Role of 1D edge in the elasticity and fracture of h-BN doped graphene nanoribbons

Recent achievement of BN-graphene alloy material has enabled the potential of bandgap tuning through both sub-10 nm width control and BN concentration variation. However, its mechanics, which is necessary for prediction of stability in functional applications, is not well studied. Here, molecular dynamics simulation is performed to conduct uniaxial tensile test for BN-doped graphene nanoribbons (BN-GNR) with varying widths and BN atom fractions. Efforts are made to study the constitutive relations for the edges and the whole BN-GNR and explore the fracture mechanisms of the hybrid nanoribbons. The substantial softening effect of the edges induced by wrinkling alters the impact of BN concentration on the stiffness in the sub-20 nm regime deviating from the linear behaviour observed in the bulk case. Fracture properties are unexpectedly independent of BN concentrations unlike in the bulk and the failure behaviour is rather decided by the graphene ribbon edge structure. Here the armchair edges serve as the source of crack nucleation at an early stage leading to weakened strength and reduced stretchability, whereas zigzag edges do not promote early crack nucleation and leads to the size dependence of fracture properties.


Introduction
Graphene's exotic Dirac-particle like feature leads to zero-band gap semi-metal behavior, which substantially limits its application towards semiconductor based nanodevices.An important strategy to band gap opening and general physical property tuning is chemical substitution (or doping) of the graphene carbon atoms with a variety of dopants [1][2][3][4].Due to the small lattice mismatch and its insulating nature, substitutional doping using h-BN is a highly promising approach for generating graphene band gap as well as providing extensive electronic property tunability.Experimental advances have enabled synthesis of graphene nanoribbons BN hybrid [5], two dimensional (2D) lateral heterojunction [5][6][7] and h-BN/graphene segregated domains [8][9][10].Recently, BNC ternary alloy of uniformly distributed h-BN cells has just been achieved [11].The hybridized bonds between B, N and C across an extended range can result in non-linear mixing of graphene and h-BN properties, enabling application for a rich variety of electronic, optical and thermal devices [3,4,12,13].
The performance of these nanodevices depends largely on the structural integrity, stability, deformability, and electromechanical coupling of BN-doped graphene.The mechanics of these hybrid BNC lateral junctions or large interweaving domains, nanotubes and BNC compounds has been keenly investigated [14][15][16][17][18][19][20][21] for the estimation of elastic, fracture and electromechanical properties and prediction of the stable operating range for the nano-electronics, optical-electronics and nanocomposites based on hybrid BNC.First-principles and molecular dynamics (MD) simulations have shown the controlling influence of h-BN cell doping concentration on mechanics where the stiffness reduces monotonically with increase in the fraction of h-BN cells and the strength is weakened by the B-C bonds relative to pure graphene [22][23][24].So far, the majority of examination have focused on infinitely large BN doped graphene hybrid structures, while bare edge structures, which often display distinct mechanical responses at small width [3,25], has yet to be examined in-depth.Recent successful fabrication of h-BN alloyed graphene (where hexagonal BN cells or aggregates are doped into the graphene substrate in a randomly distributed fashion) opens a pathway for controlled electronic (or other physical) property tuning of an arbitrary graphene-based structure.Considering that graphene ribbon of small width show sizable electronic band gap opening of 2.0 eV while retaining high carrier mobility and sustaining edge state spin-orbit effects, it is of great applicative benefit [12,13] to explore the mechanical characteristics of uniformly doped h-BN graphene nanoribbons.The fundamental issues of significance are (1) the dependence of the elastic modulus and fracture properties on the edge structure and size of the hybrid nanoribbons together with (2) the mechanistic origin of the aforementioned factors in the mechanical responses.
Here, both first-principles and MD simulations can be employed to examine these issues.The former based on quantum mechanics can yield more accurate results but needs more computational resources and thus is limited to small-scale nanostructures.In order to consider relatively large-scale nanoribbons and improve the efficiency of calculations, in this study we perform MD simulations in exploring the mechanical properties of the 2D h-BN doped graphene nanoribbons.Under uniaxial tension, the elasticity and its size-dependence are examined for the nanoribbon with different widths.The edge influences are revealed through the competing effects of bond contraction stiffening, contraction induced edge wrinkling and the structural relaxation induced softening.Emphasis is placed on understanding the mechanisms behind the negative edge Young's modulus and the edge residual stress impact.Then the fracture behaviors are characterized to evaluate the role of BN atom fraction and edge in the size-dependence of the mechanical behaviors and properties beyond elasticity.

