Research Article  Open Access
Xiaowen Li, Xiaobin Qiang, Zhenhao Gong, Yubo Zhang, Penglai Gong, Lang Chen, "Tunable Negative Poisson’s Ratio in Van der Waals Superlattice", Research, vol. 2021, Article ID 1904839, 11 pages, 2021. https://doi.org/10.34133/2021/1904839
Tunable Negative Poisson’s Ratio in Van der Waals Superlattice
Abstract
Negative Poisson’s ratio (NPR) materials are functional and mechanical metamaterials that shrink (expand) longitudinally after being compressed (stretched) laterally. By using firstprinciples calculations, we found that Poisson’s ratio can be tuned from near zero to negative by different stacking modes in van der Waals (vdW) graphene/hexagonal boron nitride (G/BN) superlattice. We attribute the NPR effect to the interaction of orbitals between the interfacial layers. Furthermore, a parameter calculated by analyzing the electronic band structure, namely, distancedependent hopping integral, is used to describe the intensity of this interaction. We believe that this mechanism is not only applicable to G/BN superlattice but can also explain and predict the NPR effect in other vdW layered superlattices. Therefore, the NPR phenomenon, which was relatively rare in 3D and 2D materials, can be realized in the vdW superlattices by different stacking orders. The combinations of tunable NPRs with the excellent electrical/optical properties of 2D vdW superlattices will pave a novel avenue to a wide range of multifunctional applications.
1. Introduction
Negative Poisson’s ratio (NPR) material shrinks laterally when axially compressed or laterally expands when subjected to axial stretching. Compared to positive Poisson’s ratio (PPR) materials, NPR material has higher indentation resistance [1], larger impact resistance, more superior sound absorption performance, and better crack propagation resistance [2]. In addition, when subjected to outofplane bending moments, the NPR material will exhibit a dome shape rather than the PPR material tending to saddle shape. These excellent properties indicate that the NPR materials have broad application prospects in the automotive, aerospace, marine, and other industrial fields [3].
The NPR phenomena have been found in many natural and artificial materials, such as cubic metals [4], porous polymers [5], honeycombs [6], perovskite [7, 8], silicon oxides [7, 8], ceramic aerogels [9], reentrant crystal structures [10–17], and liquid crystal elastomer [18]. Recently, the NPR effect has also been found in some twodimensional (2D) materials [19], such as black phosphorus [20, 21], δphosphorene [22], borophene [23, 24], graphene [25], hBN [26], 1 Ttype transition metal dichalcogenides [27], groupIV monochalcogenides [28], Be_{5}C_{2} [29], silicon dioxide [30, 31], FeB_{6} [32], B_{4}N [33], and Ag_{2}S [34]. In addition, there are hundreds of thousands of materials in the inorganic crystal structure database (ICSD); however, the number of NPR materials reported in the study is a few hundred, which is relatively small. Among them, artificial materials and structures often have a very high NPR, while crystal materials have a relatively small NPR. Especially in 2D materials, NPR is smaller. In 2014, NPR was first found in 2D materials, namely, black phosphorus, with a value of 0.027 [20]. After that, the NPR of the 2D materials found in the later study mostly remains near 0.1. Based on our theoretical calculations, unlike pure 2D materials, the NPR in the vdW heterostructure can only be maintained around 0.1, which is due to the expansion amplitude of orbital under inplane strain.
Moreover, although many indepth studies have examined the possible existence of NPR effects in 3D and 2D materials, few studies have reported the NPR effect in vdW superlattice. With the development of manufacturing technology, graphenebased superlattices showed enhanced stability in experiments [35]. Therefore, the designability and diversity of vdW superlattices provide a broad prospect for designing multifunctional NPR materials, such as NPR electrodes and molecular sieves [36]. In previous studies, the most NPR phenomena can be attributed to its unique wrinkle or reentrant structures. In addition to these special geometric reasons, we want to study the fundamental physical mechanisms that form NPR effect.
