Negative Poisson's ratios in few-layer orthorhombic arsenic: First-principles calculations

Virtually, all known materials would undergo a lateral contraction when stretched longitudinally and vice versa, as a consequence of the conservation of volume under elastic loading. This gives a normal positive Poisson's ratio (ν). In isotropic materials, the positive Poisson's ratio is theoretically in the range from 0 (cork) to 0.5 (rubber).¹⁾ If a lateral dimension expands during stretching or vice versa, the exhibited Poisson's ratio is negative and the associated material is termed auxetic.²⁾ Much intense interest in this counterintuitive feature stems from the pioneering discoveries that critical fluids, re-entrant polymer foams, colloidal crystals, laser-cooled crystals, unscreened metals, α-cristobalite, the A7 structure elements: arsenic and bismuth, and laminates are found to be auxetic.^3–6) Such an unusual mechanical property of ν < 0 makes a material potentially applicable in those special areas. For example, compression in one direction results in shrinkage rather than expansion in the transverse direction, a mechanism demonstrated by bulletproof vests.⁷⁾ The opposite situation — an expansion responding to a stretch — is manifested by artificial limbs and blood vessels.⁸⁾ This sparks a surge in research activity in new materials with negative Poisson's ratio.

In 2008, Hall and his colleagues found that negative Poisson's ratios can appear in low-dimensional carbon sheets when multiwalled nanotubes are introduced.⁹⁾ This is a huge step toward the design of nanostructured composites, artificial muscles, gaskets, and chemical and mechanical sensors. Naturally, it raises the question of whether negative Poisson's ratio can emerge in other low-dimensional materials. In fact, measuring a negative Poisson's ratio in experiments always presents a challenge as the observation of such a ratio is spurious.¹⁰⁾ In this respect, theoretical simulation seems to be a very powerful tool for predicting negative Poisson's ratio.^11,12) For example, a monolayer of black phosphorus (BP) is theoretically reported to have a negative Poisson's ratio through using an ab initio method,¹³⁾ again triggering intense research interest in negative Poisson's ratio in real two-dimensional materials. Likewise, arsenic sits below phosphorus in the same column of the periodic table. Just recently, two groups have independently demonstrated the thermal stability of its few-layer forms in the orthorhombic phase,^14,15) just as for BP. In general, the magnitude of Poisson's ratio increases with atomic number Z.³⁾ This extends the potential applications if few-layer arsenic is expected to have a larger (in magnitude) negative Poisson's ratio. However, up to now, similar studies for few-layer arsenic have been lacking. Thus, it is timely to check the possible existence of negative Poisson's ratios in few-layer orthorhombic arsenic.

In this Letter, the negative Poisson's ratio is reported to occur in few-layer arsenic through using first-principles calculations. The negative Poisson's ratio is ∼−0.09 for monolayer arsenic. As the number of layers increases, the negative Poisson's ratio becomes more negative and finally approaches the bulk value of −0.13 at four layers. To better understand the underlying mechanism, we propose a rigid mechanical model in the hope of shedding new light on our understanding of negative Poisson's ratio in those hinge-like layered materials.

First-principles calculations in this work are performed within the framework of density functional theory, as implemented in the SIESTA code.¹⁶⁾ We have used the generalized gradient approximation in the form of the Perdew, Burke, and Ernzerhof functional.¹⁷⁾ The effect of the van der Waals (vdW) interaction is taken into account by using the empirical correction scheme proposed by Cooper.¹⁸⁾ Only the valence electrons are considered in the calculation, with the core being replaced by norm-conserving scalar relativistic pseudopotentials¹⁹⁾ factorized in the Kleinman–Bylander form.²⁰⁾ We have used a split-valence double-ζ basis set including polarization orbitals with an energy shift of 100 meV for all atoms.²¹⁾ Convergence is achieved when the difference of the total energies between two consecutive ionic steps is <10⁻⁵ eV and the maximum force allowed on each atom is set to be 0.01 eV/Å.

We start our work from orthorhombic bulk arsenic with space group Cmca, which is the same as BP. The conventional unit cell includes two layers. Each layer contributes four nonequivalent atoms to the unit cell, as shown in Fig. 1(a). Each As atom within a single layer is covalently bonded with three As atoms, forming a puckered graphene-like hexagonal structure. This puckered structure, also called a hinge-like structure, consists of two orthogonal hinges (atoms 456 and 612 or atoms 456 and 432). This sets up the basis for those exotic properties in few-layer arsenic. The lattice constants are optimized to a = 4.70 Å, b = 3.77 Å, and c = 11.11 Å, generating the internal parameters r₁₂ = 2.58 Å, r₃₄ = 2.56 Å, θ₁₂₃ = 94.09°, and θ₂₃₄ = 99.05°. All these agree well with experimental and theoretical results.^14,22) A primitive unit cell and its Wigner–Seitz cell are illustrated in Fig. 1(b). It is composed of four inequivalent atoms, which is only one half of the bulk case. In principle, monolayer arsenic (arsenene), as shown in Fig. 1(c), can be obtained through exfoliating its bulk counterpart. Once it is formed, the corresponding lattice constants a and b are changed to be 4.73 and 3.71 Å, respectively. In comparison with its bulk phase, a increases while b decreases. This has a direct effect on the internal parameters: bond lengths r₁₂ and r₃₄ are decreased to 2.53 and 2.50 Å, while the bond angle θ₁₂₃ is increased to 94.54° but θ₂₃₄ = 100.36° is nearly unchanged.

