Probing the low-symmetry structure determined anisotropic elastic properties of rhenium disulphide by first-principle calculations

Materials Today Communications 21 (2019) 100684

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Probing the low-symmetry structure determined anisotropic elastic properties of rhenium disulphide by first-principle calculations

T

Yanqing Fenga,*, Hongyi Sunb,c,d, Junhui Sune, Yang Shena, Yong Youa a

School of Applied Science and Civil Engineering, Beijing Institute of Technology, Zhuhai 519085, China Institute for Quantum Science and Engineering and Department of Physics, South University of Science and Technology of China, Shenzhen 518055, China c School of Physics, Southeast University, Nanjing 211189, China d Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055, China e School of Mechanical Engineering, State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: ReS2 Elastic property First-principle calculations

As a distinct member of layered structure transition metal dichalcogenides, the anisotropic response of rhenium disulphide (ReS2) is important for the potential applications in flexible devices, while the atomic bonding structure determined mechanical properties underlying the distorted low symmetry remains to be well understood. The objective of the present work is to disentangle the atomic-scale structure determined anisotropic mechanical properties of ReS2. The elastic constants of ReS2 are studied by first-principles calculations. Based on the calculated elastic constants, the mechanical properties, such as bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio are obtained via Hill’s approximations. While having a higher in-plane elastic modulus C22 and C33 values, the layered structure has a low-strength shearing elastic constants C55 and C66, making ReS2 more flexible than most studied common transition metal dichalcogenides such as MoS2. Projected phonon density of states (PDOS) along different directions, the electronic density charge distribution as well as the Mulliken charge population are calculated underlying the anisotropic atomic bonding mechanism of this low symmetry structure, which provides an interpretation of the anisotropic mechanical properties.

1. Introduction Two-dimensional (2D) layered materials have attracted a large amount of research interest due to their excellent properties for potential applications [1,2]. The mechanical responses are essential as the materials are utilized under realistic conditions [3,4]. Usually, mechanical properties are closely related to their intralayer and interlayer structure of the crystals. And the atomic bonding is of key importance [5]. For instance, graphene-like layered structure transition metal dichalcogenides (TMDs), such as MoS2 sheets, show high elasticity and Young’s modulus (0.30 TPa for monolayer), but are several times lower than graphene due to the bonding of Mo-S from hybridization states between the p-orbital of S and the 3d-orbital of Mo atoms [6,7]. While lattice bonding plays a crucial role in determining their fundamentally mechanical properties, 2D layered materials investigated are usually inplane isotropic from isotropic covalent atomic bonding environment. Owing to the phosphorus-phosphorus sp [3] hybridization bond from its unique puckered crystal structure, phosphorene was found to demonstrate obvious anisotropic mechanical properties in the two

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orthogonal in-plane directions ∼0.17 TPa (zigzag), 0.04 TPa (armchair) with superior flexibility and much smaller Young’s modulus [8–10]. It was also discovered negative Poisson's ratio under a uniaxial stress along the zigzag direction [11,12]. The anisotropic response endows new features, which is important for wearable and polarized optoelectronics, and can be used for THz-polarized devices and the linear optical polarizers for various purposes [8,13]. Therefore, disassembling the correlation between the mechanical properties of the 2D materials and their bonding structure may be very desirable, which would allow us to tailor the flexible properties for realizing new applications. ReS2 is a distinct member of TMDs with intriguing properties due to its unique structure [14–16]. It crystallizes in a distorted triclinic (1 T) structure (shown in Fig. 1) underlying a quite low symmetry [17,18]. Each Re atom has six neighboring S sites forming Re-S bonds, but the lengths are not equal to each other. Moreover, each Re atom also has three neighboring Re sites, bonding into (Re-Re bonds) a parallelogramshaped Re4 cluster. The Re4 clusters are dimerized along one of the lattice vectors within the van der Waals plane being one-dimensional zigzag Re-Re chains in parallel. It was found that a significant amount

Corresponding author. E-mail address: [email protected] (Y. Feng).

https://doi.org/10.1016/j.mtcomm.2019.100684 Received 28 August 2019; Received in revised form 5 October 2019; Accepted 5 October 2019 Available online 23 October 2019 2352-4928/ © 2019 Elsevier Ltd. All rights reserved.

