Two-Dimensional Carbon Networks with a Negative Poisson’s Ratio

Abstract

Low-dimensional materials with a negative Poisson’s ratio (NPR) have attracted lots of attention for their potential applications in aerospace, defense, etc. Although graphene and monolayer h-BN have been reported to have NPR behavior under external strains, the mechanism is not clear, and the critical strains of the occurrence of a NPR are relatively larger. Here, we propose that the origination of the NPR phenomena in the 2D honeycomb structures can be explained by the variation of the zigzag chains under strains. Our calculations clarify that a NPR occurs along the armchair-chain direction rather than the zigzag-chain direction in these materials. Furthermore, a series of two-dimensional carbon networks including zigzag chains have demonstrated that there is NPR phenomena in them. In some of the networks, a NPR can be found under a small external strain. Our study not only deepens the understanding of the origin of NPR in honeycomb systems but also offers guidance to design auxetic nanostructures.

1. Introduction

The Poisson’s ratio, i.e., the ratio between the lateral and longitudinal strain for the stretched material, is an important parameter to describe the physical and mechanical properties of materials. In contrast to conventional materials, the materials with a negative Poisson’s ratio (NPR) (also called auxetic materials) [1] possess some exotic physical properties [2,3,4], such as high specific strength [5,6], high energy absorption [7], enhanced shear resistance [8,9], indentation resistance [10,11], and fracture toughness [12]. These remarkable properties motivate researchers to search for various NPR materials [13,14,15,16,17,18,19,20,21,22,23]. So far, the NPR phenomenon has been discovered in many three-dimensional (3D) structures, such as some metallic cubic structures [24,25] and some structures constructed by beams [26,27] or rigid units [28]. In addition, some artificial 3D auxetic materials have also been reported [29,30].

In recent years, the NPR phenomenon has been found in some two-dimensional (2D) materials, including buckled structures [30,31,32,33,34,35,36,37], such as 2D silica [31], penta-B₂N₄ [32], δ-SnS [33], penta-SiCN [34], penta-Be₅C₂ [35], penta-graphene [36], SiS [37], δ-Phosp horene [38] and some planar honeycomb materials [39,40,41,42,43,44,45], such as graphene [39,40,41,42], h-BN [43], and h-GaN [44]. Although a lot of calculations have been carried out for these 2D planar materials, the mechanism of the NPR in them is still unclear. Another issue is that, in some materials, e.g., graphene, in order to generate the NPR phenomenon, a larger critical strain is required. According to the calculations of molecular dynamics (MD) [39] and density functional theory (DFT) [40,41], 6% and 18% uniaxial strain along the armchair-chain direction in graphene are needed, respectively [46,47]. A larger strain is not easy to be realized in application. Therefore, it is natural to ask, can we find a physical mechanism for the NPR phenomenon in the 2D materials; furthermore, can we find some 2D carbon networks possessing lower critical strains than graphene?

In this paper, we find that the NPR phenomenon in 2D carbon materials [48] could originate from the zigzag chains. We first study the mechanical response of a zigzag carbon chain, and the NPR phenomenon is found. Then, we construct some 2D planar carbon structures including zigzag chains. The NPR phenomena have been observed in them when strains along the vertical direction of the zigzag chains are applied. Moreover, several 2D carbon structures have smaller critical strains than graphene. As a comparison, some 2D carbon structures with no zigzag chains have also been constructed, and no NPR is found in them. Our study deepens the understanding of the origins of a NPR in 2D materials, which would shed light on the advanced design of nanoscale electromechanical devices.

2. Materials and Methods

The calculations are mainly performed based on the classical MD method [49,50] to investigate the strain response of different materials, and the Lammps program with the Airebo potential function [51] is used for the simulation. During the MD simulation, the Airebo potential function is selected to describe the C-C interaction, which is widely used to study the mechanical properties and strain effects of carbon materials. In the calculations of the mechanical response, the relaxation of 50,000 steps is performed for the structure with temperature 10 K. The time step is selected as 1 fs, which is small enough to maintain energy conservation during the MD simulation. To verify that the Airebo potential function is appropriate, the Lammps package, together with the Airebo potential function, is adopted to stretch the y-axis (armchair) of graphene, and the obtained results are consistent with the previous study [41].

To verify the stability of the structures, the first-principle theory with the projector augmented wave (PAW) method [52] implemented in the Vienna ab initio Simulation Package (VASP) [53] is selected for the calculation. The electron exchange-correlation is described with the generalized gradient approximation (GGA) [54] of Perdew-Burke -Ernzerhof (PBE) [55], and the Monkhorst-Pack k-mesh [56] of 15 × 15 × 1 is employed to sample the irreducible Brillouin zone. The kinetic energy cutoff for the plane wave functions is set to 450 eV and the energy convergence threshold is chosen as 10⁻⁸ eV to ensure calculational accuracy. All of the atomic positions and lattice structures are completely relaxed, and the maximum force threshold is 0.02 eV/Å. In the calculation of the phonon spectrum, the structure is expanded by 4 × 4 × 1. The structure is processed by the phonopy program [57] and the phonon spectrum is calculated with the finite displacement method. The 20 Å vacuum layer is chosen to eliminate the interaction between different layers.