In our previous study, we reported nearzero Poisson’s ratio (ZPR) phenomena in G/hBN and multilayer hBN [37]. Interestingly, in this study, using firstprinciples calculations, we found that Poisson’s ratios of G/hBN superlattices are 0.109, 0.111, and 0.023 in different stacking modes. The dichotomy between NPR and PPR effects exhibited in the G/hBN superlattice, which can be explained by a special electronic structure at the interfacial layer. Although G/hBN is a kind of simple vdW heterostructure, it is convenient to make theoretical analysis and calculation clearly. It may open a beginning for the theoretical study of NPR effect in different stacking modes of vdW materials. In addition, we calculated the outofplane stiffness of the G/hBN superlattice with different stacking modes. These modes with NPR also have outofplane negative shear modulus (NSM), i.e., when shear strain is applied to NSM materials, as the shear strain increases, the corresponding shear stress tends to decrease.
Ultimately, the NPR phenomenon, which was relatively rare in 3D and 2D materials, can be realized in the vdW superlattices by different stacking orders as designed. Furthermore, studying on how to change the PPR material into the NPR material not only has important practical engineering application value but also theoretical value for indepth study on other possible related interesting physical properties, such as negative pressure electricity, negative stiffness, and negative thermal expansion.
2. Results
The unit cell of the G/hBN superlattice is composed of graphene unit cell (2 C atoms) and BN unit cell (1 B atom and 1 N atom) in the  plane. The lattice constant of the pristine monolayer graphene and BN are 2.465 Å and 2.509 Å, respectively. Then, the of G/hBN superlattice calculated by firstprinciples calculations is 2.485 Å, and the lattice mismatch between graphene and BN is less than 1%.
The interlayer binding energies () and equilibrium distances () of all stacking modes of G/BN superlattices obtained by density functional theory (DFT) approach are given in Table S1. Here, we investigated three highly symmetric stacking modes of G/BN superlattices: N atom sublattice on hexagonal C atom ring (stacking mode A), B atom sublattice on C atom ring center (stacking mode B), and N atom sublattice on C atom ring center (stacking mode C) [38]. follows the order of , while follows the order of .
2.1. Stiffness
By analyzing the strain energy, the elastic constants of G/hBN superlattices were derived from the linear fitting of the energystrain relationship (Table S2). For hexagonal crystal, the inplane mechanical properties of G/BN superlattice are isotropic (, , ) [41]. Young’s modulus is determined by elastic constants (see method section). Notably, the inplane of the 2D material is the product of the of the corresponding 3D material and the effective thickness [27], and we took the as the effective thickness for each layer of 2D material.
Table 1 shows that the outofplane of the stacking mode A, B, and C is 44.9, 45.6, and 49.0 GPa, respectively. The smaller the of the stacking mode, the greater the corresponding . In addition, we calculated the inplane of the G/BN superlattice, which is almost equal to the sum of of the monolayer graphene and BN. Therefore, this result explains the reason for the stability enhancement of carbonbased superlattices observed in experiments [35]. However, the difference in of the superlattice is mainly due to the different in .
 
For monolayer materials, the unit of Young’s and shear modulus is Nm^{1}. For G/BN superlattices, the unit of Young’s and shear modulus is GPa. 
2.2. Poisson’s Ratio
We compared G/BN superlattices in different stacking modes under different uniaxial strains along the direction () (Figure 1). For stacking modes A and B, the is auxetic for , but the same phenomenon was not found in stacking mode C. Interestingly, stacking modes A and B not only have the NPR () effect but they also have negative shear modulus (NSM) () in the outofplane direction. The shear force decreases with the increase of shear deformation, which is the NSM effect.
In order to study the anisotropy of Poisson’s ratio of these materials, orientationdependent Poisson’s ratio was calculated (Figure 2). We found that the stacking modes A, B, and C have the ZPR () effect at , 36.8°, and 18.7°, respectively. Therefore, stacking modes A and B exhibit a NPR effect in a larger crystal orientation angle range than stacking mode C.
