθ(º)

C-C C-C C-C Si-C C-C Si-C Si-Si

0 8.8 2.1 1.5 1.47 0 -

0 0 7.29 0 0 0 0

Broken Bonds

8.8 2.1

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C-C C-C

ΔD(%)

Si-C

0.67

Δθ(º)

30 22.71 23.68

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Graphene g-SiC7 g-SiC5 g-SiC3 g-SiC2 g-SiC Silicene

D and D0 are the bond lengths in g-SiCx and in their stable states, respectively.

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Particularly in g-SiC7, extremely sparse Si atoms introduce one Si-C bond at most into one hexagonal ring. Since Si-C bonds are much longer than C-C bonds, some C-C bonds have to be extremely stretched, and Si-C bonds, contracted, in order to seal the hexagonal rings. The strength of stretched C-C bonds is severely weakened and C-C bonds are

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broken at first when imposing tensile strain. Therefore, the ideal strength of g-SiC7 degrades even more severely than that of g-SiC5 containing more Si dopants. Unlike gSiC7, it is the Si-C bonds that are heavily stretched in g-SiC3.These bonds are

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undoubtedly broken under A directional loading, whereas the bond angle plays a more critical role in promoting the structural transformation without broken bonds under Z directional loading. In g-SiC2, the C-C and Si-C bonds are all slightly stretched, and the

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Si-C bonds are broken first under Z direction loading. However, the C-C bonds are broken when applying an A directional strain, most likely because the C-C bonds in 26

Page 27 of 42

armchair chains are strictly parallel to the A direction, causing them to be predisposed to

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failure. On the other hand the Si-C bonds help disperse the loading. The conjugation of π electrons in the hexagonal ring declines after introducing Si atoms, prompting the bond

angles to be more flexible. In highly symmetric g-SiC3 and g-SiC, the flexible bond angles are synergetic to easily open under Z directional tension, which avoids breaking bonds and promotes structural transformation.

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The elucidation of the underlying atomistic mechanism definitely enriches the fundamental understanding of mechanical behaviors for g-SiCx monolayers. Based on

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the analysis of the failure mechanism at the atomic level, the present results provide some hints for designing appropriate g-SiCx networks to satisfy the practical mechanical

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engineering applications.

3.3 Elastic modulus, Poisson’s ratio, and toughness The ideal strength only provides an upper limit for the irreversible lattice resistance to a large strain a material can sustain. To evaluate the mechanical properties of g-SiCx

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thoroughly, we also calculated their elastic constants, Young’s moduli, shear moduli,

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Poisson ratios as well as toughness as seen in Table 4.

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27

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Table 4 In-plane elastic constant (𝐶11), Young’s moduli (𝐸𝑥 ), shear moduli (𝐺𝑥𝑦 ), Poisson’s ratio (𝜈𝑥𝑦 ), and toughness of g-SiCx monolayers. The values with the 2D superscript are expressed in N/m

Graphene

3.48

g-SiC7 g-SiC5

3.86 3.90

g-SiC3 g-SiC2 g-SiC

4 4.1 4.19

Silicene

2.81

2𝐷 C11

ν2𝐷 𝑥𝑦

E𝑥2𝐷

2𝐷 G𝑥𝑦

Toughness A

source

Z

352.8(1013) 350(1045) 352.7 358.1 268(695) 266.6(683)

342.3(982) 330(984) 342.2 366.4 247.9(643) 245.6(629)

0.173 0.243 0.173 0.169 0.27 0.28

145.8(478) 139(414) 145.9 148.9 97.2(252) 95.8(246)

4.05

5.22

1.84 2.27

2.31 3.03

229.2(573) 217.2(530) 178.7(428) 179.7 68.1(243) 68.3

214(535) 197.1(481) 161.9(389) 163.5 60.5(220) 68.3

0.26 0.30 0.31 0.30 0.32 0.341

85(213) 75.6(184) 62.1(149) 62.9 23(84.1) 22.5

1.66 2.29 2.21

1.11 2.22 2.13

0.77

0.55

us

d0 (Å)

an

g-SiCx

cri pt

for the in-plane sheet without considering the effective thickness. The values in brackets are for in3𝐷 plane bulk considering the effective thickness expressed in GPa. Here, ν2𝐷 𝑥𝑦 =ν𝑥𝑦 .

