Auxetic B4N Monolayer: A promising 2D material with In-Plane

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Functional Nanostructured Materials (including low-D carbon)

Auxetic B4N Monolayer: A promising 2D material with In-Plane Negative Poisson’s Ratio and Large Anisotropic Mechanics Bing Wang, Qisheng Wu, Yehui Zhang, Liang Ma, and Jinlan Wang ACS Appl. Mater. Interfaces, Just Accepted Manuscript • Publication Date (Web): 22 Aug 2019 Downloaded from pubs.acs.org on August 22, 2019

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ACS Applied Materials & Interfaces

Auxetic B4N Monolayer: A Promising 2D Material with In-Plane Negative Poisson’s Ratio and Large Anisotropic Mechanics Bing Wang,† Qisheng Wu, † Yehui Zhang, Liang Ma,* and Jinlan Wang* School of Physics, Southeast University, Nanjing 211189, P. R. China

ABSTRACT: Auxetic materials, known with negative Poisson’s ratio, are highly desirable for many advanced applications but the candidates are rather scarce, especially at low dimension. Motivated by the re-entrant structure that often exposes negative Poisson’s ratio, we predict a two-dimensional (2D) planar B4N monolayer as a promising auxetic material with unusual in-plane negative Poisson’s ratio within the framework of density functional theory calculations. B4N monolayer also exhibits a highly emerged mechanical anisotropy, characterized by Young’s modulus and Poisson’s ratio. In addition, this monolayer shows superior mechanical flexibility in ideal tensile strength and critical strain values. The phonon dispersion calculations and ab initio molecular dynamics simulations further demonstrate that this monolayer also owns excellent dynamical and thermal stabilities. The fantastic mechanical properties coupled with robust structural stability render the auxetic B4N monolayer promising for future nanomechanical devices. KEYWORDS: Auxetic materials, re-entrant structure, mechanical property, B4N monolayer, first-principles method.

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INTRODUCTION Poisson’s ratio (ν) of a material is defined as the ratio of the transverse contraction strain to the longitudinal extension strain, which measures the fundamental mechanical responses of solids against external loads.1 Most materials expose positive Poisson’s ratio, which means that when experiencing a longitudinal tension (or compression) they will become thinner (or thicker) in the lateral directions.2-3 In contrast, a small group of materials with negative Poisson’s ratio (NPR), namely auextics or auxetic materials, exhibit a counterintuitive behavior in contrast to that of conventional materials, which expand when stretched and contract when compressed.4 The first auxetic material was discovered in 1982,5 and since then, auxetic materials have attracted world-wide attention because of their unique mechanical properties5-8, and highly demanding in many advanced applications, such as tissue engineering9, medicine10, tougher composites11, fasteners12, bulletproof vests, aircraft13, national security, and so on 14-16. Within the last 30 years, a variety of NPR materials and structures have been discovered, synthesized or fabricated, ranging from macroscopic down to molecular levels.17 Although most reported auxetic materials are in bulk form, the negative Poisson’s ratio is possible in two-dimensional (2D) structures according to the theory of Gibson.5 As the blooming of 2D family, several systems with NPR have been predicted theoretically18-29, and only one of them, black phosphorus19-20, has been observed experimentally. To date, the intrinsic 2D auxetic materials are still rather limited, far from meeting the wide range of needs in practice. Seeking for new 2D auextic materials is still a pressing task for modern nanoscale devices. Re-entrant structure is one of the typical auxetic structures, which has been widely used to model the deformation of auxetic foam by many researchers.2,

5, 15, 30-36

Lakes et al.

33

reported the first man-made auxetic material with a re-entrant foam structure. Wang et al.30 designed a novel re-entrant auxetic honeycomb with the in-plane impact responses computationally. Jin et al.

34

and Chang et al.

35

used the re-entrant

structure as sandwich structure core to improve the blast resistance and dynamic 2