Simulation methodology
In this study, we consider hybrid monolayers of graphene containing unit-cell sized h-BN cells, which has been synthesized recently [11].To model the h-BN doped graphene nanoribbons (BN-GNR), rectangular graphene sheets of width from 5 to 80 nm and constant length of 80 nm (figure 1(a)) are first constructed utilizing a C-C bond length of 0.142 nm.Here two nanoribbon structures are considered with either armchair and zigzag edges, which are the most prominently synthesized edge structures in experiments.The original graphene matrix is first divided into labelled hexagonal carbon rings (6 carbon atoms).Then using a random generator, we successively select individual carbon rings and replaced them with BN hexagonal cells (3 boron and 3 nitrogen atoms) as shown in figures 1(b) and (c)).Successive replacements are conducted until the total atom fraction of BN reaches the pre-set atom fraction values of 10%, 20% and 30%.Here, the atom fraction of h-BN cells in graphene is determined by where N B and N N are the number of B and N atoms and N tot is the total atom number of the doped graphene.The distance between replaced hexagonal carbon rings are not controlled in order to model a random h-BN cell distribution that is achievable by chemical vapor deposition.The simulated uniaxial tensile test is performed based on the classical MD method via the large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [23].The Tersoff empirical potential with parameterization by Kinaci et al is chosen to describe the interactions between the C, B and N atoms [26,27].This parameter set has been proven to be accurate in predicting the mechanical and thermal transport properties of a variety of hybrid BNC systems [14, 22-24, 28, 29].The periodic boundary condition is applied along the side of the nanoribbons perpendicular to the deformation direction while free boundary condition is used in all other directions to model a monolayer with free edges.Here, the local energy minimized BN-GNR structures are first found utilizing the conjugate gradient method.The local energy minimized structures are then thermalized under the Nosé-Hoover (NVT) thermostat at 300k for 10 ps and then under the NPT ensemble with the periodic zig-zag or armchair direction barostated to 0 bar for 15 ps.Deformation in the periodic direction is conducted by displacement loading at a constant strain rate of 10 9 s −1 under the NVT thermostat.This strain rate is selected as the strain rates from 10 8 s −1 to 10 10 s −1 were shown to yield reasonably accurate results in simulating the mechanical responses of nanomaterials [30].The uniaxial tension test is simulated by applying a variable displacement to the two ends of nanoribbon along the deformed direction such that a constant engineering strain of 0.001 is experienced after each loading.Here zero stress condition is maintained in the direction perpendicular to the applied displacement load.A time step of 0.1 fs is utilized, and the simulation is performed until global fracture of the BN-GNR hybrid structure occurs.

Results and discussions
Based on the uniaxial tensile deformation conducted via the MD process presented in section two, the mechanical behavior and properties of the hybrid nanoribbons are discussed in the context of characteristic size variation, edge properties and edge structural influence.