To explore the properties of NPR, the ${\epsilon }_x$ and ${\epsilon }_y$ are introduced to indicate the relative strain along the x- and y-axis, respectively, and the ${\epsilon }_i$(i$\in \left(x, y\right)$) is defined as ${\epsilon }_i=\left(L_i-L_{i_0}\right)/L_{i_0}$, where $L_{i_0}$ and $L_i$ are the initial and deformation lengths of the unit cell along the i-axis (i$\in \left(x, y\right)$). Then, the Poisson’s ratio can be defined as $v=-\partial {\epsilon }_x/\partial {\epsilon }_y$ [58]. One thing we would like to emphasize is that in some previous works, the Poisson’s ratio is defined as $v=-{\epsilon }_x/{\epsilon }_y$ [2,4,41,59]. With the latter definition, until the failure of the structure, no NPR occurs in graphene [2], which is not adopted in this work.

3. Results

3.1. Mechanical Properties of a Zigzag Carbon Chain

Graphene can be regarded as a combination of zigzag carbon chains (illustrated in Figure S1), so it is meaningful to study the mechanical response of the zigzag chain to explain the mechanical properties of graphene. To ensure the coordination number of the studied carbon atoms is 3 in the zigzag carbon chain, the structure of the zigzag carbon chain is chosen, as shown in Figure 1a. The blue dotted box represents the unit cell of the zigzag carbon chain, and the 20 Å vacuum layers are applied in the y- and z-axis. In Figure 1a, the bond angle, bond length, and lattice constant are marked by $d$,$\theta$ and $a$, respectively, where the optimized lattice constants are $a=2.285Å$, $d=1.42Å$, and $\theta =115.82°$. To investigate the mechanical response of the zigzag carbon chain, it is expanded 40 unit cells along the x-axis, and the strain is applied along the y-axis. To simulate the zigzag chain under a vertical compressive or tensile strain, the upper and lower brown carbon atoms (in the black dotted boxes) are shifted toward or backward to the red atoms, respectively, and then they are fixed. The Lammps program with the Airebo potential function is used to simulate the mechanical response of the red atoms in the zigzag chain.

The mechanical response of the zigzag chain is shown in Figure 1b, in which the red line indicates the variation of the lengths of the unit cell under a strain along y-axis. It is found that a turning point appears in the case of ${\epsilon }_y$ = 0.007 with the structure parameters $d=1.421Å$ and $\theta =115.82°$. The purple line is the Poisson’s ratio curve. One can see that a NPR appears when ${\epsilon }_y$> 0.007. The whole evolution of the carbon chain under stretch is available in the Supplementary Materials.

To explore the mechanical mechanism of the NPR phenomenon, we investigate the relation between the bond length, bond angle, and structural energy of the zigzag chain, shown in Figure 1a. Figure 2a presents the zigzag chain extracted from Figure 1a. The distance of two next-nearest atoms is labeled as x, and the thickness of the zigzag chain is labeled as y. The bond angle is labeled as $\theta$. The black dots represent the intrinsic zigzag chain without strain. The optimized parameters of the intrinsic zigzag chain is $x_0=2.40Å$, $y_0=0.75Å$, ${\theta }_0=115.82°$. When tensile strain is applied on the zigzag chain along the y-axis, the atoms in the zigzag chain shift away from the equilibrium positions, as shown by the red dots in Figure 2a. With the increase in $y$, i.e., $\mathsf{\Delta }y=y-y_0>0$ (${\epsilon }_y>0$), if $\mathsf{\Delta }x=x-x_0>0 ({\epsilon }_x>0)$, a NPR occurs. Otherwise, the Poisson’s ratio is positive.

In Figure 2b,c, we calculate the $\mathsf{\Delta }$E/atom of the zigzag chain as a function of $\mathsf{\Delta }x$, when $\mathsf{\Delta }y=0.05y_0$ and $\mathsf{\Delta }y=0.1y_0$. Here, $\mathsf{\Delta }$E refers to the difference in the energy between the changed structure and the initial structure (the relaxed structure without any strain). The variation of $\mathsf{\Delta }x$ is also related to the variation of the bond angle $\theta$. The relation between the structural parameters is:

$$\mathsf{\Delta }x=x-x_0=2y\left(\tan \frac{\theta }{2}-\tan \frac{{\theta }^{\prime }}{2}\right)$$

(1)

where the angle ${\theta }^{\prime }$ refers to the hypothetical situation that $\mathsf{\Delta }x=0$ and $\mathsf{\Delta }y>0.$ Seen from Figure 2b, when $\mathsf{\Delta }y=0.05y_0$, the lowest energy occurs at $\mathsf{\Delta }x$= 0.006$Å$. This means that the bond angle $\theta$ is larger than ${\theta }^{\prime }$ and the zigzag chain needs to be extended to reduce the total energy. Figure 2c presents the relation between $\mathsf{\Delta }y$ and $\mathsf{\Delta }x$ when a tensile strain is applied along the y-axis with $\mathsf{\Delta }y$ = 0.1$y_0$. One can obtain a conclusion similar to the case in Figure 2b, i.e., a tensile strain along the vertical direction of the zigzag chain will extend the chain.