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Y. Feng, et al.

Vienna ab initio simulation package (VASP) code [27]. For the exchange and correlation functionals, local density approximation (LDA) of the Perdew and Zunger (CAPZ) [28] was used. The van der Waals (vdW) correction was included by using the DFT-D2 method of Grimme [29]. The Brillouin-zone (BZ) integration was done on uniform Monkhorst-Pack [30] grids of 18 × 18 × 18. The plane-wave cutoff energy was set to be 550 eV. During the structural optimization process, the positions of ions were relaxed towards equilibrium until the pressure was less than 0.5 K Pa and the Hellman-Feynman forces became less than 0.01 eV/Å. The elastic constants are calculated from the strainstress relationship by performing six finite distortions of the lattice using the volume-conserving strain technique [31,32]. The displacement of ions during each step is less than 0.001 nm. Phonon dispersion curves are calculated from the finite displacement method as implemented in the PHONOPY package of VASP. The Mulliken charge population are calculated using Materials Studio [33,34]. 3. Results and discussion As shown in Fig. 1, layered ReS2 crystallizes in a triclinic structure with the space groupP1 − Ci1(No.2) [16–19]. Due to the extra valence electron of rhenium atoms, ReS2 exhibits Re-Re bonds creating the zigzag rhenium chains along the lattice vector b-axis (Fig. 1b). The angle between the in-plane unit-cell vectors (b-axis and c-axis) is121.1o from distortion of the crystal structure. The primitive cell (Fig. 1a) consists of eight chalcogen atoms surrounding the Re4 cluster, and the only symmetry operation relating the atoms is inversion with the center in middle of Re1-Re3 bond. The optimized lattice parameters and the independent fractional coordinates are shown in Table 1, which agree very well with the experimental data [18]. The Re-S bond lengths and Re-Re bond lengths are not equal to each other, respectively. All these special bonding environments can give bulk ReS2 special elastic properties, indicating that ReS2 can be anisotropic. Based on theoretical relaxed lattice structure, we perform elastic constants calculations. For triclinic crystal structure, there are 21 independent values and the calculated results are presented in Table 2 in excellent agreement with the recent theoretical values of monolayer ReS215 (here, we make a transformation of coordinates and the interlayer direction is x). C11, C22, and C33 indicate the resistance to linear compression from x, y, and z direction, as shown in Fig. 1c. The values of all the elastic constants satisfy Born’s mechanical stability criteria [35], implying that the system is in a mechanical stable state. And they are comparable to that of distorted 1 T-ReSe2 [36]. We see that the inplane elastic modulus C22, C33 are calculated to be 252.13 GPa, 243.79 GPa, which are higher than other 2D TMDCs with hexagonal lattice [37,38], reflecting the stiff in-plane covalent bonding properties in ReS2 [18]. Moreover, C22, C33 are about ten times larger than the outof-plane elastic modulus C11, in reasonable agreement with the quite weak interlayer van der Waals forces. The low-strength shearing elastic constants C55 = 8.13 GPa, C66 = 7.90 GPa can make ReS2 good solidstate lubricants for future nanoscale mechanical systems.