Size-dependence of young's modulus
Under tensile deformation, the linear elastic response of the BN-GNR is characterized by Young's modulus (Y) and its variation with characteristic width.Here, the BN doped armchair edged graphene nanoribbon (BN-AGNR) is subjected to a strain applied along the armchair direction.The BN-AGNR samples examined are from 5 to 80 nm in width, where the stress-strain relation is achieved in figure 2 via polynomial regression of the MD simulation data and the Young's modulus Y is obtained by calculating the slope of the stress-strain curve in the strain regime of 5%.Here, the results obtained for bulk graphene are also shown in figure 2 where Young's modulus Y of 0.9 TPa and fracture strength 129 GPa (associated with the effective thickness 0.34 nm) are achieved in an excellent agreement with the data reported in the literature [31,32].
The trend of Y variation with the width of the doped-graphene nanoribbon is shown in figure 3(a) for the BN-AGNR of 20% BN doping or atom fraction.As variation in the BN distribution may lead to fluctuation in Y, the simulation is repeated for three groups of BN-AGNR at the same BN atom fraction of 20%.It is observed from figure 3(a) that Y monotonously increases as the nanoribbon characteristic width enlarges.A clear inverse relation exists where Y initially rises from ∼691 to 793 GPa by ∼15.6% when the width grows from 5 to 20 nm and then asymptotically approaches the bulk limiting Y o value of 821 GPa for width greater than 20 nm.This bulk Y o value is the Young's modulus in the armchair direction for an infinitely large BN-AGNR at the same BN atom fraction.
Therefore, the overall size-dependence of Young's modulus can be effectively captured through equation (1) for the BN-AGNR.Here, W is the sample width, C is found to be 580 GPa • m and n, the degree of width dependence, is one showing the strong inverse proportionality.Based on the core-shell paradigm [33,34] for describing size (or surface) effect of 1D material properties, C here is the product of the effective edge width h s and the difference between the Young's modulus (Y o ) of the bulk material and that of edge modulus (Y s ).
Having shown a strong, well-behaving Y size-dependence at 20% BN atom fraction, subsequent simulations were performed for BN atom fraction of 10% and 30% to evaluate the stability of the observed size effect.Using a similar computational procedure, it is evident from figure 3(b) that the distinctive C W relationship is preserved amongst all BN atom fractions.
In addition, the influence of the edges on the BN fraction-dependence of Young's modulus is also observed.For the BN-AGNR with width smaller than 20 nm, Y varies non-linearly with the BN atom fraction as seen through the larger Y variation between 20% to 30%.For those with width greater than 20 nm, the BN atom fraction-dependence regains approximate linearity, which is consistent with the results obtained for infinitely large nanofilms [22,24].This non-linearity at small width reflects significant edge influence on BN-AGNR elasticity while beyond the critical width of 20 nm interior atom elastic property becomes the controlling factor on the overall Y value, which in turn yields a linear BN atom fraction relation in agreement with the bulk BN-GNR case.These results clearly show that no matter which faction of h-BN is considered the presence of the edges will significantly soften the BN-AGNR.Specifically, the softening effect enlarges at smaller width leading to the size-dependence of Young's modulus.
For BN doped zigzag-edged graphene nanoribbon (BN-ZGNR) the elasticity size effect generally exhibits a similar inverse dependence on width (figure 3(a)).Y initially increases from 755 to 836 GPa when the width rises from 5 to 20 nm.It finally reaches its maximum value of ∼830 GPa (near bulk value) at the width greater than 20 nm.Here it is found that n is 1.5 and C is 750 GPa • m 1.5 in equation (1).Although C is apparently higher for the zig-zag structure, its width sensitivity turns out to be much lower than the armchair-edged nanoribbon due to substantial increase of n.Here, it should be pointed out that for the zig-zag structure with n = 1.5 equation (2) does not hold true as the simple theory of composite mechanics based on which equation (2) is derived is not valid in this case.Here, it is worth mentioning that the monotonical change of Young's modulus with rising width obtained for the h-BN doped-graphene nanoribbon was also reported for other nanoribbons in previous studies [35,36].