To our knowledge, it is very difficult to measure Poisson’ s ratio of several layers of twodimensional (2D) material with the existing experimental method. Because for these ultrathin films, when the inplane strain is applied, the outofplane deformation is very small and difficult to observe. However, for multilayer 2D materials, Xray diffraction can be easily used to measure Poisson’ s ratio when the thickness is close to 10 nm [42]. The NPR effect is generated in the interfacial layer. Therefore, both multi and singlelayer vdW materials can exhibit the same NPR. It is relatively easy to measure Poisson’ s ratio for the multilayer vdW materials with a certain thickness.
3. Discussion
3.1. Interlayer Binding Energy
Assuming that the interaction between the two layers of the superlattice is additive, the binding potential can be expressed as the cumulative interaction of atoms between different layers [43]. The binding energy of two atoms combined by vdW forces can be expressed by the LennardJones potential: Here, represents the distance between the two atoms. The and are fitting constants. The first term represents the vdW attraction, and the second term represents Pauli’s repulsion [44]. Therefore, the interlayer potential of the vdW superlattice can be expressed as where and are the mass densities of two layers of vdW superlattice, respectively. The distance is obtained from the geometric relationship between the coordinate and . For G/BN superlattice, . In Figure 2, the fitting curves of ~ below the horizontal coordinate axis represent vdW attraction, and the curves of ~ above the horizontal coordinate axis represent Pauli repulsion. Therefore, the loworder vdW term plays a major role in the large , while highorder Pauli’s repulsion term plays a major role in the small . According to the firstprinciples calculation, Eq. (2) can well describe the of the vdW superlattice.
Figures 2(d) and 2(e) show that when , the Pauli repulsion energy increases significantly, while the vdW attraction energy has a negligible change. Therefore, with the decrease of , of the G/BN superlattice expands, while the lowest point of the energy curve moves forward along the positive direction of the axis, resulting in a negative of 0.109 and 0.111. For the stacking mode C, vdW and Pauli repulsion show negligible changes under (see Figure 2(f)). The binding energy increases slightly; so, the G/hBN superlattice exhibits the ZPR effect. According to the firstprinciples calculation, Poisson’s ratio of the material has a relationship with . If the material can exhibit the NPR effect, the value of under tensile strain () is greater than the initial value of (Table S4). However, we calculated that the increase of is not a necessary condition for the NPR effect, i.e., plays a major role in the NPR effect.
3.2. Relationship between Orbitals and NPR
The Bloch wave function of the orbital electron in a periodic lattice under the tight binding (TB) approximation can be expressed as For G/BN superlattice, C atoms in graphene and N atoms in hBN have orbitals. When the atom A is used as the origin of coordinates (Figures 3(a) and 3(b)), let the inplane strain be the perturbation . The wave function of orbitals of the atom A is . Meanwhile, the wave function of orbitals of the atom B is . When the A and B atoms in the lattice are bonded, the wave function of the bonded p_{z} orbital can be expressed by the linear combination of atomic orbitals as The constant C is a normalization constant, which should satisfy the normalization condition . The densityweighted length of the electrons in the outofplane direction can be expressed as Here, is the length of the electrons with momentum in the outofplane direction. Therefore, the length () of electrons with all momentum should be the integral of in the first Brillouin zone (BZ). Finally, the charge densityweighted length of the orbital in the outofplane direction can be obtained: where and , which is the length of the isolated orbital (Figure 3(c)). Therefore, according to Eq. (6), we got the analytical solution of the relationship between and (Figure 3(d)). The calculation details can be found in the Supporting Information. Meanwhile, partially differentiate to the inplane perturbation is . Therefore, orbitals will extend outofplane under inplane tensile strain.
Since the length of the orbital has auxetic effect under inplane strain, we quantitatively studied the charge density distribution along the outofplane direction of G/BN superlattice by using firstprinciples calculations. In the outofplane direction, the charge density at coordinate can be expressed as Here, is the charge density at the coordinate (, , ) with the energy of , and is the Fermi level of the system. In order to quantify the change of the charge density in the outofplane direction under stress, we calculated the weighted length of the electron density in the outofplane direction according to the following formula: Notably, the charge of the graphene in the outofplane direction is mainly contributed by the orbital. The of each layer in the G/BN superlattice under inplane strain and is shown in Table S4. In each layer of the G/BN superlattice, the is elongated. When an inplane strain was applied, the bond angle ∠NBN increased from 120 to 122.36, resulting in the charge localization (Fig. S2).