This work [59] [74] [75] This work This work This work This work This work [70,74] This work [75]

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The elastic modulus represents the full reversible response of a material in a linear elastic Hookean spring [76]. According to Wei’s description [57], the effective thickness (d0) must be considered for the in-plane elastic constants. We performed van der Waals density functional (vdW-DF) of optB86b-vdW function [77,78], where the effective

pte

thickness of g-SiCx was chosen as follows. For graphene calculation, d0=3.45 Å; it should be noted that 3.5 Å is used for describing a-few-layered graphene when calculating thermal transport properties [79]. For g-SiC, d0=4.19 Å ; this is in the 3.8 to

ce

4.5 Å range for the SiC nanotube measured in experiment [80] and is slightly greater than 4.138 Å, the distance between double SiC monolayers stacked in a Si-on-Si fashion [81]. For silicene, d0=2.81 Å which is close to 3 Å, the value for stacking silicene layers

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Page 28 of 42

[82,83]. The thicknesses for other g-SiCx are listed in Table 4, and are highly consistent

28

Page 29 of 42

with previous studies that guarantee their numerical reliability.

cri pt

For the in-plane elastic constants, C11 equals to C22 and C12 equals to C21 for all gSiCx monolayers studied here. This indicates that Young’s moduli and Poisson’s ratios

for g-SiCx are isotropic. Young’s modulus for graphene was calculated to be 0.982TPa

(342.3N/m), comparable with previously reported data of ~1TPa [84] and other simulated values [59,74,75]. The moduli for g-SiC7 and g-SiC5 reach 247.9 and 245.6

us

N/m, representing approximately 72.4and 71.7% of graphene’s modulus, respectively.

The modulus for BN is 267N/m [17]. The g-SiCx (x = 7, 5) sheets even rival the modulus

an

of the BN monolayer, suggesting that 2D g-SiCx (x = 7, 5) sheets are as flexible as BN. Albeit Young’s moduli in the 161.9 to 247.9 N/m range are lower than 342.3 N/m for

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graphene, g-SiCx are superior to those allotropes of graphene, such as graphyne and graphdiyne, in regards to their elastic moduli. The allotropes of graphene have extremely

ce

pte

poor elastic moduli ranging from 7 to 10 N/m [85].

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29

AUTHOR SUBMITTED MANUSCRIPT - NANO-117311.R2

Fig. 8 Curves of (a) Young’s modulus and shear modulus; (b) Poisson’s ratio; (c) toughness versus the concentration of Si-dopants under strains along the A direction. It should be noted that the calculated mechanical parameters are isotropic in the A and Z directions.

cri pt

Furthermore, the relationships between mechanical parameters and concentration

of Si-dopants are depicted in Fig. 8. Young’s modulus and shear modulus in Fig. 8(a) decline from g-SiC7 to g-SiC. This reflects the association between the elastic moduli and the content of the Si-C bond, which is intrinsically weaker than the C-C bond.

us

Poisson’s ratio, defined as the ratio of the strain in the transverse direction to that

in the longitudinal direction, measures the fundamental mechanical response of material

an

against external loads. The calculated Poisson’s ratio exhibits an increasing trend with the Si concentration except for a distinct pit for g-SiC3, as seen in Fig. 8(b). Large values of Poisson’ ratio show that the contraction increases in transverse to the direction of

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stretching. C-rings are stable and invariant due to the strong C-C bonds and rigid bond angles as discussed above. Thus, we can deduce that the decrease of C-rings in the hexagonal network is responsible for the increase in Poisson’s ratio. As seen in Fig. 1, C-rings barely survived in g-SiC7 and g-SiC3. Particularly, in g-SiC3, C-rings remain

pte

undeformed, which can reasonably explain that g-SiC3 has the lowest Poisson’s ratio among the g-SiCx sheets.