Mechanisms underlying edge softening effects
It was shown previously [22,24] that in general, h-BN doping softens infinitely large BN-GNR in accordance with the classical Voigt laminate composite rule.Thus, the elastic modulus of BN-GNR depends on the atom fraction of the h-BN cells.In section 3.1 we further revealed the edge effect on the elasticity of the finite width BN-GNR.However, the underlying mechanism of the edge effect have yet to be ascertained.It is well known that crystal facets in nanostructures can be subjected to significant pre-compressive or tensile stress due to the tendency for surface structural relaxation or reconstruction that lowers the energetic costs incurred by production of dangling bonds [37,38].For 1D nanowires, a high compressive residual stress has been directly associated with surface stiffening in the outmost atom chain, which leads to the size-dependence of elasticity [39].
To shed some light on this issue for 2D BN-GNR, the atomic stress in the armchair direction is computed at 0 K for an equilibrated BN-AGNR of 10 nm width as shown in figure 4(a) (upper figure).It is evident that, at 0 K the chain of edge atoms in finite width BN-AGNR sustains visibly greater contraction of the bonds in the armchair direction (blue) and the adjacent interior atom chain experiences bond extension (red).Due to the strong contraction of the edge bonds, the outermost horizontal bonds have achieved a bond length of 1.26 Å close to that of a carbon triple bond.As is well known, this may have a stiffening effect in C-C bonds [25,37,40], which however is in contradiction with the observation at 300 K that the edge atoms have a softening effect on BN-AGNR.Moreover, the tendency for the edge chains to contract (figure 1(a) at 0 K) is restricted by the nearest interior chain which tends to stretch out, causing residual tensile stress on the edge chain and vice versa, a compressive stress on the nearest interior chains.When the temperature is raised to 300 K, periodic edge wrinkles of significant amplitude are observed in the armchair direction in figure 4(a) (lower figure) for the zerostrain structure primarily due to (1) the compressive force on the edge of the BN-AGNR exerted by the contracted outmost atom chain and (2) the thermal perturbation on a one-atom thick BN-AGNR layer at 300 K. Due to the localization of compression to a single atom chain, wrinkling is mostly restricted to a shallow region spanning about two atom chains in the zigzag direction, i.e., the outmost chain and the nearest interior one.Such edge wrinkling induced by the edge atom bond contraction was also observed previously for pure graphene sheets [41].
To exam the effect of the wrinkled edge, a chain-wise analysis at 300 K is carried out to calculate the relation between the strain energy E per atom and uni-axial strain ε for the outmost (edge) atom chain, three adjacent interior atom chains and an atom chain at the nanoribbon center (figure 4(a)).As shown in figure 5(a), the two wrinkled atom chains show unique E-e curves (blue and red curves) differing substantially from those of the next two adjacent interior chains and the chain in the nanoribbon center, which almost coincide with one another.In figure 5(a), a concave curve is obtained at ε <8% for the wrinkled outmost atom chain (blue) associated with e ¶ ¶ E 2 2 < 0 or a negative elastic modulus.In this atom chain, deformation during tensile test occurs primarily through the reduction of the wrinkle amplitude.At higher strains (e > 8%) the curve regains convex curvature corresponding to e ¶ ¶ E 2 2 > 0 or positive modulus since the structure is now stretched out with largely reduced wrinkle amplitude.As a result, the deformation of the outmost atom chain is now mainly in-plane stretching via changes in the bond lengths and angles.For the wrinkled interior chain nearest to edge chain (i.e., interior atom chain 1), a convex E-e curve (red) is achieved associated with a positive modulus or e ¶ ¶ E 2 2 > 0, which is close to the typical convex curves of other interior atom chains without wrinkling.These results indicate that the softening effect of the edge results primarily from the wrinkling caused negative equivalent modulus on the outmost atom chains.Moreover, it is noted in figure 5(a) that, at e < 8% the E-e curves of the two wrinkled atom chains are nearly symmetric to the axis E = 0, highlighting the opposite responses to the uniaxial load.As mentioned before, at 0 K bond contraction is observed for the outmost atom chain (figure 4(a)), which exerts compression on the adjacent interior atom chain.This compression does not lead to wrinkling at 0 K as the structure without any thermal perturbation is stable.When the temperature rises to 300 K wrinkling occurs for outmost atom chain and the adjacent interior chain because of the thermal perturbation.The softening effect of wrinkling eventually overpowers the possible stiffening effect of the bond contraction leading to the softer edges and the more compliant BN-AGNR.In section 3.1, the softening effect is observed in the linear elastic region with strain less than 5% while the results in this section further confirm that the edge softening remains substantial also in nonlinear elastic region with strain up to 12% or even larger (figure 5(a)).
On BN-ZGNR a similar behavior is observed but the overall Young's modulus is around 20% higher than that of BN-AGNR (figure 3(a)).Contraction of atom bonds is also observed at 0 K along the zig-zag direction on BN-ZGNR edge atoms (lower figure in figure 4(b)), which, at 300 K causes edge wrinkles with longer wavelength and lower amplitude than those of the wrinkles found on armchair edges (figure 4(b)).Therefore, wrinkle softening effect is reduced significantly.Indeed, as shown in figure 5(b), a less prominent edge effect is observable from the concave curve or negative modulus of the outmost atom layer at small strain 1.5% (blue).At larger deformation, strain energy continues to be stored in the zig-zag C-C bonds causing inflection to the convex curvature or positive modulus similar to the interior chains 1-3 and the center chain (figure 5(b)).Here it should be pointed out that for BN-ZGNR only the outmost atom chain exhibits a distinctive mechanical response while all the interior atom chains have nearly the same mechanical behavior (figure 5(b)).

