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. 2021 May 28;6(23):14896–14902. doi: 10.1021/acsomega.1c00718

Auxetic Carbon Honeycomb: Strain-Tunable Phase Transitions and Novel Negative Poisson’s Ratio

Yanchun Li 1, Shuaiwei Wang 1,*, Baocheng Yang 1,*
PMCID: PMC8209820  PMID: 34151071

Abstract

graphic file with name ao1c00718_0008.jpg

Auxetic structure and tunable phase transitions are fascinating properties for future application. Herein, we propose two three-dimensional (3D) carbon honeycombs (CHC), known as Cmcm-CHC and Cmmm-CHC. Based on first-principles calculations, these novel 3D materials exhibit auxeticity with a fascinating negative Poisson’s ratio, which stems from (i) the puckered structure of Cmcm-CHC along the tube axis and (ii) significant change of angle-dominant deformation for Cmmm-CHC in the armchair direction. In addition, the moderate strain drives semimetal to semiconductor phase transition in CHCs, which thoroughly establishes its C–C bond formation. In the meantime, two new phases, namely P63/mmc-CHC and P6/mmm-CHC, form and exhibit semiconductor characteristics. Our results also show that Cmcm-CHC and P63/mmc-CHC are superhard materials. The outstanding negative Poisson’s ratio and phase transition properties make CHCs highly versatile for innovative applications in microelectromechanical and nanoelectronic devices.

Introduction

Since the discovery of graphene in 2004,1 the field of carbon-based material has attracted tremendous interest. Due to the unique ability of carbon to form sp1 and sp2 as well as sp3 hybridization states, a series of two-dimensional (2D) and three-dimensional (3D) carbon allotropes were identified and studied.2−7 These materials exhibit a variety of structural and remarkable electronic, thermal, optical, and mechanical properties.8−13 Especially, 3D carbon honeycomb5,6,14 materials have been extensively researched.

Recently, the 3D carbon honeycomb (CHC) has been successfully synthesized by experimental research.6 The CHC structure has a regular channel structure with sp2-bonded carbon atoms and exhibits high levels of physical absorption than carbon nanotubes and fullerenes. However, imaginary phonon modes were observed in CHCs by theoretical calculations.14 Therefore, some new structures of CHCs with Dirac15 and Weyl semimetal5 characteristics were predicted with ultrahigh Fermi velocity. Besides the fascinating electronic band structure, the carbon materials also display outstanding mechanical properties, especially a negative Poisson’s ratio (NPR). NPR materials with typical auxetic features exhibit a counterintuitive behavior: they expand when stretched and contract when compressed. Due to their unique mechanical properties, auxetic materials have been attracting increasing attention from researchers. The first auxetic material was manufactured in 1987.16 In the past years, the computational model of an auxetic molecular system was first computed by Monte Carlo simulations,17 and then the static version of a two-dimensional molecular model was rigorously solved by Wojciechowski.18 Evans et al. proposed the re-entrant 2D auxetic networks.19 Tretiakov et al. reported a rapid decrease of Poisson’s ratio in 2D hard body systems by Monte Carlo simulation.20 For the carbon-based materials, either modifying the cell shape or altering the bond length can possibility induce the NPR phenomena in single-walled carbon nanotubes.21 Furthermore, the NPR phenomena occur in graphene at a small strain and independent of the size and temperature,22 or by introduction of vacancy defects.23 The NPR for single-layer graphene ribbons was found at widths smaller than 10 nm.24 In addition, metals with periodic porous structures can also show a NPR.25 Hence, it is of great interest to explore the mechanical properties of novel 3D CHC materials, especially for their structural and property response by applying an external strain, which may offer key insights into their fundamental mechanisms and potential for application in nanodevices.

In this paper, the mechanical properties of two 3D CHCs are investigated by first-principles calculations. Our results reveal that the CHCs exhibit auxeticity (negative Poisson’s ratio) under tensile loading and surperhard properties. The NPR phenomena in CHCs originate from the puckered structure and angle bending. Remarkably, the strain-induced semimetal to semiconductor transformation in CHCs is mainly due to the new C–C bond forming. The excellent mechanical coupled with highly tunable electronic properties make 3D CHCs a promising material for nanoscale devices.