It is noted that the bond angle $\theta$ of the intrinsic zigzag chain is 115.82°, while the bond angle in graphene is 120°. Graphene is the most sTable 2D carbon allotrope, which indicates that in a 2D carbon structure, a smaller angle will increase the structural energy. To reduce the total energy of the structure, the bond angle will increase. This is the reason that a zigzag chain extends along the chain direction when a vertical tensile strain shrinks the bond angle.

3.2. NPR in Some 2D Carbon Networks

It is demonstrated above that a zigzag carbon chain holds a NPR phenomenon when a tensile strain is applied perpendicular to the chain. Next, we sought to investigate whether the zigzag chain plays an important role in the occurrence of NPR in 2D carbon materials. To answer this question, except for graphene, three types of 2D planar carbon materials, constructed by zigzag carbon chains, are chosen to investigate the mechanical response. The schematic structures of these three materials are shown in Figure 3a,c,e, and the proportion of the zigzag carbon chains (marked by red color) in these structures are 66.67%, 80%, and 57.14%, respectively.

The stability of these materials is verified by the calculated phonon dispersion based on the DFT calculation, which shows that there is no imaginary frequency (see Figure S2 in the Supplementary Materials). Among three structures, the C12, also called OPG-Z, has already been discovered by Su et al. [60]. Based on the MD simulation carried out by Lammps, the ${\epsilon }_{x }-{\epsilon }_{y }$ curve (red line) and the Poisson’s ratio curve (purple line) are plotted in Figure 3b,d,f. These red and purple lines indicate that a NPR occurs when ${\epsilon }_{y }$ is equal to 3.85%, 2.37%, and 3.28%. It is demonstrated that under a strain along the y-axis, C10 is the first one to incur a NPR, which has the highest proportion of zigzag chains; while C14 is the last one to show a NPR, which has the lowest proportion of zigzag chains. Moreover, by comparing the bond lengths and bond angles of the carbon chains in these materials, we found that these chains are similar to those in Figure 1 (more details can be found in Tables S1 and S2 in the Supplementary Materials). These results generate a hypothesis that the proportion of carbon chains has a positive correlation with the occurrence of a NPR.

To verify this hypothesis, more than 600 structures are searched according to our structure search program of 2D carbon materials [61]. In addition to these carbon structures, shown in Figure 3, three sTable 2D carbon structures, C36, C20, and C24, are obtained (the phonon dispersion is shown in Figure S2 of the Supplementary Materials). The percentage of zigzag chains (marked by red color) in C36, C20 and C24 are 44.44%, 40% and 33.33%, respectively. We have carried out MD simulations on these structures and investigated the mechanical response, as shown in Figure 4. In these three materials, a NPR occurs when the values of ${\epsilon }_{y }$ are 9.725%, 11.548%, and 13.31%, which is consistent with our hypothesis.

Moreover, as shown in Figure 5, two carbon networks without zigzag carbon chains, marked as C16 and C32, are chosen to compare the mechanical response under strain. The phonon dispersions of C16 and C32 are shown in Figure S2 of the supplementary materials. It is clear that the zigzag carbon chain is absent in both the x- and y-axis. The mechanical response of these two structures is studied when the uniaxial stretch is applied in both the x- and y-axis. As shown in Figure 5c–f, until the structure arrives at mechanical failure, there is no NPR. To return to the structures discussed in Figure 3 and Figure 4, there is no zigzag chain perpendicular to the x-axis. The Poisson’s ratios of these structures along the x-axis are calculated and no NPR occurs, as shown in Figure S3 of the Supplementary Materials. Thus, we conclude that the zigzag carbon chain plays a crucial role in the mechanical properties of 2D carbon materials, and it has a positive effect on the occurrence of NPR. The simulation is processed by the MD method with the Airebo function, and the calculated Poisson’s ratio may be difference under other methods, e.g., DFT [46,47]. However, as shown in Figure S5, these quantitative differences do not influence the qualitative results.

The Poisson’s ratio is not only determined by the direction, but is also influenced by the temperature. The structure C10 is chosen to test the influence of the temperature, and more details are shown in the supplementary material. It is noted that the mechanism of a NPR in our work is different from the occurrence of a NPR in vacancy-defected graphene structures [39], whose NPR is aroused by the rippled structure.

4. Conclusions

In summary, we investigate the mechanical response of some 2D planar carbon networks. First, the NPR in the zigzag carbon chain is uncovered. We found that the zigzag carbon chain has a strong relationship with the occurrence of a NPR. Some 2D planar carbon structures, including zigzag chains, indeed possess NPR phenomena when strains are applied along the vertical direction of the zigzag chains in the structures. Several 2D carbon structures have smaller critical strains to obtain a NPR compared with graphene. In contrast, the carbon structures with no zigzag chains were not found to have a NPR in the structures. Our study can not only be used to explain the occurrence of a NPR in 2D carbon structures, but can also be extended to explain a NPR in all 2D honeycomb systems or 2D structures, including zigzag chains.