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The calculated band structure of bulk arsenic is displayed in Fig. 1(d). It clearly shows that the valence band maximum (VBM) and the conduction band minimum (CBM) are located at the same crystal momentum point Z, demonstrating a direct bandgap semiconductor. However, when the structure becomes a monolayer, it behaves as an indirect bandgap semiconductor, as shown in Fig. 1(e). The obtained bandgap is ∼1 eV. The underlying mechanism of the bandgap transition from direct (bulk) to indirect (monolayer) is dominated by the mutual competition between the two interlayer bondings r₁₂ and r₃₄ (see Ref. 14 for more details). In the following, we pay attention to the mechanical properties, particularly the negative Poisson's ratio in few-layer arsenic.

When strain is applied along the x-direction, strain occurs in the y-direction, as shown in Fig. 2(a). Note that the positive (negative) ε means a tensile (compressive) strain. The calculated data (solid circles) follow a strongly nonlinear curve and are well fitted by the function $y = - \nu _{1}x + \nu _{2}x^{2} + \nu _{3}x^{3}$. The linear parameter ν₁ is fitted to be 0.35 and can be regarded as the linear Poisson's ratio in the theory of infinitesimal deformations. Similarly, we can obtain the linear Poisson's ratio in the z-direction, as shown in Fig. 2(b). The corresponding linear Poisson ratio is ν = ν₁ = 0.13, which is nearly a factor of 3 smaller in magnitude than that in the y-direction. This means that monolayer arsenic is harder in the z-direction than in the y-direction when responding to the strain applied along the x-direction. It should be noticed that, in the hinge-like structure of arsenic [see Fig. 1(a)], the deformation in the z-direction is dominated by the bond length r₃₄ and the bond angle θ₂₃₄, whereas that in the y-direction responds to changes of r₁₂ and θ₁₂₃. Perturbed by the same external field, such as the strain in our case, the bond angle is more easily affected compared with the bond length. Thus, we can infer that the bond angle θ₁₂₃, in comparison with θ₂₃₄, is largely changed when the stress is applied along the x-direction, in good agreement with our first-principle calculations.

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If the strain is applied along the y-direction, the situation dramatically changes. ε_x linearly depends on ε_y, as displayed in Fig. 2(c). The fitted Poisson's ratio is ν = 1.07, which is much larger than the Poisson's ratio when the strain is applied along the x-direction. This is a direct manifestation of anisotropy in the mechanical properties, which also explains the anisotropic transport reported in Ref. 14. Up to now, all the obtained Poisson's ratios are normal and positive. However, when we check the response of the strain along the z-direction to the y-direction loading, the law exhibited as in Fig. 2(d) is opposite: ε_z increases with ε_y, leading to an unusual Poisson's ratio, namely, ν < 0. The fitted linear Poisson's ratio is $\nu = - 0.093$. This magnitude is quite large compared to the negative Poisson's ratio of ∼−0.027 in monolayer BP.¹³⁾ This makes monolayer arsenic more suitable for applications in special areas such as defense and medicine where larger negative Poisson's ratios are desirable. For monolayer arsenic, the internal bond lengths are larger than those of 2.42 and 2.38 Å in monolayer BP.¹³⁾ In principle, under the same loading, a bond with a larger length would bear a stronger deformation. This explains why monolayer arsenic holds a larger (more) negative Poisson's ratio compared to monolayer BP, being consistent with ν mostly increasing with atomic number Z.³⁾