Fig. 1. (a) Perspective drawing of a primitive cell of bulk ReS2 down the interlayer lattice vector a, The vectors b and c are in-plane lattice vectors. (b) Periodical lattice structure of bulk ReS2. The parallelepiped is the primitive cell of bulk ReS2. Re chains are indicated in red colour. (c) ReS2 bilayer structure. Each layer repeats in x direction.

of bond charges are between the Re-Re dimers, but charge differences between the neighbouring Re and S planes are small [17,19]. The lowsymmetry structure with exceptional bonding makes bulk ReS2 behave as electronically decoupled layers, and remain a direct-band gap to monolayers for photoluminescence, field-effect transistors applications [14,15,19]. Due to the triclinic symmetry, ReS2 are optically and electrically biaxial, leading to considerable anisotropy of electronic, mechanical, transport and optical properties, allowing the potential applications as photodetectors [19,20], solar cell material in photovoltaics [14,21]. Previous theoretical and experimental studies identified the Raman active modes for this lower symmetry and extraordinary weak interlayer coupled structure [19,22]. The mechanical properties of this distorted 1 T structure of ReS2 play an especially critical role in their utilization. Density functional theory (DFT) calculations [15] anticipated the low Young’s modulus of monolayer ReS2 for applications of flexible optoelectronic devices. Yu et al. [21] revealed the elastic modulus of ReS2 for the out-of-plane direction is nearly three times larger than the in-plane ones. Yagmurcukardes et al. [23] demonstrated that fully hydrogenated single layer ReS2 not only enhances the flexibility of the single-layer ReS2 crystal but also increases anisotropy of the elastic constants. Despite of significant progress, the mechanical properties and the anisotropic mechanism of ReS2 underlying this low symmetry structure remain largely unexplored. In this paper, the mechanical responses and the underlying mechanism for bulk ReS2 are investigated by DFT calculations. From the calculated elastic constants, the mechanical properties, such as bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio of polycrystalline ReS2 are obtained via Voigt-Reuss-Hill (VRH) approximations [24,25]. PDOS along different directions, the electronic density charge distribution as well as the Mulliken charge population are analyzed to provide a comprehensive interpretation its mechanical properties of this low-symmetry structure for various applications.

Table 1 The relaxed lattice parameters and independent fractional coordinates of bulk ReS2. The experimental data [18] are listed for comparison. The lengths are in units of Å.

2. Computational details The calculations were performed in the framework of DFT, using the projector augmented-wave (PAW) [26] method as implemented in the 2

Lattice parameter

a

b

c

α

β

γ

Cal. Exp. Fractional coordinates Cal. X Y Z Exp. X Y Z

6.315 6.417 Re1 0.503 0.513 0.298 0.503 0.511 0.297

6.482 6.510 Re2 0.492 0.058 0.247 0.493 0.056 0.248

6.415 6.461 S1 0.210 0.250 0.366 0.217 0.250 0.368

121.4° 121.1° S2 0.276 0.774 0.383 0.277 0.771 0.384

88.3° 88.4° S3 0.696 0.754 0.119 0.698 0.753 0.117

106.6° 106.5° S4 0.761 0.279 0.119 0.756 0.273 0.118

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Table 2 The single-crystal elastic constants of bulk ReS2 (in unit of gigapascals). C11

C12

C13

C14

C15

C16

C22

C23

C24

C25

C26

C33

C34

24.00 C35 −0.77

9.23 C36 0.86

10.17 C44 96.98

0.20 C45 0.13

−0.77 C46 0.59

1.50 C55 8.13

252.13 C56 0.41

61.63 C66 7.90

2.46

0.02

1.10

243.79

−6.23

From the calculated single-crystal elastic constants, the bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (ν) of polycrystalline ReS2 are calculated using the VRH approximations [24,25]. In VRH approximations, the Voigt bulk modulus (BV), the Reuss bulk modulus (BR), the Voigt shear modulus (GV), the Reuss shear modulus (GR) are defined as follows:

BV =

1 2 (C11 + C22 + C33) + (C12 + C13 + C23) 9 9

BR =

1 (S11 + S22 + S33) + 2(S12 + S13 + S23 )

GV =

1 1 (C11 + C22 + C33 − C12 − C13 − C23) + (C44 + C55 + C66) 15 5

GR =

15 4(S11 + S22 + S33) − 4(S12 + S13 + S23) + 3(S44 + S55 + S66 )

