Engineering the flexibility of graphene/h-BN lateral heterojunctions

Employing the first-principles calculations, the mechanical properties of graphene/hexagonal boron nitride (h-BN) lateral heterojunctions were studied, including the Young’s modulus and bending modulus. It was found that by varying the ratio of graphene in the graphene/h-BN heterojunction, both the Young’s modulus and bending modulus of can be effectively engineered. Particularly, the bending modulus increases with the ratio of graphene to h-BN, providing a way to tailor the flexibility of two-dimensional materials.


Introduction
Heterojunction is effective in tailoring the properties and expanding the applications of materials, especially in two-dimensional (2D) materials [1,2].Benefited from the similar structure and low lattice mismatch, graphene and hexagonal boron nitride (h-BN) are commonly hybridized to form heterojunctions with great practicality and applications [3,4].Generally, there are three main methods to fabricate graphene/h-BN (G/h-BN) heterojunctions, i.e. layer-by-layer transferring individual layers [5], co-segregation from precursors [6], and chemical vapor deposition (CVD) [7,8].For G/h-BN heterojunctions, mechanical properties are attractive since both the graphene and h-BN present excellent mechanical strength.The Young's modulus (E) of monolayered graphene is about 1.0 TPa [9][10][11], making graphene one of the strongest materials among the known materials.Meanwhile, the E of monolayered h-BN is about 0.87 TPa [12,13], suggesting that BN is promosing in superhard materials.However, connecting different chiral graphene and h-BN may result in grain boundaries (GBs) composed of pentagons and heptagons at the junctions.Usually, GBs can affect the mechanical properties of heterojunctions.For example, Ding et al. [1 4] studied the mechanical properties of G/h-BN heterojunctions, focusing on the interface details, and suggested that the Young's moduli of G/h-BN heterojunctions decreased as the inflection angles increase.Both the connection type and defects at the interfaces will contribute to the mechanical properties of G/h-BN heterojunctions.Ge and Si [13] revealed the influence of grain boundaries on the mechanical properties of 2D lateral G/h-BN heterostructures through simulations.It was found that similar to the GBs in graphene heterojunctions, the mechanical strengths of G/h-BN heterojunctions also rely on the disorientation angle.Without misorientation, the G/h-BN heterostructure show a comparable intrinsic strength to that of pristine graphene.With misorientation, the intrinsic strengths of G/h-BN heterostructures dominated by armchair (AC)-type interfaces show different dependences on the disorientation angle from the structures dominated by zigzag (AC)-type interfaces.Particularly, compared with structures with non-Clar's connections, the heterojunctions with the Clar's connections present larger mechanical strengths under tensile strain, showing better mechanical stability.In addition, by comparing stress contour of GBs with different mismatch angles, Li et al. [15] found that the arrangement of pentagon-heptagon deviations along the GBs is critical to the strength of G/h-BN heterojunctions.They found an extraordinary increase of intrinsic strength as the mismatch angle increases, which is originated from the decreased concentration of stress at the interfaces as the differentiation density enlarges.Compared with lots of studies on the mechanical strengths of G/h-BN heterojunctions as introduced above, few is known about the flexibility of G/h-BN heterojunctions.For the multilayered structures, it was found that despite that the graphene layers have larger elastic moduli than the h-BN samples with similar thickness, the bending stiffness is ordered inversely where the bending stiffness of h-BN samples is larger than that of graphene due to the strong shear interactions between the layers in h-BN samples [16,17].However, the case is different for the monolayered h-BN and graphene.The bending modulus of graphene monolayer is about 1.5 eV [18,19], but 0.86~1.4eV for h-BN [16,20].So how about the bending modulus of monolayered G/h-BN heterojunction when connecting the graphene and h-BN?Here, we connect the graphene and h-BN by GBs to form a lateral G/h-BN heterojunction.Density functional theory (DFT) calculations suggested that both the Young's modulus and bending modulus of the G/h-BN heterojunctions can be tailored by changing the ratio of graphene to h-BN (G:h-BN).

