Da
Ke
,
Jinquan
Hong
and
Yubo
Zhang
*
Minjiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou, 350108, China. E-mail: yubo.drzhang@mju.edu.cn
First published on 17th April 2023
Grain boundaries (GBs) with low misorientation angles are interfacing lines connecting sparsely distributed dislocation cores, but high-angle GBs could have amorphous atomic arrangements with merged dislocations. Tilt GBs in two-dimensional materials frequently emerge in large-scale specimen production. In graphene, a critical value for differentiating low and high angles is quite big because of its flexibility. However, understanding transition-metal-dichalcogenide GBs meets additional complexities regarding the three-atom thickness and the rigid polar bonds. We construct a series of energetic favorable WS2 GB models using coincident-site-lattice theory with periodic-boundary conditions. The atomistic structures of four low-energy dislocation cores are identified, consistent with the experiments. Our first-principles simulations reveal an intermediate critical angle of θc ≈ 14° for WS2 GBs. Structural deformations are effectively dissipated via W–S bond distortions especially along the out-of-plane direction, instead of the prominent mesoscale buckling in one-atom-thick graphene. The presented results are informative in studies of the mechanical properties of transition metal dichalcogenide monolayers.
The critical value for differentiating low and high misorientation angles is material-dependent, and crystal dimensionality can be the most fundamental structural determinant. GBs in three-dimensional (3D) bulk materials have been extensively studied, and a small angle of θc ≥ 5–10° is generally cited.2 The smallness of the critical angle is an indication of strong mutual interactions among the nearby dislocation cores. At the microscopical level, these interactions are bridged by long-range stress fields (with mechanical energies exerted on the underline material), which cannot be effectively dissipated in a 3D confined environment. The GB-associated long-range effects in 3D bulk materials can significantly impact material properties, such as sinks for dopants and vacancies.3
Understanding GBs in atomic thin 2D materials is driven by large-scale specimen production via chemical-vapor-deposition techniques, from which GBs frequently emerge. They are tilt GBs since the two misorientated grains are placed on the same plane. 2D materials are generally mechanically soft and have an additional pathway for dissipating stress fields owing to the open geometry along the out-of-plane direction. As a result, graphene GBs were predicted to have a much higher critical angle, which was attributed to the significant buckling induced by the GBs.4 In addition, dislocation structures are significantly reconstructed due to compelling attraction between dislocation cores, resulting in unique asymmetric hillocks. The mesoscale buckling and local reconstruction make a particular high-angle system extraordinarily stable, which changes the overall landscape of GB formation energies.4,5 GB-induced buckling has been experimentally confirmed in graphene6 and hexagonal BN.7
Tilt GBs in transition-metal dichalcogenides (TMDs) have shown various interesting functions relating to the thermodynamic,8,9 mechanical,10 electrical,11 magnetic,12,13 and optoelectronic properties14,15 if rationally engineered.16 Since 2D TMD systems have three atomic layers and polar bonds, they have intermediate flexibility between 3D bulk materials and one-atom-thick graphene. Therefore, a medium critical angle for TMD-GBs is expected from a straightforward geometric analysis. This was previously inferred from an experimental study on their electrical transport properties11 and also suggested by a density-functional-theory simulation.17 However, comparative investigations on TMD-GBs covering both low-angle and high-angle systems are still lacking. Particularly, previous simulations of low-angle systems usually relied on either over-simplified or inappropriate structural models. To contain GBs within a manageable cell size, GB models were incorporated into non-periodic nanoribbons with undesired dangling bonds at the cell edge.
In this work, we carry out density-functional-theory calculations on WS2 GBs that are constructed by using coincident-site-lattice theory with periodic-boundary conditions. Twenty-three misorientation angles are considered, from the lowest angle θ = 1.297° to a high angle of θ = 42.103°. The energy-favorable GB configurations are carefully located. Systematic investigations reveal a critical value of θc ≈ 14° for separating WS2 GBs into low- and high-angle families.
