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Negative Poisson's Ratio in Single-Layer Black Phosphorus

The Poisson's ratio is a fundamental mechanical property that relates the resulting lateral strain to applied axial strain. While this value can theoretically be negative, it is positive for nearly all materials, though negative values have been observed in so-called auxetic structures. However, nearly all auxetic materials are bulk materials whose microstructure has been specifically engineered to generate a negative Poisson's ratio. In the present work, we report using first principles calculations the existence of a negative Poisson's ratio in a single-layer, two-dimensional material, black phosphorus. In contrast to engineered bulk auxetics, this behavior is intrinsic for single layer black phosphorus, and originates from its unique, puckered structure, where the pucker can be regarded as a re-entrant structure that is comprised of two coupled orthogonal hinges. As a result of this atomic structure, a negative Poisson's ratio is observed in the out-of-plane direction under uniaxial deformation in the direction parallel to the pucker, with the Poisson's ratio becoming increasingly negative with both increased tension and compression. The puckered structure also results in highly anisotropic in-plane Poisson's ratios, which are found to be 0.4 in the direction perpendicular and 1.28 in the direction parallel to the pucker.

1 Nearly all materials have a positive Poisson's ratio, which means that when stretched longitudinally, they become shorter in the lateral directions. In contrast, the list of materials that exhibit a negative Poisson's ratio is small, and thus researchers have shown significant interest in finding new materials that exhibit this distinctive property. Negative Poisson's ratio materials are also of great interest because they typically have enhanced toughness and shear resistance, along with enhanced sound and vibration absorption, which has been exploited in fields as varied as medicine 1 , fasteners, 2 tougher composites, 3 tissue engineering, 4 and national security and defense, i.e. for bulletproof vests and armor enhancement. 5 Most reports of negative Poisson's ratio have been from bulk-scale, engineered structures.
For example, the first foam with a negative Poisson's ratio was reported in 1987 by Lakes 6 ; such materials that exhibit a negative Poisson's ratio were eventually termed auxetics by Evans et al. in 1991. 7 The negative Poisson's ratio in these bulk structures is derived from engineering of the material microstructure. 8 For example, the negative Poisson's ratio in the foam of Lakes 6 is due to a particular microstructure containing a re-entrant unit cell with two orthogonal hinges. 9 During the mechanical tension of a unit cell in the foam, one of the hinges opens along the tensile axis, while the other hinge expands in the lateral direction.
This re-entrant mechanism provides a physical explanation for the negative Poisson's ratio seen in auxetic foams.
While bulk materials with engineered microstructure have been the focus for generating materials exhibiting a negative Poisson's ratio, this is not a requirement for the phenomenon to occur. For example, negative Poisson's ratio has been observed in cubic materials 10,11 and also crystalline SiO 2 . 12 Many theoretical studies of auxetics have been performed on a variety of model systems, for example as done by Wojciechowski. 13 Until recently, all auxetics that were reported were bulk materials. However, Ho et al. recently reported that negative Poisson's ratios can be found in metal nanoplates due to a combination of free surface effects, and a phase transformation under uniaxial loading. 14 While significant effort has been expended in studying negative Poisson's ratio in various bulk materials and structures, we are not aware of such a report in one-atom-thick, or two-dimensional materials such as graphene or boron nitride, which have already been the focus of extensive research over the past decade due to their unique mechanical, electronic, thermal, optical and other physical properties. [15][16][17][18][19][20][21] Therefore, we are interested in examining possible negative Poisson's ratio effects in these two-dimensional nanomaterials.
In this work, we focus for three reasons on a recent arrival to the world of two-dimensional materials, single-layer black phosphorus (BP). First, few-layer BP has recently been explored as an alternative electronic material to graphene, boron nitride, and the transition metal dichalcogenides for transistor applications 22-24 . This is because single-layer BP has a direct bandgap that can be modulated through thickness and strain, 23,25,26 and exhibits a high carrier mobility 23 and large optical conductivity. 27 Furthermore, single-layer BP has a characteristic puckered structure as shown in Fig. 1 (a), which should lead to substantial anisotropy in the mechanical behavior and properties, i.e. the Poisson's ratio, along both the two in-plane and the out-of-plane directions. Finally, the pucker can be thought of as consisting of two orthogonal hinges θ 546 and θ 214 (or θ 314 ). Because these two coupling hinges strongly resemble those seen in the re-entrant unit cell that enable the negative Poisson's ratio in auxetic foams, 6,9 we wish to investigate their impact on the Poisson's ratios in single-layer BP.
In this letter, we report using first principles calculations the existence of a negative Poisson's ratio in a single-layer, two-dimensional material, black phosphorus. In contrast to previous reports on negative Poisson's ratio in bulk auxetic materials with specifically engineered microstructures 10,28 or more recently in metal nanoplates due to surface and large deformation effects, 14 this behavior is intrinsic for single layer black phosphorus, and originates from its unique, puckered structure, where the pucker can be regarded as a reentrant structure that is comprised of two coupled orthogonal hinges. As a result of this atomic structure, a negative Poisson's ratio is observed in the out-of-plane direction under uniaxial deformation in the direction parallel to the pucker, which becomes increasingly negative with both increased tension and compression.