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response. Although this structure is allowable in traditional engineered materials, to our knowledge, 2D auxetic material with re-entrant structure has yet to come. Very recently, various 2D boron monolayers (BMs) were predicted theoretically 37-41,

and the χ3 phase (χ3-BM) was successfully synthesized on Ag(111) substrate by

two different groups.42 It is interesting to note that a re-entrant-like structure can be extracted from the χ3-BM. However, the expected NPR is actually absent without negative inclined and vertical strut angle. In this work, we predict a new planar auxetic 2D B4N monolayer, which can be viewed as nitrogen embedded χ3-BM. Our density functional theory (DFT) computations show that the B4N monolayer exposes large anisotropic NPR and high in-plane Young’s modulus. It also owns ultrahigh tensile strains (19% and 18% along x and y directions, respectively), which are comparable to those of graphene. Moreover, the planar B4N monolayer shows excellent thermal, dynamic, and mechanical stabilities as well. RESULTS AND DISCUSSION The typical hexagonal and re-entrant honeycomb structures, as shown in Figure 1a and 1b, respectively, can be numerically distinguished by the length of the horizontal strut h, the length of the inclined strut l, and the angle θ (ϵ[-π/2, π/2]) between the inclined and vertical strut. In such a representation, the Poisson's ratio along x direction (νx) for both hexagonal and re-entrant honeycomb structure with mechanical loading along y-direction, as an example, can be expressed by the Eq. (1):43-45 𝜈𝑥 = ―

𝜀𝑦 𝜀𝑥

=

(ℎ + 𝑙sin 𝜃)sin 𝜃 𝑙(cos 𝜃)2

(1)

Markedly, the sign of νx is determined by the sinθ term in the numerator, and negative angle θ (ϵ [-π/2, 0]) will produce negative νx, that is, the NPR effect. As can be clearly seen from Figure 1, the angle θ is always positive for the typical hexagonal structure but negative for the re-entrant structure, which means that the NPR effect would definitely appear in the re-entrant structure of B4N monolayer but not the regular hexagonal structure. 3



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Figure 1. Sketch for the hexagonal (a) and re-entrant (b) honeycomb structure. The lengths (in angstroms) of the representative chemical bonds of χ3-BM (c), and the atomic configuration for B4N monolayer (d). The unit cells are denoted with the black lines. The blue and green balls represent N and B atoms, respectively. The synthesized χ3-BM consists of waved and double-chained boron atoms that connect each other with regular hexagonal honeycomb hole in-between (Figure 1c). Geometrically, the angle θ for χ3-BM is always positive due to its hexagonal honeycomb structure. Thus, only positive Poisson’s ratio is expected for the χ3-BM monolayer. It is known that B and N atoms can also form strong B-N bonds but their lengths (~1.45 Å in h-BN) would be much shorter than the B-B bonds. It would be very interesting to figure out whether the honeycomb structure can be altered by introducing shorter B-N bonds into the 2D χ3-BM monolayer and even give rise to the negative angle θ with charming NPR. To this end, we design a planar 2D re-entrant structure by injecting N atoms into χ3-BM, namely B4N monolayer, as shown in Figure 1d. After structure relaxation, it is shown that the length of B-N bonds (1.33 Å) in B4N monolayer is much shorter than the B-B bonds in χ3-BM (1.64 ～ 1.72 Å, Figure 1c) and even shorter than that in h-BN ~1.45 Å, which is able to convert the honeycomb structure into a stable re-entrant structure with NPR. More detailed 4


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discussion will be given in the following sections. The structure exposes Cmmm symmetry (space group No. 65) with orthogonal lattice. The optimized lattice parameters are a=2.9693 Å and b=10.7035 Å with B1 at 4j (0, 0.2212, 1/2), B2 at 4j (1/2, 0.1246, 1/2), and N at 2c (1/2, 0, 1/2), respectively. N atoms act as ‘bridges’ by bonding with two B atoms in B4N monolayer. Young’s modulus (E), reflecting the flexibility or rigidity of materials, and Poisson’s ratio (ν), reflecting the mechanical responses of solids against external loads, are two main mechanical parameters of a material. For 2D linear elastic solid materials, E and ν along x and y directions can be derived from the elastic constants as below: Ex = (C11C22-C12C21)/C22,

(2)

νx = C21/C22,

(4)

Ey = (C11C22-C12C21)/C11, νy = C12/C11,

(3) (5)

Table 1. The elastic constants Cij (N/m), Young’s modulus E (N/m), and Poisson’s ratio ν of B4N, χ3-BM, graphene, h-BN, and MoS2. The present results are also compared with previous literature.

structures

reference

C11

C22

C12

C66

Ex

Ey

νx

νy

B4N

Our work

153

267

-4.9

28

153

268

-0.018

-0.032

χ3-BM

Ref. 38

196

208

0.11

0.12

334

334

0.206

0.206

340

340

332

332

Graphene

h-BN MoS2

Our work

349

349

72

138

Ref. 46 Ref. 26

343

343

62

Our work

290

290

64

112

276

276

0.220

0.220

Ref. 47

289.8

289.8

63.7

113.1

275.8

275.8

0.220

0.220

Our work

128.8

127.9

28.8

50

122.3

121.5

0.224

0.225

123

123

0.250

Ref. 48 Ref

46:

Experiment; Ref

26, 38, 47, 48:

Theory.