Results and Discussion

Figure 1 presents the top, side, and perspective views of the atomic structure of Cmcm-CHC and Cmmm-CHC, in which the sp3-hybridized carbon atoms (pink) are linked with three nanoribbons by the sp2-hybridized carbon atoms (blue). The optimized structural parameters are shown in Figure 1. For the Cmcm-CHC, the carbon atoms occupied the 8g (0.70 0.22 1.25) and 4c (0.5 0.07 1.25) positions. For the Cmmm-CHC, the carbon atoms occupied the 8q (0.78 0.81 0.5), 8p (0.82 0.87 0.0), 4g (0.61 0.5 0.0), and 4h (0.55 0.5 0.5) positions. In our previous study,15 the Cmcm-CHC and Cmmm-CHC structures showed either thermal stability or mechanical stability. The strain is defined as εi = (a – a0i)/a0i (i = x, y, z), where a0i is the lattice constant. The Poisson’s ratios under tensile loading along the x direction were calculated26 using the formula νxj= –∂εj/∂εx, (j = y, z). The details of the computational method are given in Section I of the Supporting Information.

Figure 1.

Figure 1

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Optimized atomic structures of (a) Cmcm-CHC and (b) Cmmm-CHC from top, side, and perspective views along x (zigzag), y (armchair), and z (tube) directions. The blue/red box represents the basic unit cell/calculation model of the CHC structure. The pink and blue carbon atoms are sp3-hybridized and sp2-hybridized, respectively.

Negative Poisson’s Ratio

Having established the structure configurations of CHCs, we now focus on the structural response under uniaxial loading and evaluate the resulting Poisson’s ratios. The Poisson’s ratios of CHCs are shown in Figure 2 for uniaxial deformation along the x (zigzag) and z (tube axis) directions. The strain of Cmcm-CHC along the y (armchair) axis (Figure 2a) exhibits approximate linearization in the tensile deformation region, as indicated by the positive Poisson’s ratio. Remarkably, the lattice constant along the z (tube axis) direction first shrinks and then expands by the tensile strain in the x (zigzag) direction. By using a quartic fit, the Poisson’s ratio can be obtained from Figure 2a. The value of Poisson’s ratio is shown in Figure 2b as a function of the strain. The Poisson’s ratio along the tube axis is negative at strains larger than 0.27 in the zigzag direction. At large strains, with growing strain the absolute value of the negative Poisson’s ratio becomes smaller. For instance, the Poisson’s ratio along the tube axis is 0.034(−0.015) at a strain of 0.1(0.3). It should be noted that the negative Poisson’s ratio (NPR) phenomenon also appeared along the tube axis (zigzag direction) during its deformation along the armchair direction (tube axis) (Figure S1) and no NPR phenomenon occurs along the tube axis. Different from Cmcm-CHC, the strain for Cmmm-CHC along the y axis (armchair) reveals a monotonic increase, while the x axis shows a positive Poisson’s ratio during its deformation along the z axis. Figure 2d shows that the NPR of Cmmm-CHC expands along the y (armchair) axis when stretched along the z (tube) axis, whereas the Possion’s ratio undergoes changes in three stages (first decreases at strains <0.05, then tending to keep a constant value at strains from 0.05 to 0.14, and then dropping sharply at strains >0.14). The NPR is about −0.632 at a strain of 0.21 along the armchair direction, which is larger than that of Cmcm-CHC. As shown in Table 1, the NPR of Cmcm-CHC is lower than most of other materials,27−34 whereas the NPR of Cmmm-CHC is larger than that of other 3D carbon-based materials,28,29 similar to 2D carbon-based materials27,30,33 and other materials.31,32,34 In short, the Poisson’s ratios are dependent on the deformation, which means that the Poisson’s ratio of CHCs is not a constant in the elastic stage and that it is a function of strain.

Figure 2.