Edge effects on fracture properties and mechanisms
Softening effect of the edges on elasticity is revealed in prior sections.In this section we shall further investigate the edge effect on the fracture properties and fracture mechanisms of the BN-GNR.To this end, the dependence of the fracture stress and strain is calculated in figure 6 for the BN-AGNR with width of 5 to 60 nm and at 20% BN atom fraction.It is noted that in the presence of the edges the BN-AGNR experiences a decrease of fracture stress from the bulk values of 107 GPa to around 97 GPa by 10% and the reduction of fracture strain from the bulk value 19.3% to around 17% by 12%.However, different from the softening effect of the edge on Young's modulus (figure 3), the weakened fracture strength and reduced stretchability do not show strong dependence on width (figures 6(a) and (b)).In contrast to the BN-AGNR, the BN-ZGNR exhibits clear sharp drop of fracture stress (figure 6(c)) and strain (figure 6(d)) with increasing width.The highest fracture properties of 122 GPa and 24.7% are found at 5 nm, which are even higher than the bulk values.They then fall towards to the bulk values of 109 GPa and 19.5% at the width of 20 to 40 nm and dip slightly below the bulk values at the width of 50 to 60 nm.The distinctive size effect trends for fracture stress and strain of BN-AGNR and BN-ZGNR is also preserved for BN atom fraction of 10% and 30% in figure 7 while no clear repeatable dependence on BN fraction is detectable.The apparent size-dependence of the BN-ZGNR fracture properties is unanticipated, which, combined with the observation of apparent size-insensitivity for the BN-AGNR fracture properties points to the dominance of edge properties on overall failure behavior and justifies the further examination of the underlying mechanisms in terms of the edge influences.
To exam these issues, we shall first compare the fracture mechanisms of the BN-AGNR with finite width against the mechanisms observed for infinite BN-AGNR (without bare edges).It is noted that simulation results in literature show that infinite BN-AGNR exhibits elastic deformation without bond rapture or crack nucleation until the peak of its stress-strain curve where fracture stress of 107 GPa and fracture strain of 19.5% are obtained (figures 6(a)-(b)).With such high fracture stress, the critical length of cracking is small.Once a crack nucleates it will lead to immediate final fracture where the crack propagates across the entire structure at extremely high speed.In the presence of edges, the BN-AGNR with finite width (e.g., 10 nm) shows substantially different behavior where the first crack nucleates on the edges at much lower stress of 51.8 GPa and strain of 7.4% via the early rapture of the C-C bonds and formation of the decagonal rings on the nanoribbon edge (the inset in figure 8(a)).With stress/strain increase, more decagon rings form along the nanoribbon edges to release the stored strain energy in the structure.When stress and strain approaches 96.1 GPa and 17.5%, one of the cracks starts propagating from the edge towards the interior of the BN-AGNR, typically along a zigzag direction and in a general course perpendicular to the uniaxial stress/strain (figure 8(b)).The length of the crack initially increases at relatively low speed as the external load is raised.When the crack length reaches its critical value the crack propagates at extremely high speed and the final fracture occurs at a fracture stress of 96.6 GPa and fracture strain of 17.8% much lower than the fracture properties (107 GPa, 19.5%) of the infinite BN-AGNR.The above comparison of the fracture mechanisms between bulk BN-AGNR and those with finite width clearly evidences that the low fracture stress and strain of BN-AGNR with finite width are a result of (1) the crack nucleation on edges at an early stage and (2) the directed crack propagation towards the interior of the BN-AGNR.This fracture mechanism observed for the 2D films is found to be consistent with the understanding that fracture at the nanoscale is usually attributed to edges or surfaces as they are effective defect nucleation sources and initiates or affects the final failure behavior in nanostructures with ultra-low defect density [42].Specifically, the decagon rings (i.e., the cracks) on the edges of the BN-AGNR are formed via the rapture of the C-C bonds that connect the outmost atom chain and the adjacent interior atom chain (the inset of figure 8(a)).
As mentioned in section 3.2, at 0 K there is residual tensile strain between the two outmost atom chains due to their different deformation tendency (figure 4(a)) and the compatibility of their deformations.As demonstrated before, at 300 K, the two atom chains are found to exhibit distinct responses to the uniaxial strain (figure 5(a)).Specifically, the total stresses of the two atom chains (i.e., the applied stress plus the internal stress between the two atom chains) represented by the slopes of the E -e curves, i.e., e ¶ ¶ , E have opposite signs.This observation implies that similar to what happens at 0 K, there could be a high internal tensile strain on the C-C bonds connecting the two outmost atom chains, which finally results in the early rupture of these C-C bonds on the edges.This fracture mechanism is similar to the mechanism of 1D nanostructures with high surface area-tovolume ratio, where inelastic deformation is determined primarily by the number of the sources of surface dislocation nucleation and the associated activation energy.These factors in turn determines the eventual fracture properties of the nanostructures [43].In this view, analogously, the independence of 2D BN-AGNR fracture properties on size may be attributed to the fact that the number of crack nucleation sites and the associated nucleation energy on the two edges are not sensitive to the variation of the width.Next let us take a look at the BN-ZGNR.As observed from figure 9(a), cracking and global fracture is initiated from the interior region instead of starting on the edge structures.Specifically, cracking nucleation occurs by the rupture of the weakest C-B bonds (figure 9(a)) and immediately leads to the fracture of the BN-ZGNR perpendicular to the edges (figure 9(b)).Such fracture mechanisms observed are found to be similar to those of the infinitely large BN-GNR but substantially different from the mechanism of the BN-AGNR.From figure 5(b), structurally, the BN-ZGNR response to the uni-axial strain of the outmost two atom chains (the blue and red curves show positive curvature for strain energy) are close to each other in contrast to the opposing trends found on the BN-AGNR edge chains (figure 5(a)).In other words, the outmost two atom chains of the BN-ZGNR show similar mechanical responses to the external strain.This suggests that the high internal stress achieved between the two outmost atom chains of the BN-AGNR does not exist in the BN-ZGNR.As a result, Another observation of significance is that for the BN-ZGNR, the fracture properties (figures 7(c)-(d)) and elastic modulus (figure 3) show the opposite trends of change with rising width, i.e., the fracture stress of the BN-ZGNR increases with the decreasing width whereas its Young's modulus decreases in the same process.Thus, when the width decrease the rising fracture stress should be as a result of the rising fracture strain.In other words, a smaller BN-ZGNR width in general leads to greater stretchability or larger fracture strain which in turn results in higher fracture stress.It can be seen from the fracture mechanism of the BN-ZGNR that its fracture strain is ultimately controlled by the deformability of C-B bonds.If we assume that the fracture strain of the C-B bonds remains a constant, then the width-dependence of the fracture strain and stress obtained for the BN-ZGNR can be explained based on the understanding that slimmer BN-ZGNR will lead to enhanced compliance.This feature can be characterized by the increased ratio a between overall strain of the nanoribbon and the local strain of the C-B bond, i.e., with smaller width or greater ratio a it requires higher fracture strain and accordingly higher fracture stress to generate sufficient strain to break the C-B bonds and trigger the fracture of the BN-ZGNR.This reveals the rationale behind the similar size-dependence of the fracture stress and strain of the BN-ZGNR.