The calculated mechanical properties of B4N monolayer are shown in Table 1, together with those of other well-studied 2D materials for comparison. The calculated results for known 2D materials are in good agreement with previous experimental/theoretical reports 26, 38, 46-48, demonstrating the reliability of our present 5



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theoretical method. Based on eqs. (2) and (3), the in-plane Young’s moduli along x (Ex) and y (Ey) directions of B4N monolayer are calculated to be 153 N/m and 268 N/m, respectively, exposing strong anisotropy. To gain a comprehensive description of the anisotropic mechanical properties of B4N monolayer, the in-plane Young’s moduli as functions of the arbitrary direction α (α is the angle relative to the positive x direction in this monolayer) were calculated and presented in 2D polar representation curve (Figure 2a). Obviously, the Young’s moduli of B4N monolayer are highly anisotropic in the whole plane. The Young’s modulus E(α) first decreases to its minimum value of 80 N/m at α=45o from a value of 153 N/m at α=0°(x direction) and then increases to a maximum value of 268 N/m at α=90o (y direction). The maximum value (268 N/m) is comparable to that of h-BN monolayer (275.8 N/m)47 and graphene (340 N/m)46 and is about two times of that of MoS2 (123 N/m) 48.

Figure 2. Calculated orientation-dependent (a) Young’s modulus E(α) in N/m and (b) Poisson’s ratio ν(α). (c) Total energy with respect to lattice response of the other direction when the B4N monolayer lattice is under 5% tensile strain along x and y directions, respectively. The arrows indicate the equilibrium magnitude of εx and εy, respectively. (d) The stress-strain relationship of B4N/graphene along x (0o)/zigzag and y (90o)/armchair directions. The arrows denote the strain at the maximum stress. 6


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The diagram of calculated Poisson’s ratio as functions of the arbitrary direction shows that the Poisson’s ratio of B4N monolayer is also spatially varying and highly anisotropic (Figure S1). More remarkably, an intriguing unconventional NPR phenomenon is observed in this Figure for B4N monolayer along both x and y directions. Due to the small values of negative Poisson’s ratio, a zoom of Figure S1 near the center point is plotted in Figure 2b. According to eqs. (4) and (5), the NPRs of B4N monolayer along x and y direction are -0.018 and -0.032, respectively, due to the negative value of C12. The Poisson’s ratio calculated according to the uniaxial strain method is also in good agreement with the results obtained through elastic constants (Figure S2). To confirm this interesting point, we also applied a uniaxial strain of 5% in x and y direction of B4N monolayer, respectively. Just as expected, we find that the equilibrium lattice constants of y and x directions are expanded by ~0.28% and 0.52% (Figure 2c), respectively. The calculated values of NPR are comparable to that of Pmmn-borophene21 (νx= -0.04, νy= -0.02) and black phosphorus19 (−0.027 along z direction). Along the directions other than x and y, B4N monolayers expose the normal positive Poisson’s ratio. The NPR value indicates that B4N monolayer will actually expand in the transverse direction when stretched along x/y direction. In addition to Young’s modulus and Poisson’s ratio, the ideal strength is another important mechanical property of material. We thus investigate the ideal strength of B4N monolayer by calculating the stress-strain curves of x and y directions as depicted in Figure 2d. With applied small strains, the sheet exhibits linear stress–strain relationship with notable elastic anisotropy. As the strain increases, the stress–strain behavior becomes more and more nonlinear and shows enhanced anisotropy. The maximum stress along y direction is higher than that along x direction. The peak stress reaches 25 N/m along the ridges at εy = 18% and 11 N/m at εx = 19% across the ridges, which is secondary only to graphene (34 N/m)

38

but higher than black phosphorus

(16 N/m)49. For comparison, our calculated maximum tensile strains for graphene is 19% and 25% along armchair and zigzag directions, respectively, in good agreement 7



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with the previous report50 (19.4% and 26.6% along armchair and zigzag directions), further confirming the reliability of our computational method.