Figure 2

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Poisson’s ratio as a function of the uniaxial deformation of Cmcm-CHC and Cmmm-CHC structures. (a) Strain εx versus εy and εz for the Cmcm-CHC structure subjected to uniaxial tensile loading in the zigzag direction. (b) Poisson’s ratio of the Cmcm-CHC structure along the tube axis. (c) Strain εz versus εy and εx for Cmmm-CHC structure subject to uniaxial tensile loading along the tube axis. (d) Poisson’s ratio of the Cmmm-CHC structure in the armchair direction.

Table 1. Comparison of the Poisson’s Ratio of CHCs with Those of Some Previously Reported Materials.

structures υx υy υz  
3D carbon-based materials Cmcm-CHC     0.038∼−0.042 strain-induced
Cmmm-CHC   0.14∼−0.63  
bco-C2028   ∼−0.13  
carbon honeycomb29     –0.32
2D carbon-based materials Penta-graphene27 –0.068 –0.068   intrinsic
Xgraphene30 0.18∼−0.053     strain-induced
tetrahex C33 0.04∼−0.081 0.036∼−0.127  
graphene22   0.3∼−0.04  
other materials B4N31 –0.018 –0.032   intrinsic
Ag2S32   –0.12 –0.54
gallium thiophosphate34   –0.033 –0.62
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Mechanisms of Negative Poisson’s Ratio

We next unveil the NPR mechanisms by examining the atomic configurations. Figure 3 shows the deformation modes to gain further insight into the relationship between the structure and NPR mechanisms, where the arrows represent the direction of atomic movement. Two deformation processes are found for Cmcm-CHC. In process I (εx < 0.27), atoms 1, 2, 5, and 6 move along the x axis (blue arrow) as shown in Figure 3a; meanwhile, the increasing angles of ∠132 and ∠546 induce atoms 1, 2, 5, and 6 to move up/down along the y axis (Figure 3b) and the distances d1 and d2 decrease/increase as shown in Figure 3g. So, no NPR phenomenon is found in process I. In process II (εx> 0.27), d1 increases with increase in the strain, which induces the relative motion of atoms 3 and 4 (red arrow) as shown in Figure 3c along the y axis, so d2 decreases. The effect of the movement of atoms 3 and 4 results in the NPR appearing along the z axis. These results indicate that the angle and bond length elongate to accommodate the external tension, leading to the NPR phenomenon along the tube axis for Cmcm-CHC. Compared with Cmcm-CHC, only one process is observed for Cmmm-CHC. Atoms 1, 2, 5, and 6 move along the z axis as shown in Figure 3d, with the tensile strain along the tube axis, whereas the surrounding atoms 3, 4, and 7 adjust their position along the x and y axis (red arrows in Figure 3e,f). Therefore, the lattice constant along the y axis increases, inducing the NPR phenomenon. For example, the angles of ∠546 and ∠132 are 121.71 and 112.32° at a strain εz = 0.0; a larger increase (with d1 decreasing) of the angles ∠546 = 128.09° and ∠132 = 118.75° is found at a strain εz = 0.10. Therefore, the length from atom 3 to atom 10 slightly changes from 5.89 to 5.83 Å (atom 10 in Figure 1b). As a consequence of the movement of atoms 3, 4, and 7 (red online), the angle ∠437 decreases (with d2 increasing) from 101.34 to 98.38°. In contrast, the variations for the angles are larger than for the bond length, so the angle-dominant deformation process leads to the NPR phenomenon.

Figure 3.

Figure 3

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Deformation mechanism for NPR. The blue arrows indicate the movement direction of the atoms by stretching along the zigzag direction of (a) Cmcm-CHC and tube axis of (d) Cmmm-CHC structure. The atomic positions along the y and z axis for (b, c) Cmcm-CHC and x and y axis for (e, f) Cmmm-CHC are shown. The red arrows display the movement direction after tensile loading. (g, h) Distances d1 and d2 as a function of strain.