Quantitatively, we found that the value of of the monolayer BN and graphene in G/BN superlattices increased by 1.8 ~ 1.9% and 2.3 ~ 2.4%, as the inplane tensile strain increases by 8% (Table S5), explaining the NPR effect in stacking modes A and B along the outofplane direction. This is consistent with the analytical solution obtained by TB approximation (Figure 3(d)). For stacking mode C, the N atom sublattice is on the C atom ring center. In BN, the N atom has a fully filled orbital, while the B atom has an empty orbital. The orbitals of G/BN superlattices hardly overlap (Figure 4(c)); so, the change of the orbitals has little effect on the Pauli repulsion between the interfacial layers, resulting in no significant NPR effect.
3.3. Relationship between Electronic Band Structures and NPR
Figure 5 shows the DFT and TBbased band structure of G/BN superlattices in different stacking modes. To further understand the firstprinciples calculation results, we adopted the TB model to describe the electrons in G/BN superlattices with different stacking modes near the Fermi level. During the TB calculation, a unit cell contains two C_{1} and C_{2} carbon atoms at different positions and one N atom. Since the electronic states of the three bands around the Fermi level are completely contributed by the orbitals of C_{1}, C_{2}, and N atoms, only the orbitals of C_{1}, C_{2}, and N atoms are included in the TB model. The Hamiltonian matrix can be written as where subscripts 1, 2, and 3 represent C_{1}, C_{2}, and N atoms, respectively. Because the interlayer distance is longer than the CC bond length, the nearestneighbor interaction between C and N atoms and the next nearestneighbor interaction between C and C atoms are considered (detailed Hamiltonian matrix elements can refer to the Supporting Information). The distancedependent hopping integral is determined by the formula Here, represents the interfacial layer equilibrium distance, and is the hopping integral between orbitals at . is the distance between the th and th atoms, and is the decay constant for the integral [45]. For the G/BN superlattices, the values of and can refer to the Supporting Information Table S6.
The distancedependent hopping integral () describes the intensity of the interaction between the orbitals of C and N atoms. Therefore, is a powerexponential function of the interlayer spacing and is proportional to the NPR (Table 2). Furthermore, the higher the value of , the greater the value of the corresponding NPR. Note that after Taylor expansion of the , the quadratic term is the previous research results [46, 47].

Consequently, the vdW superlattice can exhibit an NPR effect only if they have orbitals in the outofplane direction, and the orbitals overlap between the interfacial layers. Meanwhile, the NPR effect in all vdW materials can be explained by the same physical mechanism given in this section. For example, for latticematched materials, a previous study showed that AAstacked BN (a N atom on a N atom in another layer) can exhibit an NPR effect [26] (Table 3). In addition, for latticemismatched vdW materials, the orbitals between the interfacial layers overlap; so, these materials should exhibit an NPR effect. For example, it has been observed that WS_{2}/WSe_{2} heterostructure expands abnormally under engineering tensile strain [48]. Therefore, according to this physical mechanism, the NPR phenomenon should exist in a large number of vdW materials, which was considered as a rare phenomenon in bulk and monolayer 2D materials.
Moreover, in the experiment, the isolated atomic layers can also be reassembled into the designed heterostructure layer by layer in a precisely selected order [49]. Therefore, for the same kind of investigated material, it can also be switched in different stacking modes through experimental methods. For example, the G/BN superlattices may also be tuned among stacking modes A, B, and C. Similarly, we can change the material without strong interlayer orbital interaction into a material with orbitals strongly overlapping between the interfacial layers, thus exhibiting an NPR effect.