Figure 3. (a) The lengths (in angstroms) of the representative chemical bonds of B4N monolayer. (b) Charge density of B4N monolayer with the iso-surface of 0.99 e/Å3. (c) Explanation of negative Poisson’s ratio via hinging of ribs in response to applied stretching in x direction. (d) Band structure and 2D Brillouin zone of B4N monolayer. To probe the origin of the NPR, we further analyze the bond lengths of B4N monolayer (Figure 3a). Take the left part of B4N monolayer as an example, the bond lengths of B4-B5 (1.65 Å), B2-B5 (1.61 Å) and B2-B7 (1.65 Å) are shorter than those of B4-B2 (1.81 Å) and B5-B7 (1.81 Å) by ~9% - 11%. The shorter lengths indicate the B4-B5, B2-B5, and B2-B7 B-B bonds are stronger than those B4-B2 and B5-B7 B-B bonds. These bonding strength differences can give rise to the typical re-entrant honeycomb-like structure with negative angle  and the NPR phenomenon (ν < 0). The charge density of B4N monolayer is also plotted to elucidate its bond properties shown in Figure 3b. The higher charge density between B4-B5, B5-B2, and B2-B7 also indicate stronger bond strength than that between B4-B2 and B5-B7 bonds, which is in good agreement with the analysis of bond lengths. Figure 3c displays the detailed structure deformation along the hinging of the diagonal ribs for re-entrant structure in response to an applied uniaxial load in x direction. Under a global tensile loading in x direction (red arrows), the atoms B5 and B6 of this re-entrant structure 8


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move apart, while atoms B2 and B1 maintain their separation. Meanwhile, the tensile strain (red arrows) in x direction of B4N monolayer causes the ribs (B2-B5-B2 and B1-B6-B1 chains) aligned along y direction to separate more (black arrows). As a result, the overall structure expands in y direction, and its area is also apparently expanded while the structure is stretched, leading to negative Poisson’s ratio. Our analyses show that the auxeticity of our novel B4N monolayer is mainly originated from its particular atomic arrangement. We also calculate the projected band structures and partial density of states to probe the electronic properties of B4N monolayer. As shown in Figure 3d, the band lines of B4N monolayer travel across its Fermi level, indicating that it is a nonmagnetic metal. The PDOS analysis shows that the B-p states and N-p states dominate around the Fermi level, which is the principal cause of metallicity (Figure S3). The revealed apparent hybridization between both B-p and N-p states are consistent with the strong B-N bonds. The orbital electronic band structure shows the main contribution to the states around the Fermi level arises from B_pz and N_px orbitals. The metallic property is also maintained and the band structures change slightly under the different external strains (Figure S4).

9



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Figure 4. (a) Phonon dispersion of B4N monolayer. (b) Fluctuation of total energy during AIMD simulation of B4N monolayer at 500 K. The inset is the structure of B4N monolayer at the end of the AIMD simulation after 5 ps. (c) Schematic illustration of preparing B4N. To assess the dynamical stability of B4N monolayer, we examined its phonon dispersion as shown in Figure 4a. The absence of imaginary modes confirms dynamically stability of the B4N monolayer. The highest frequency of B4N monolayer reaches up to 50.84 THz (1695 cm-1), which is comparable to that of graphene (1586 cm-1), 51 indicating the robust B-B and B-N interactions in B4N monolayer. Ab initio molecular dynamics (AIMD) simulations were also conducted to probe the thermal stability of B4N monolayer. Figure 4b shows that the average value of the total potential energy during the AIMD simulations is oscillating within a narrow range and the integral configuration of B4N monolayer is well maintained after 5 ps AIMD simulation, confirming its thermal stability. The mechanical stability was also investigated by using the strain-stress method. For a mechanically stable 10


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orthorhombic 2D sheet, the elastic constants need to satisfy that C11C22-C12C21 > 0 and C66 > 0. 47 Considering the symmetry of B4N, we have C12=C21. Table 1 shows that the calculated elastic constants of B4N monolayer satisfy stability criteria, confirming the mechanical stability of this sheet. Therefore, we can conclude that the B4N monolayer has very good dynamical, thermal, and mechanical stability. Although the demonstrated dynamical and thermal stabilities have already suggested great feasibility to synthesize B4N monolayers, it would be much great to propose any practical synthesis approach. The χ3-BM was successfully synthesized on Ag(111) surface by molecular beam epitaxy.27 In the experimental process, one-dimensional boron chains or nanoribbons were also realized in the process of BM synthesis.17,