Strain-Induced Phase Transformation

The stress–strain response of CHCs is shown in Figure 4 for different directions. At smaller strains, a linear relationship is found for Cmcm-CHC in the armchair, zigzag, and tube directions. As the strain increases, the stress moves to a plastic region until the structure breaks in the armchair and zigzag directions. The tensile strengths (strain) are 62 GPa (0.28) along the armchair direction, 88 GPa (0.35) along the zigzag direction, and 137 GPa (0.26) along the tube axis. It should be noted that the tensile strength of Cmcm-CHC in the tube axis is larger than that of graphene (113 GPa) in the armchair direction.35 Compared with Cmcm-CHC, the stress–strain response of Cmmm-CHC exhibits a nonlinear relationship in the armchair and zigzag directions due to the angle increase being greater than that of bond length at small strains. The tensile strengths are about 68 GPa (0.35), 53 GPa (0.49), and 81 GPa (0.21) along the armchair, zigzag, and tube axes, respectively. Interestingly, the stress quickly drops for Cmcm-CHC at a strain of 0.28 in the armchair direction and 0.22 for Cmmm-CHC in the tube axis. These intriguing stress variations indicate the structure reconstructions and transitions in CHCs.

Figure 4.

Figure 4

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Strain–stress curve of (a) Cmcm-CHC and (b) Cmmm-CHC.

In order to explore the structural transition under uniaxial loading, the bonding configuration and structural evolution of CHCs under strain loading are illustrated in Figure 5. The distance d1 of the C1–C2 atom decreases linearly at strains from 0.0 to 0.28. The distance d1 is 1.51 Å at a strain of 0.30, which is close to the C–C single bond distance, indicating a new bond formation. The space group of Cmcm-CHC changes from the original Cmcm to P63/mmc. This new structure has been predicted in a previous study.36 On further loading in the armchair direction, the bond length d2 breaks at strain 0.46 and the structure spontaneously transforms into the initial phase (Cmcm-CHC). The evolution of the bond lengths d1, d2, and angle θ are shown in Figure 5c. For uniaxial tension along the tube axis for Cmmm-CHC, both the bond lengths d1, d2, and angle θ show a significant change at strain 0.21, which is similar to the case for Cmcm-CHC. The structure transforms to another phase at strains larger than 0.21. In this process, the distance d1 shrinks as the strain increases in the zigzag direction and leads to the movement of C1 and C2 atoms close to each other, thus forming a new C–C bond in the zigzag direction. This is the main reason for a sharp drop in the stress as shown in Figure 4b. Different from Cmcm-CHC, there is no new phase at larger strains for Cmmm-CHC, except for the Cmmm phase to P6/mmm phase transformation. The phonon dispersions of two phases (P63/mmc-CHC and P6/mmm-CHC) are calculated to examine their stability. As shown in Figure S2, the absence of an imaginary frequency demonstrates the dynamic stability of P63/mmc-CHC and P6/mmm-CHC after the structural transition.

Figure 5.

Figure 5

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Structure evolution under uniaxial tensile loading for (a) Cmcm-CHC and (b) Cmmm-CHC structures at different strains. The bond length and bond angle as a function of strain for (c) Cmcm-CHC and (d) Cmmm-CHC structures. The upper inset picture in (c) and (d) shows the unit cell of a new structure under tensile loading.

We illustrate the crystal structures of P63/mmc-CHC and P6/mmm-CHC in the inset of Figure 5c,d. The details of the crystal structures, including lattice parameters, space group, bond length, and cohesive energy, are shown in Table S1. However, the most fascinating feature of P63/mmc-CHC and P6/mmm-CHC is their electronic property. The electronic band structures of P63/mmc-CHC and P6/mmm-CHC at its equilibrium state and strain are presented in Figure 6. Figure 6a demonstrates a semiconducting phase with a direct band gap of 2.256 (1.554) eV at a strain of 0.0 (0.31) at the Γ point, which is close to the band gap of silicon37 and black phosphorene.38 There are three low energy bands at the band edges. One nondegenerate conduction band and two valence bands degenerate at the Γ point. Under the tensile strain, the conduction moves down, while the two valence bands slightly change. Interestingly, P6/mmm-CHC shows metallicity at a strain of 0.0, while an indirect-gap semiconductor shows an indirect band gap of about 0.887 eV at a strain of 0.31. As shown in Figure 5c,d, both Cmcm-CHC and Cmmm-CHC transition from semimetal to semiconductor phase under external strain and remain stable and robust against environmental perturbations. The results will definitely motivate and provide useful guidance for designing and applying these exciting phenomena in nanodevices.