Results
The structure of the single-layer BP is optimized using the first-principles calculations; the computational details are given in the Methods section. Fig. 1 shows via plotting with XCRYSDEN 29 the relaxed structure for a single-layer of BP of dimension 17.69 × 16.74 × 5.29Å. Fig. 1 (a) shows a perspective view that displays the puckered configuration of singlelayer BP. In this puckered structure, each P atom is connected to three neighboring P atoms.
There are two inequivalent P-P bonds in the relaxed structure, i.e r 12 = r 13 = 2.4244Å and r 14 = 2.3827Å, and also two inequivalent bond angles θ 213 = 98.213 • and θ 214 = θ 314 = 97.640 • . The blue box in Fig. 1 (b) shows the unit cell with four P atoms. The two lattice constants are a 1 = 4.1319Å and a 2 = 3.6616Å, and we chose 5.29Å to be the thickness of single-layer BP as it is the inter-layer spacing in bulk BP. 25 These structural parameters are close to the experimental values. 30 The top view depicted in Fig. 1 (b) shows a honeycomblike lattice structure for single-layer BP. Note that two edges in the honeycomb are much shorter than the other four edges. The Cartesian coordinates are set with the x-direction perpendicular to the pucker and the y-direction parallel with the pucker.
The geometric anisotropy introduced by the pucker implies that single-layer BP may exhibit significantly larger ductility, and thus mechanical anisotropy in the x-direction than the y-direction. Because the mechanical behavior of single-layer BP has not been reported in the literature, and to illustrate the anisotropic deformation that is related to the existence of the negative Poisson's ratio, we first discuss the changes in atomic structure that results in single-layer BP from uniaxial deformation in the both the x and y-directions.
We compare the initial relaxed configuration for single-layer BP with its deformed configuration for loading both parallel and perpendicular to the pucker. Having qualitatively illustrated how single-layer BP responds structurally to uniaxial deformation in the x and y-directions, we now focus on the resulting Poisson's ratios. The Poisson's ratios of single-layer BP are shown in Fig. 4 for uniaxial deformation in the xdirection (perpendicular to the pucker). Fig. 4 (a) shows the strain in the y-direction during its deformation in the x-direction. A strongly nonlinear behavior is observed in the figure, especially in its tensile deformation region. The data are fitted to the function y = −ν 1 x + ν 2 x 2 + ν 3 x 3 , where ν = ν 1 = 0.40 can be regarded as the elastic Poisson's ratio, which we define to be the value for infinitesimal deformations, or excluding nonlinear contributions.
Similarly, Fig. 4 (b) shows that the elastic Poisson's ratio in the z (out-of-plane) direction is ν = ν 1 = 0.046. Fig. 5 provides information for the Poisson's ratio of single-layer BP due to uniaxial deformation in the y-direction (parallel to the pucker). Fig. 5 (a) shows that the elastic Poisson's ratio in the y-direction is ν = ν 1 = 1.28, which is about three times larger than the Poisson's ratio in the x-direction. Hence, the elastic Poisson's ratio is highly anisotropic in single-layer BP.
Up to now, we have seen that single-layer BP will contract (expand) in the other two lateral directions, when it is stretched (compressed) in the x or y-direction. However, Fig. 5 (b) shows that single-layer BP expands (contracts) in the z-direction when it is stretched (compressed) in the y-direction; we can calculate that the negative elastic Poisson's ratio is ν = −0.043 in the out-of-plane direction during its deformation in the y-direction. The data is again fitted to the function y = −ν 1 x + ν 2 x 2 + ν 3 x 3 . From the first derivative of this function, i.e ν = −∂y/∂x, we get the Poisson's ratio including nonlinear effects, as shown in Fig. 6. The Poisson's ratio is also negative when nonlinear effects are included for large strains, and becomes increasingly negative with increasing tensile or compressive strain.