Computational methods
The h-BN and graphene are matched by GBs composed of pentagons and heptagons based on the Clar's rule, as shown in Figure 1, forming hybrid heterojunctions.To simulate the effect of G:h-BN ratio, the width of h-BN and graphene is changed in a supercell with 296 atoms, where the G:h-BN ratio is defined by the ratio of h-BN atoms to graphene carbon atoms.The simulations were carried out by using Vienna ab initio simulation package (VASP) [21].Perdew-Burke-Ernzerhof functional was used to describe the exchange-correlation interaction within the general gradient approximation [22].The projector augmented wave method [23,24] was adopted to deal with the ion-electron interaction.The energy cutoff of 600 eV, vacuum thickness of 15 Å, and Monkhorst-Pack k-mesh grid [25] of 0.02 Å -1 were adopted.Convergence criteria of 0.01 eV Å -1 for force and 10 -5 eV for energy were used in simulations.

Young's modulus
The Young's modulus E is calculated from the linear part of strain-stress curves.Starting from the equilibrium structures, the heterojunctions with different G:h-BN ratios were uniformly stretched at a strain step of 0.5% in the linear elastic region.Then, it was found that the E of the graphene in zigzag direction is 325 N/m, while the E of h-BN in armchair direction is 270 N/m.The calculated E of graphene and h-BN agree well with previous reports [17], and graphene monolayer shows higher E than h-BN.When connecting graphene and h-BN to form lateral heterojunctions, the E of G/h-BN heterojunction is in between the values of graphene and h-BN and closely depends on the G:h-BN ratio, as shown in Figure 2. It can be noticed that the E of G/h-BN heterojunction increases with the G:h-BN ratio since the graphene exhibits larger E than that of h-BN, which finally reaches the limit of Young's modulus of graphene.

Figure 2. Relationship between the Young's modulus E and G:h-BN ratio of G/h-BN heterojunctions.
Generally, the E depends closely on the interactions between the atoms, which can be characterized by the bond length.It can be noticed in Figure 3, as the G:h-BN ratio increases, the average bond length of G/h-BN heterojunction decreases inversely, leading to an enhancement of atom interactions.Therefore, the E of G/h-BN heterojunction increases as G:h-BN ratio enlarges.

Bending modulus
Bending modulus is a parameter to determine the flexibility of a material, which can be calculated by analysing the relationship between the bending energy and bending curvature.To study the bending modulus, the G/h-BN heterostructures were curled the into nanotubes to simulate the bending behaviors [26][27][28].From our calculations, the pristine graphene has a bending modulus of 1.501 eV in AC direction, and pristine h-BN exhibits a bending modulus of 1.16 eV in ZZ direction, agreeing well with previous reports [18,20,27].Similar to the Young's modulus, when connecting the h-BN and graphene to form a G/h-BN heterojunction, the bending modulus of G/h-BN heterojunction could be also engineered by G:h-BN ratio.As presented in Figure 4, it can be noticed that the bending modulus of G/h-BN heterojunction varies in between the values of graphene and h-BN, and increases monotonously with the increase of G:h-BN ratio.As the G:h-BN ratio increases, the bending modulus of G/h-BN heterojunction will finally reach the limit of graphene.Since higher bending modulus indicates lower flexibility, to enhance the flexibility of G/h-BN heterojunction, one can increase the ratio of h-BN.For the 2D structures, the planar atomic density is a key factor contributing the bending modulus of a 2D material [27].The larger the atomic density is, the higher the bending modulus will be.Thus, we plotted the relationship between the atomic density and G:h-BN ratio of G/h-BN heterojunctions in Figure 5 to understand the change of bending modulus.It can be noticed from Figure 5, the atomic density of G/h-BN heterojunction increases with the G:h-BN ratio, which can directly account for the engineering of bending modulus of G/h-BN heterojunction by G:h-BN ratio.

Conclusion
Graphene/h-BN lateral heterojunctions are generated by connecting h-BN and graphene with grain boundaries.DFT calculations suggest that by varying the G:h-BN ratio, both the Young's modulus and bending modulus of G/h-BN heterojunction increase with the G:h-BN ratio.Change of bond length and atomic density account for the engineering of Young's modulus and bending modulus by the G:h-BN ratio.Generally, the current study provides a feasible approach to engineer the flexibilty of 2D materials, which could be promising in flexible materials.

Figure 1 .
Figure 1.G/h-BN heterojunction formed by connecting h-BN and graphene through GBs with a G:h-BN ratio of 216/80.

Figure 3 .
Figure 3. Relationship between the bond length and G:h-BN ratio of G/h-BN heterojunctions.

Figure 4 .
Figure 4. Relationship between the bending modulus and G:h-BN ratio of G/h-BN heterojunctions.

Figure 5 .
Figure 5. Relationship between the atomic density and G:h-BN ratio of G/h-BN heterojunctions.