To further evaluate fracture resistance quantitatively, we calculated the fracture

ce

toughness (Γ) of g-SiCx using Eq. (3). Toughness, measured as the energy needed to crack a material, serves as the key to understanding the capability for a material to fracture. Similar to the ideal strength, toughness generally decreases as the Si

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Page 30 of 42

concentration increases. Exceptionally, the toughness values for g-SiC7 and g-SiC3

30

Page 31 of 42

located in the valleys of the curves in Fig. 8(c) are very small. The singularities for the

cri pt

two cases are due to the abnormal stretching of C-C bonds that embrittles g-SiC7. Moreover, structural transformation decreases the toughness of g-SiC3. In general,

strength and toughness are conflictive [86,87]. Lower-strength corresponds to highertoughness. This is somewhat surprising since toughness follows a semblable variation

that profiles the relationship between strength and the concentration of Si-dopants. This

us

means that the ideal strength and toughness are positively correlated. Studies on

monolayer graphene oxide also showed a similar phenomenon [66]. This may

an

characterize the 2D monolayer with monolithic structure. The decreasing Young’s moduli, increasing Poisson’s ratios and relatively increasing toughness values observed

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as the Si content increases, are always intermediate between those for graphene and silicene. This happens because the strength of bonds forming the network is intermediate between the strongest C-C bond in graphene and the weakest Si-Si bond in silicene, as follow: C-C> Si-C>Si-Si. Thus, silicene (graphene) has the lowest (highest) elastic

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modulus and highest (lowest) Poisson’s ratio. 4. Conclusions

We have systematically investigated the mechanical properties of the hexagonal

ce

monolayers, g-SiCx (x = 7, 5, 3, 2, and 1), where mechanical parameters of the g-SiCx, such as the ideal strength, the critical strain, Young’s modulus, shear modulus, Poisson’s ratio as well as fracture toughness have been thoroughly evaluated, and the failure

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AUTHOR SUBMITTED MANUSCRIPT - NANO-117311.R2

mechanics of g-SiCx elaborated in detail. The effect of Si substitution on the mechanical properties of C-based hexagonal networks was revealed as a function of Si concentration 31

AUTHOR SUBMITTED MANUSCRIPT - NANO-117311.R2

therefrom. We concluded the following:

cri pt

1. Mechanical parameters (obtained ideal strength, critical strain, elastic modulus, Poisson’s ratio, and toughness) are generally dependent on the concentration of the Si dopant except for g-SiC7 and g-SiC3. The decrease in ideal strength, Young’s

modulus and toughness, and the increase in Poisson’s ratio originate from the increasing number of weak Si-C bonds and the decreasing number of strong C-C

us

bonds.

2. For g-SiC7 and g-SiC3, the mechanical properties of the network depend on the

an

structure not the concentration of Si. In g-SiC7, the C-C bond strength is severely weakened due to abnormal stretching, which causes the ideal strength and other

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mechanical parameters to degrade even more severely than those of g-SiC5. 3. Because g-SiC3 features a conjugation structure, pairs of symmetric C-Si-C angles are able to synchronously increase under Z-axial tension and structural transformation occurs, which leads to a low ideal strength value compared to g-SiC2. 4. Poisson’s ratio is closely associated with the C-ring content in the network since C-

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rings are stable and invariant. Unlike other g-SiCx, perfectly hexagonal C-rings survive in g-SiC3, thus facilitating a somewhat small Poisson effect for g-SiC3. 5. The detailed failure mechanism of g-SiCx lies in the comprehensive effects from

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bond length and angle, and their variations induced by Si doping, as well as the spatial distribution of the Si dopant.

6. Strength and toughness are mutually exclusive for most engineering materials. In

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Page 32 of 42

our study on the 2D g-SiCx monolayers, the toughness and the ideal strength are 32

Page 33 of 42

positively correlated, which may characterize the 2D monolayers of monolithic

cri pt

structure. 7. All g-SiCx monolayers analyzed in this work preserve rigorously isotropic Young’s moduli and Poisson’s ratios.

8. The versatile mechanical performances of g-SiCx monolayers have been described.

We hope that our study can provide some suggestion for designing 2D sheets with

us

satisfactory properties for practical engineering applications.

an

Acknowledgements

This work was supported by the Natural Science Foundation of China under Grant

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