Sij are the elastic compliance constants and [Sij] =[Cij]−1. Based on these calculated quantity, according to the Hill empirical average, B, G, E, ν are estimated by:

B=

1 (BR + B V ) 2

G=

1 (GR + G V ) 2

E=

9BG 3B+G

v=

3B-2G 2(3B + G )

The calculated results are presented in Table 3, which are also in good agreement with the recent theoretical values of monolayer ReS2 [15]. The Young’s modulus of ReS2 is found to be 81.6 GPa, much lower than other 2D TMDCs, such as MoS2 [39,40] for the flexible optoelectronic devices. Moreover, according to Pugh’s ratio [41], a B/G ratio above (below) 1.75 is associated with ductility (brittleness). It is seen that, the calculated B/G value is 1.47, indicting a brittle nature of this polycrystalline material. The Poisson’s ratio ν is about 0.2, which is comparable to that monolayer ReS2 [15]. The small ν indicates that the bonding is more directional. To explore the bonding mechanism underlying the mechanical properties of this low symmetry structure, the phonon spectrum (shown in Fig. 2a) along high-symmetry lines connecting high-symmetry points of BZ and the projected phonon density of states (shown in Fig. 2b) along the in-plane (b and c) and out-of-plane (a) lattice vector directions are calculated. No imaginary frequencies are observed throughout the whole phonon spectrum, confirming dynamical stability of the relaxed structure. The unit cell contains 12 atoms, resulting in 36 phonon

Fig. 2. (a) Phonon dispersion of bulk ReS2. (b) PDOS along the interlayer lattice vector a direction, the in-plane lattice vector b and c directions. (c) PDOS along the in-plane different directions.

dispersion branches including 3 acoustic and 33 optical ones, extending up to 14 THz. And due to this low symmetry, the 33 optical modes at Г point are all non-degenerate. Fig. 2b shows that the a-, b- and c-polarized vibrations are well separated from each other, and the popular frequency of b and c-polarized vibrations is visually higher than that of a-polarized vibration, implying a stronger in-plane bonding than the out-of-plane direction. The popular frequency of basal plane b- and cpolarized vibrations is also different, indicating anisotropy between these two directions. Moreover, the projected phonon density of states of in-plane different directions are also calculated, which is shown in ⇀ ⇀ c is c ( b and⇀ Fig. 2c. Here, the direction is labeled by (m, n) asm b + n⇀ the in-plane lattice vector directions). Due to the inversion symmetry, PDOS of the (m, n) direction is same to (-m, -n). From Fig. 2c, we can see that the projected phonon density of states of in-plane different directions are all different to each other, characterizing this in-plane anisotropic bonding environment. To better understand the relationship between the bonding characteristics and the mechanical properties of this low symmetry structure, the difference charge density and the Mulliken overlap population are calculated. The electron accumulation corresponds to the

Table 3 Calculated values of the polycrystalline ReS2 Voigt bulk modulus (BV), Reuss bulk modulus (BR), Voigt shear modulus (GV), Reuss shear modulus (GR), and elastic modulus (G), bulk modulus (B), Young’s modulus E, Poisson’s ratio ν (in unit of gigapascals). Parameter

BV

BR

B

GV

GR

G

B/G

E

ν

75.8

22.4

49.0

51.9

14.8

33.3

1.47

81.6

0.2

3

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Fig. 3. The top (a) and side (b) view of electron density differences of the ReS2 structure. The color scale is equivalent in the panels. Red and blue represent electron accumulation and depletion, respectively. The yellow and cyan spheres are the S and Re atoms, respectively.

characterizing this distorted low symmetry structure. The Re–Re has larger Mulliken population than Re-S, indicating the stronger covalent bonding. But the negative Re-Re antibonding character gives a negative contribution to their structural stability. Those anisotropic atomic bonding mechanism provides an interpretation of the anisotropic mechanical properties of this low symmetry structure.