Figure 6.

Figure 6

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Band structures of (a) P63/mmc-CHC and (b) P6/mmm -CHC structures at different strains.

Bulk Moduli (B), Shear Moduli (G), Young’s Moduli, and Vickers Hardness

The mechanical properties, including bulk moduli (B), shear moduli (G), and Young’s moduli (E), of the CHCs are calculated (Table 2). The B and G values for Cmcm-CHC are 313 and 289 GPa, respectively, which are derived using the Voigt method39,40 and are given by B = (1/9)[C11 + C22 + C33 + 2(C12 + C13 + C23)] and G = (1/15)[C11 + C22 + C33 + 3(C44 + C55 + C66) – (C12 + C13 + C23)]. The results show that Cmcm-CHC has high resistance to fracture and weak plastic deformation. Note that the B and G of Cmcm-CHC and Cmmm-CHC are lower than those of diamond (B 432 GPa and G 519 GPa).5 The Young’s modulus E is evaluated using the equation E = −9BG/(3B + G).39,40Cmcm-CHC possesses the highest E among all these stable CHC structures (Table 2). The Vickers hardness Hv, which reflects elastic and plastic properties, is determined by the formula41Hv = 2(k2G)0.585 – 3, where k = G/B is Pugh’s modulus ratio. The Hv for Cmcm-CHC, Cmmm-CHC, P63/mmc-CHC, and P6/mmm-CHC are 47, 30, 49, and 17 GPa, respectively. It should be emphasized that the ultrahigh Hv values for Cmcm-CHC and P63/mmc-CHC are 47 and 49 GPa, which are higher than the critical value (40 GPa42) for a superhard material. These values are much lower than those of diamond (93 GPa)5 and also lower than those of carbon materials (84–96 GPa)43 (Table 2). The results suggest that Cmcm-CHC and P63/mmc-CHC as superhard materials can be widely used in cutting and polishing wear-resistant coatings.

Table 2. Bulk Modulus (B), Shear Modulus (G), Young’s Modulus (E), and Vickers Hardness (Hv) of Cmcm-CHC, Cmmm-CHC, Graphene, Diamond, and Other Carbon Materialsa.

  Cmcm-CHC Cmmm-CHC P63/mmc-CHC P6/mmm-CHC graphene diamond other carbon materials
B 313 190 324 195   432c  
G 289 164 302 126   519c  
E 663 383 691 312 1024b 1220d  
Hv 47 30 49 17   93c 84–96e
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a

The units of B, E, G, and Hv are GPa.

b

Density functional theory (DFT) calculations in ref (44).

c

DFT calculations in ref (5).

d

Ref (45).

e

Ref (43).

Conclusions

In this work, we unveil the phase transition and negative Poisson’s ratio mechanism of CHCs by first-principles calculations. The NPR phenomenon in Cmcm-CHC and Cmmm-CHC occurs along the tube axis and armchair direction, respectively. It occurs mainly due to elongation of the angle and bond length to accommodate the external tension for Cmcm-CHC and the angle-dominant deformation for Cmmm-CHC. Meanwhile, a semimetal to semiconductor phase transition of Cmcm-CHC and Cmmm-CHC is observed by external strain along the armchair direction and tube axis, which stems from the new C1–C2 bond formation under tensile loading. Interestingly, the elastic properties show that Cmcm-CHC and P63/mmc-CHC are superhard materials. These outstanding properties render CHCs as an outstanding 3D nanomaterial with great promise for application in novel nanodevices.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (No. 51872110).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c00718.

  • Details of the computation details, validation of the theoretical calculations, structure parameters of CHCs, Poisson’s ratio, and phonon dispersions (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao1c00718_si_001.pdf (577.1KB, pdf)

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