Layer stacking has emerged as a new degree of freedom for tuning electronic properties of vdW hetero-materials,^23,24) but its impact on the negative Poisson's ratio remains unclear. In the following, we will demonstrate how it affects the negative Poisson's ratios in few-layer arsenic. Once an additional layer is added, the negative Poisson's ratio varies accordingly as the effect of layer stacking is involved, as displayed in Fig. 3(a). In comparison with a monolayer, the nonlinearity of ε_z versus ε_y is significantly enhanced for a bilayer. This is represented by the larger fitted parameters for the cubic terms, which are at least twice that of a monolayer. In the vicinity of zero, L₁ and L₂ are nearly identical. When the strain is beyond a ∼±4%, L₁ and L₂ largely separate from each other. This is because the two layers are no longer equivalent in the bilayer because of the AB stacking. The separated feature in these curves L₁ and L₂ under larger strain reflects the asymmetry of the layers, supporting the validity of our simulations. The linear Poisson's ratio is fitted to be ∼−0.14, which is ∼50% greater in magnitude than the value for the monolayer. Based on the above discussion, we conclude that the larger the bond length, the larger the negative Poisson's ratio. The bond length r₃₄, compared to r₁₂, is almost parallel to the z-direction, which is directly related to the negative Poisson's ratio. Owing to the interlayer vdW interaction introduced by an additional layer, r₃₄ is increased to be 2.54 Å for the bilayer, resulting in a more negative Poisson's ratio.

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In the case of a trilayer, as shown in Fig. 3(b), the curves L₁ and L₃ are identical, leaving the curve L₂ deviated from L₁ and L₃ in the whole strain range. All these coincide with the symmetry of a trilayer, in which layers L₁ and L₃ are symmetrically equivalent but not layer L₂. This leads to two negative Poisson's ratios ∼−0.13 for layers L₁ and L₃ and ∼−0.098 for layer L₂, which is slightly smaller in magnitude than those of L₁ and L₃. This is because the length r₃₄ of layer L₂ is 2.51 Å, which is smaller than the value of 2.55 Å for layers L₁ and L₃. When any additional layers are added, the internal bond lengths, in particular r₃₄, will largely remained unchanged. For example, in the case of a four-layer structure, the bond lengths r₃₄ are 2.55 and 2.54 Å for L₁ (or L₄) and L₂ (or L₃), respectively, which are very close to the value of 2.55 Å in bulk arsenic. The obtained negative Poisson ratio for four-layer arsenic is ∼−0.121 (averaged value), as shown in Fig. 3(c). This value approaches the bulk value of −0.125, as shown in Fig. 3(d). This means that the quantum effect from layer stacking on the negative Poisson's ratios is limited within four layers.

To further understand the effect of layer stacking on the negative Poisson's ratios, we demonstrate it from a rigid mechanical model. As pointed out above, the appearance of negative Poisson's ratio stems from the hinge-like structures in few-layer orthorhombic arsenic. When few-layer arsenic is stretched in the y-direction, the layer contracts in the x-direction, protected by the normal Poisson's ratios of the in-layer direction, as shown in Figs. 1(c) and 4(a)–4(d). In other words, atoms 3 and 4 as well as atoms 1 and 6 move inward along the x-direction when stretched in the y-direction [see Fig. 1(a)]. This directly causes the bond angles θ₂₃₄ and θ₂₁₆ to become smaller compared to the initial values. By taking into account all the values of θ₂₃₄ and θ₂₁₆ (with and without strain) being >90°, the layer thickness along the z-direction (the projected distance of r₃₄ or r₁₆ along the z-direction) is thus increased, leading to a negative Poisson's ratio. Therefore, bond angles θ₂₃₄ and θ₂₁₆ being >90^o is another necessary condition for the negative Poisson's ratios in few-layer arsenic. This is also true for BP because the bond angle θ₂₃₄ is 97.64° in BP.¹³⁾ If a material possesses bond angles θ₂₃₄ and θ₂₁₆ that are <90°, the negative Poisson's ratio should disappear. Future research can test this prediction directly.

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Owing to layer stacking, the normal Poisson's ratios of the in-layer direction increase from 1.07 for the monolayer [Fig. 1(c)] to 1.19 for a four-layer structure [Fig. 4(c)]. This implies that the decrease of θ₂₃₄ or θ₂₁₆ under the same stretching in the y-direction is enhanced as the number of layers increases. As a result, the magnitude of the negative Poisson's ratio increases with increasing number of layers. For four-layer arsenic, the normal Poisson's ratio is very close to the bulk value, as shown in Figs. 4(c) and 4(d). Thus, the negative Poisson's ratio approaches a limit at four layers. This tells us that the effect of layer stacking is exhibited in the negative Poisson's ratio in the y-direction, which is indeed through the normal Poisson's ratio in the x-direction.

In conclusion, the negative Poisson's ratio is for the first time reported in few-layer arsenic through using first-principles calculations. The magnitude of the negative Poisson's ratio is ∼0.09 for a monolayer. As the number of layers increases, the negative Poisson's ratio become more negative. A limit of ∼−0.12 is predicted for four layers; this value is very close to the bulk value. Studies like ours should shed new light on the layer-dependent negative Poisson's ratio in those hinge-like layered materials, which will evolve into an active field.

Acknowledgments