Conclusions
The elastic and fracture behavior of 2D BN-GNR is investigated utilizing molecular dynamics simulation for the most prevalent low energy armchair and zig-zag graphene ribbon edge structures.The presence of edges is shown to alter BN atom fraction effect on stiffness versus bulk behavior and is the determining factor of the fracture properties in the suspended hybrid nanoribbons.
For armchair BN-GNR, the strong internal strain of the edges leads to the contraction of the outmost atom chain which in turn exerts compression on the edges and results in the edge wrinkling spanning two atom chains in the zigzag direction.The wrinkled outmost atom chain shows negative equivalent modulus and exerts substantial softening effect on the nanoribbon.In particular, in the linear elastic region, the width (w) −dependence of the Young's modulus Y is achieved for the BN-GNR with different BN atom fractions, where the Y-reduction relative to its bulk value obeys the power law - w 1 (w denotes the width of the nanoribbons) The cracks of the armchair BN-GNR nucleate on the edges at an early stage via the rupture of the C-C bonds connecting the outmost atom chain and the adjacent interior chain.The early crack nucleation followed by the crack propagation eventually weakens the strength and reduces the stretchability of the BN-GNR.Since the edges serve as the nucleation defect sources that control the fracture properties, the edge-induced decrease of the fracture properties is determined primarily by the number of the defects/crack sites on the edges but do not show detectable size-dependence.Here the early crack nucleation on the edges may be attributed to the large local strain on the C-C bonds connecting the outmost atom chain and the adjacent one due to the deformation compatibility between them.
The mechanics of BN-GNR show significant sensitivity to edge structure (i.e., the armchair or zigzag structures) or the crystalline direction.Thus, for zigzag BN-GNR the contraction of the outmost atom chain becomes less pronounced, and the edge wrinkles show longer wavelength and smaller amplitude than those of armchair edges.As a result, the softening effect of the zigzag edge is weaker and accordingly, the decrease of the Young's modulus relative to the bulk value satisfies a power law of w −1.5 .
Moreover, the zigzag edges do not promote the crack nucleation on the edge at early stage.Instead, the crack nucleates in the interior of the nanoribbon via the breaking of the C-B bond.Specifically, decreasing the width will lead to slender and thus more compliant nanoribbons where higher fracture strain and fracture stress are required to generate sufficient local strain on the C-B bond to finally break the bonds.The size-dependence found for the fracture stress and strain of zigzag BN-GNR therefore is opposite to the size dependence of its Young's modulus.
The results summarized above are of importance for the study of novel two-dimensional hybrid nanomaterials and the design of the nanodevices and nano electromechanical systems based on the h-BN doped graphene.