In conclusion, we studied Poisson’s ratios and the binding energies of G/BN superlattices in different stacking modes by using the firstprinciples method. We found that the stacking mode C has a ZPR effect at the interfacial layer, while the stacking modes A and B show NPR effects. The NPR effect is mainly due to the interaction of the orbitals between the interfacial layers. Furthermore, the distancedependent hopping integral () calculated by analyzing that the electronic band structure can be used to describe the intensity of this interaction. The is a powerexponential function of the interlayer spacing and is proportional to the NPR. Moreover, we calculated their Young’s and shear modulus and found that the stacking modes A and B also have NSM effect in the outofplane direction. These materials with negative index coexistence will provide broad prospects for multifunctional and multipurpose materials. Finally, we expect that the theory can be verified by experiments and provide a solid foundation for the largescale searching and predicting NPR materials in the future.
4. Methods
Based on density of functional theory, all firstprinciples calculations were implemented by the planewave projector augmented wave (PAW) method in Vienna ab initio simulation package (VASP) code [50]. The exchange correlation functional adopted the generalized gradient approximation (GGA) of the Perdew−Burke−Ernzerhof (PBE) functional [51]. In order to test the robustness of our results, the vdWcorrected functionals proposed by Grimme DFT + D2 [52], DFT + D3 [53], manybody dispersion (MBD) [54], and vdWcorrected functional optB88vdW [55] methods were used in firstprinciples calculations. In this paper, the calculation results of functional optB88vdW are given because of its good agreement with the experimental results, and the results obtained by different vdWcorrected methods are only slightly different in numerical value (the detailed results are in the Supporting Information).
The G/BN superlattice was calculated by using Monkhorst–Pack Kpoint mesh. The energy cutoff value is 500 eV, and the structures were completely relaxed until their atomic Hellmann–Feynman forces were less than 0.005 eV/Å. The convergence criterion of energy in the selfconsistency process is 10^{6} eV. We also calculated electronic band structures for G/BN superlattices by using the HSE06 hybrid functional [56].
To quantitatively characterize the mechanical properties of the interface, the interlayer binding energy () between the monolayer graphene and BN is as follows: where , , and are the energies of the G/BN superlattice, graphene, and hBN, respectively. represents the inplane area of the superlattice.
The elastic constant is defined by expanding the internal energy into Taylor series in elastic strain at constant entropy. The expansion coefficient in the Taylor series is the elastic constant [57]: where and are the initial mass density and the Lagrangian strains of the material [58]. In this work, we use contracted notations (1) as tensor indices. In addition, we define Lagrangian strains and . The compliance coefficients are defined as Young’s modulus for the material is computed by Poisson’s ratio is defined as where is the strain in the direction of uniaxial loading (in the direction), and is the resulting strain in the transverse direction (the direction). In our calculations, we applied different uniaxial strains to the lattice. This strained structure was then completely relaxed to evaluate the magnitude of the strain in the outofplane direction. The detailed calculation process of the relationship between θ and is provided in the Supporting Information.
Data Availability
All data needed to evaluate the conclusions in the paper are presented in the paper and supplementary materials. And additional data are available from the corresponding authors upon reasonable request.
Conflicts of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work is supported by the Center for Computational Science and Engineering of Southern University of Science and Technology, NSFC (National Natural Science Foundation of China) grant number 51972160, the Science and Technology Research Items of Shenzhen grant numbers JCYJ20170412153325679, JCYJ20180504165650580, JCYJ20190809142603695, and JCYJ20170817110302672, and Highlevel Special Funding (Nos. G02206303 and G02206403).
Supplementary Materials
1: interlayer binding energies and equilibrium distances resultsSupplementary Table S1 2: elastic constants resultsSupplementary Table S2 3: volume modulus and shear modulus resultsSupplementary Table S3 4: interlayer binding energySupplementary Table S4 5: relationship between orbitals and NPRSupplementary Figure S1 6: charge densitySupplementary Figure S2, Table S5 7: charge density. Table 8: the relationship between Poisson's ratio and stiffness. The relationship between θ and 9. The relationship between electronic band structures and NPRSupplementary Table S6. (Supplementary Materials)
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