Discussion
Mechanistic explanation for the negative Poisson's ratio. We offer two explanations to explain the origin of the negative Poisson's ratio, starting with a mechanistic summary. Fig. 7 illustrates the relationship between the puckered structure and the negative Poisson's ratio, where the motion of atoms for a representative strain increment in the y-direction is illustrated. Specifically, single-layer BP is stretched in the y-direction, i.e the four surrounding atoms are moved along the attached arrows (blue online) in Fig. 7 (a).
The in-plane angles θ 546 and θ 213 increase and strain energy is stored in these two angles. To accommodate the elongation in the y-direction, single-layer BP contracts in the x-direction, i.e atoms 1 and 4 move inward along the x-direction along the attached arrows (red online) in Fig. 7 (b), while the four surrounding atoms 2,3,5 and 6 follow the movement of atoms 1 and 4 (see green arrows). Furthermore, we note that due to the large Poisson's ratio 5 (ν = 1.28) in the y-direction, single-layer BP will undergo substantial contraction in the xdirection during tension in the y-direction. The inward movement of atoms 1 and 4 reduces the two in-plane angles θ 546 and θ 213 , which were increased by the applied mechanical tension in the y-direction. As a consequence, the 1-4 bond becomes more closely aligned with the vertical (z)-direction, which alters the out-of-plane bond angles θ 214 and θ 314 , and which causes strain energy to be shifted from the in-plane angles to the out-of-plane angles via the arrangement of bond 1-4. For ǫ y = 0.1, we find θ 214 = θ 314 = 95.820 • , which is smaller than the initial value (97.640 • ). As a consequence of the movement of atoms 1 and 4 (red online), the single-layer BP thickness (i.e projection of bond length r 14 to the z-direction) is increased, and single-layer BP expands in the z-direction, leading to a negative Poisson's ratio in this direction.
Re-entrant mechanism for the negative Poisson's ratio. We have established that the negative Poisson's ratio in single-layer BP emerges due to its unique puckered atomic structure. Structurally, we note that the pucker can be regarded as two coupling hinges formed by the angles θ 546 and θ 214 (or θ 314 ). These two hinges are orthogonal to each other.
When the hinge θ 546 is opened in the y-direction, the other hinge θ 214 (or θ 314 ) closes along the orthogonal (x)-direction, which results in the expansion and negative Poisson's ratio in the out-of-plane (z)-direction. This coupling hinge mechanism is similar to the re-entrant mechanism previously exploited in auxetic foam. 6,9 In the re-entrant foam, the idealized unit cell is also built based on coupling orthogonal hinges. When one of the hinges is opened in a particular direction, the other orthogonal hinge will expand in a lateral direction, leading to negative Poisson's ratio. In this sense, the pucker in single-layer BP functions as a nanoscale re-entrant structure. We thus expect that negative Poisson's ratios will be intrinsic to other puckered two-dimensional nanomaterials.
In conclusion, we have performed first-principles calculations on a single-layer of black phosphorus to reveal for the first time that a two-dimensional material with puckered structure exhibits a negative Poisson's ratio in the out-of-plane direction. This phenomena should be intrinsic to other puckered two-dimensional nanomaterials due to the fact that the pucker functions as a nanoscale re-entrant structure, which has previously been exploited in bulk materials to induce a negative Poisson's ratio. We close by noting that two-dimensional auxetic materials may lead to novel material functionality as compared to those seen in bulk auxetics due to the unique material properties and functionalities that are intrinsic to 6 two-dimensional nanomaterials.

Methods
First-principles calculation technique. In this section, we detail the computational methodology used in this work. For the ab initio calculations, we used the SIESTA package 31 to optimize the structure of single-layer BP. The local density approximation was applied to account for the exchange-correlation function with Perdew-Burke-Ernzerhof (PBE) parametrization 32 and the double-ζ basis set orbital was adopted. During the conjugategradient optimization, the maximum force on each atom is smaller than 0.01 eVÅ −1 . A mesh cut off of 120 Ry was used. Periodic boundary conditions were applied in the two in-plane transverse directions, while free boundary conditions were applied to the out-ofplane direction by introducing sufficient vacuum space of 15Å. Gamma point k sampling was adopted for wave vector space. Acknowledgements