Figure 1 .
Figure 1.Construction of h-BN doped graphene model.h-BN doping occurs through direct replacement of the hexagonal rings of graphene at randomly selected location where spacing between the h-BN cells are not controlled.

Figure 2 .
Figure 2. Stress-Strain relationship under uni-axial tension in armchair direction for BN-AGNR of width 5 to 80 nm and BN atom fraction fixed at 20%.The Bulk curve indicates the stress-strain relation obtained for the bulk BN-AGNR and GPE curve represents the stress-strain relation of bulk pure graphene sheet.

Figure 4 .
Figure 4. Atomic stress distribution for energy minimized structure at 0 K (upper figure) and edge wrinkled morphology for the 300 K relaxed structure (lower figure) for (a) BN-AGNR and (b) BN-ZGNR.Here all the hybrid nanoribbons have width of 10 nm and BN atom fraction of 20%.The color bar used in (a) and (b) indicates the magnitude and directions of the atomic stress found in the 10 nm nanoribbons where the top (light yellow) marks max tension and the bottom (dark red) marks max compression.The insets show the detailed structures of the edge atom chains, interior chain 1 (the interior atom chain adjacent to the edge chain), interior chain 2 (the interior atom chain next to interior chain 1) and interior chain 3 (the atom chain next to interior chain 2).

Figure 5 .
Figure 5. Atom number normalized strain energy to strain relation for the edge atom chain, nearest atomic chains 1 to 3 adjacent to edge and the central atom chain for deformations along (a) armchair direction of BN-AGNR and (b) zig-zag direction of BN-ZGNR.Here the samples have a width of 10 nm and BN fraction of 20%.The insets in (a) and (b) show a zoomed-in view of the overlapping curves at selected strain states for clarity and the corresponding structures of atom chains examined are displayed previously in figure 4.

Figure 6 .
Figure 6.(a) Fracture stress variation and (b) fracture strain variation with the width for BN-AGNR at 20% BN fraction.(c) Fracture stress variation and (d) fracture strain variation with the width for BN-ZGNR at 20% BN fraction.The results shown are averages of three sets of BN-AGNR and BN-ZGNR with the same BN atom fraction but different BN cell distribution.

Figure 7 .
Figure 7. (a) Fracture stress variation and (b) fracture strain variation with the width for BN-AGNR.(c) Fracture stress variation and (d) fracture strain variation with the width for BN-ZGNR.Here 10%, 20% and 30% BN fraction are respectively colored in red, black and green.At each BN fraction, the average fracture property values for three sets of BN-AGNR or BN-ZGNR with different BN cell distribution is computed.

Figure 8 .
Figure 8.(a) Crack nucleation from armchair edge due to rupture of C-C bond and (b) crack propagation along zigzag direction obtained for armchair edges of BN-AGNR with the width 10 nm and BN atom fraction 20%.(Here the C atoms is colored in orange and the B and N atoms are respectively colored in blue and yellow).

Figure 9 .
Figure 9. Structure of width of 10 nm for boron nitride doped graphene with zig-zag edge (BN-ZGNR) (a) crack nucleation from interior upon rupture of C-B bond and (b) crack propagation.BN atom fraction is at 20%.(Here the C atoms is colored in orange and the B and N atoms are respectively colored in blue and yellow).