39, 41, 21, 52

If the N atoms are injected among the boron chains or

nanoribbons, the strong interaction between B and N atoms may form the B4N monolayer with re-entrant structure. A possible synthesis route for 2D B4N monolayer is therefore proposed in Figure 4c. To evaluate the relative stability between the χ3 B sheet and B4N monolayer, we calculated the formation energy, which is defined as Ef = (EB4N - 4μχ3B - 1/2μN2)/5. EB4N is the total energy per formula unit, and μχ3B and μN2 are chemical potentials of 2D χ3-BM and nitrogen, respectively. The calculated formation energy for B4N monolayer is -0.16 eV per atom. The negative formation energy suggests that the synthesis of the B4N monolayer is exothermic. We also performed a global search for the lowest-energy stable structure for B4N monolayer by using the particle-swarm optimization method as implemented in CALYPSO code.53 The results show that our predicted 2D monolayer is the metastable state which has very close energy to the most stable one (only 18 meV/atom high). More details are given in Supporting Information (Figure S5). Considering the excellent dynamical and thermal stability, the experimental fabrication of B4N monolayer might be feasible although it would be very challenging. CONCLUSIONS To summarize, we have successfully predicted a 2D auxetic B4N monolayer material with attractive negative Poisson’s ratio by using first-principles calculations. 11



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It is revealed that the auxeticity of B4N monolayer mainly originates from its particular re-entrant structure. The intriguing robust in-plane NPR and high Young’s modulus of auxetic B4N monolayer are both highly anisotropic. Besides, B4N monolayer can sustain tensile strains of up to 19% and 18% with maximum stresses of 11 and 25 N/m in the x and y directions, respectively, which is comparable to that of graphene. The thermal, dynamical, and mechanical stability evaluations demonstrate the feasibility of the synthesis of this 2D B4N monolayer and a viable experimental route for the synthesis of B4N monolayer is proposed as well. In view of the novel properties, we wish this new monolayer could be synthesized in near future and applied to the nanomechanical devices. COMPUTATIONAL METHODS. The structure relaxations and self-consistent energy calculations were carried out by using Perdew-Burke-Ernzerhof parameterization of the generalized gradient approximation (PBE-GGA)54 as implemented in the Vienna ab initio simulation package (VASP)55. The projector-augmented plane wave (PAW) approach56 was used to describe the ion-electron interaction. The energy cutoff of the plane wave was set to 500 eV with the energy precision of 10−6 eV and the force precision of 10−2 eV/Å. The Brillouin zone was sampled with a 21 × 7 × 1 Γ-centered k point grid. Phonon dispersions were calculated based on density functional perturbation theory (DFPT) as embedded in phonopy program.57 Van der Waals interactions were considered by using the DFT-D2 method.58 The vacuum length was set to 20 Å. The ab initio molecular dynamics (AIMD) simulations were carried out to evaluate the thermal stabilities of B4N monolayer. At each temperature, AIMD simulation lasted for 5 ps in NVT ensemble with a time step of 1 fs. The temperature was controlled by using the Nosé-Hoover method59. The orientation-dependent Young’s modulus E(α) and Poisson’s ratio ν(α) were calculated as,28, 60, 61 𝐶11𝐶22 ― 𝐶212

𝐸(α) = 4

4

𝐶11𝑠 + 𝐶22𝑐 +

(

𝐶11𝐶22 ― 𝐶212 𝐶44

)

― 2𝐶12 𝑐2𝑠2 12


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ν(α) =

𝐶11𝐶12 ― 𝐶212 2 2 (𝐶11 + 𝐶22 ― )𝑐 𝑠 ― 𝐶12(𝑐4 + 𝑠4) 𝐶44 4

4

𝐶11𝑠 + 𝐶22𝑐 +

(

𝐶11𝐶22 ― 𝐶212 𝐶44

)

― 2𝐶12 𝑐2𝑠2

where c = cosα, and s = sinα. ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI:XXX The

details

about

the

derivation

process

of

Equation

1;

alculated

orientation-dependent the Poisson’s ratio ν(θ) in xy plane (Figure S1); Correlations between applied strain along the x (y) direction and the resultant transverse strains along the y (x) direction (Figure S2); The projected of density of state for B4N monolayer (Figure S3); Band structure of B4N monolayer under different strains (Figure S4); The structure ponon dispersion and B4N monolayer (Figure S5). (PDF) AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]; [email protected], Author Contributions †Bing Wang and Qisheng Wu contributed equally to this work. Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS This work is supported by the National Key Research and Development Program of China (2017YFA0204800), Natural Science Funds of China (21773027,21525311), Jiangsu 333 project (BRA2016353), the Fundamental Research Funds for the Central Universities (2242019R10021), the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1773). The authors thank the computational resources at the SEU and National Supercomputing Center in Tianjin.

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Auxetic B4N Monolayer: A promising 2D material with In-Plane

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