Table 4 Calculated Mulliken overlap population and bond length (Å). Bond Type

population

bond length

S 3 – Re 1 S 7 – Re 3 S 5 – Re 4 S 1 – Re 2 S 5 – Re 3 S 1 – Re 1 S 5 – Re 2 S 1 – Re 4 S 6 – Re 1 S 2 – Re 3 S 2 – Re 2 S 6 – Re 4 S 3 – Re 3 S 7 – Re 1 S 3 – Re 2 S 7 – Re 4 S 4 – Re 4 S 8 – Re 2 S 4 – Re 1 S 8 – Re 3 S 4 – Re 2 S 8 – Re 4 S 6 – Re 3 S 2 – Re 1 Re 1 – Re 3 Re 3 – Re 4 Re 1 – Re 2 Re 2 – Re 3 Re 1 – Re 4 Re 2 – Re 4

0.54 0.54 0.42 0.42 0.39 0.39 0.41 0.41 0.45 0.45 0.47 0.47 0.39 0.39 0.36 0.36 0.47 0.47 0.41 0.41 0.49 0.49 0.41 0.41 −0.66 −0.64 −0.64 −0.61 −0.61 −0.59

2.32926 2.32926 2.35412 2.35412 2.36043 2.36043 2.36147 2.36148 2.38123 2.38123 2.38266 2.38266 2.41763 2.41764 2.42473 2.42473 2.45134 2.45135 2.46313 2.46314 2.47211 2.47212 2.51632 2.51632 2.70858 2.80216 2.80216 2.81435 2.81435 2.89616

4. Conclusion In conclusion, the elastic properties of the distorted triclinic ReS2 has been probed using DFT calculations. We further discussed atomic bonding mechanism from projected phonon density of states, the electronic density charge distribution as well as the Mulliken charge population analysis, which provides an interpretation of the anisotropic mechanical properties. CRediT authorship contribution statement Yanqing Feng: Writing - original draft, Writing - review & editing, Validation, Investigation, Conceptualization, Methodology, Software. Hongyi Sun: Data curation, Software. Junhui Sun: Software, Validation. Yang Shen: Supervision. Yong You: Supervision. Declaration of Competing Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. There is no professional or other personal interest or any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

localization of electrons, whereas the electron depletion indicates the delocalization of electrons. The localized hybridizations form covalent bonds. The Mulliken bond population characterizes the overlap of electrons between two atoms, which provides an alternative criterion for the ionic and covalent bond characteristic between two atoms. A positive value indicates the bonding character, and the larger bond population, the stronger covalent bonding. The negative one demonstrates antibonding character [42,43]. A zero Mulliken population indicates the ionic bonding [44]. The calculated electron density differences are showed in Fig. 3. The red color represents the accumulation of electrons, and the blue color represents the depletion. We can see that, many accumulated electrons are present between the Re and Re, Re and S atoms of the parallelogram-shaped Re4 cluster, indicating stronger covalent bonds along the zigzag Re-Re chains (the in-plane b direction) in the system. The Mulliken electron population analysis for bond population and bond length are listed in Table 4. It can be observed that the Re–Re and Re-S population are all tiny different to each other,

Acknowledgments This work was supported by Guangdong Province Science Foundation Project (No. 2018A030310001), Beijing Institute of Technology, Zhuhai, Science Foundation Project (No. XK-2018-03), National Key R&D Program of China (No. 2017YFB0702303), the National Natural Science Foundation of China (Grant No. 21373249), Key Program of the Chinese Academy of Sciences (No. QYZDY-SSWJSC009), and Guangdong Province Colleges and Universities Characteristic Innovation Natural Science Project (No. ZX-2017-001). References [1] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009) 109–162. [2] S.Z. Butler, S.M. Hollen, L.Y. Cao, et al., Progress, challenges, and opportunities in two-dimensional materials beyond graphene, ACS Nano 7 (2013) 2898–2926. [3] B.J. Kim, H. Jang, S.-K. Lee, B.H. Hong, J.-H. Ahn, J.H. Cho, High-performance

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