Second, the misorientation angles θ are the only input parameter for generating the models. According to CSL theory, the hexagonal lattice has 26 possible misorientation angles (0° < θ ≤ 60°). Table 1 collects 25 models with 23 misorientation angles up to 42.103°, missing the other three systems having even higher angles. All the structures are fully relaxed and shown in Fig. S1 in the ESI.†
Dislocation cores nd | Misorientation angle θ (degree) | CSL cell size Σ | Formula account N | Lattice constant a (Å) | Lattice constant b (Å) | Dislocation rings |
---|---|---|---|---|---|---|
a The structural relaxation is not well converged because there are too many atoms involved. b The model difference is due to the different thresholds for removing the overlapped atoms. c The model difference is due to the relative translations between the grains. | ||||||
n d = 1 | 1.297a | 1951 | 3853 | 277.35 | 139.81 | 4|6 |
2.134 | 721 | 1430 | 169.08 | 85.01 | 5|7 | |
3.481b | 271 | 537 | 103.75 | 52.10 | 4|6, 6|8 | |
3.481b | 271 | 540 | 104.15 | 52.10 | 5|7 | |
5.086 | 127 | 507 | 142.75 | 35.65 | 4|6, 4|5|7 | |
7.341c | 61 | 243 | 98.92 | 24.74 | 4|6, 6|8 | |
7.341c | 61 | 243 | 98.60 | 24.75 | 5|7 | |
9.430 | 37 | 147 | 76.83 | 19.26 | 4|6, 6|8 | |
13.174 | 19 | 113 | 82.58 | 13.81 | 4|6, 6|8 | |
21.787 | 7 | 56 | 67.55 | 8.38 | 5|7 | |
n d = 2 | 4.723 | 589 | 1162 | 152.44 | 76.83 | 5|7 |
6.609 | 301 | 596 | 109.10 | 54.89 | 4|6, 6|8 | |
8.256 | 193 | 376 | 86.45 | 44.02 | 4|6, 6|8 | |
10.993 | 109 | 430 | 131.18 | 33.08 | 4|6, 6|8 | |
16.426 | 49 | 192 | 87.74 | 22.19 | 5|7 | |
32.204 | 13 | 102 | 90.42 | 11.41 | 4|6, 6|8 | |
n d = 3 | 8.613 | 399 | 518 | 145.32 | 63.25 | 4|6, 5|7, 6|8 |
11.635 | 219 | 284 | 107.77 | 46.89 | 4|6, 5|7, 6|8 | |
15.178 | 129 | 164 | 82.37 | 36.01 | 4|6, 5|7, 6|8 | |
17.897 | 93 | 118 | 69.97 | 30.59 | 4|6, 5|7, 6|8 | |
27.796 | 39 | 74 | 67.97 | 19.76 | 4|6, 5|7, 6|8 | |
38.213 | 21 | 80 | 99.66 | 14.49 | 4|6, 5|7, 6|8 | |
n d = 4 | 18.734 | 151 | 294 | 76.79 | 38.60 | 4|6, 5|7, 6|8 |
26.008 | 79 | 310 | 111.38 | 28.13 | 4|6, 6|8 | |
42.103 | 31 | 182 | 104.65 | 17.62 | 4|6, 5|7, 6|8 |
Third, GB models are classified into four families that are characterized by the number of dislocation cores nd along the period length d of the supercell [Fig. 1(b–e)]. The parameter d is equivalent to the lattice constant b, and they are used interchangeably in this work. The separation of the dislocation cores along the boundary direction equals b/nd, a quantity reflecting the coupling tendency between the dislocations.
Fourth, a unitless parameter Σ is defined to measure the size of each grain (Ωgrain) but is normalized by the WS2 unit cell (Ωunitcell), i.e.,
Σ = Ωgrain/Ωunitcell. | (1) |
The formula account N is in principle twice of Σ (i.e., N = 2Σ), since the GB model contains two grains. The tiny deviations found in Table 1 (e.g., N = 1.975Σ for θ = 1.297°) occur when removing the overlapped atoms between the two grains. Coupling between dislocations perpendicular to the GB direction is a spurious effect in our models. It is prevented by increasing the CSL cell size perpendicular to the GB direction, which leads to more complicated relationships between N and Σ (e.g., N = 5.947Σ for θ = 13.174°).
The relative translation of the two CSL grains along the boundary direction is unfixed by CSL theory. Fig. 1(b–e) show five GB models with the same misorientation angle of θ = 7.341° but different translations. The misorientation angle defined in Fig. 1(d) is the angle between S–S bonding directions in each grain. All the crystal structures are fully relaxed, and their formation energies (defined in eqn (2)) are compared in Fig. 1(a). Obviously, translation is a key degree-of-freedom in determining the dislocation structures, which in turn locates the energetically favorable configurations. For example, dislocation cores in S1 and S3 models are composed of many homo-elemental bonds, i.e., W–W bonds and S–S bonds, which leads to high energies of the corresponding GB models. S2 and S4 models have much lower energies because their dislocation cores are less distorted. In fact, the non-hexagonal distorted rings in S2 and S4 models are the most stable structures found for all the misorientation angles, as shown in the last column of Table 1. It is worth noting that S5 has high formation energy, although it is solely composed of low-energy rings. This is because two dislocation cores are found along the GB direction in S5, indicating that the θ = 7.341° model is accidently pushed from the anticipated family nd = 1 to the family of nd = 2 under a peculiar translation.
We discuss the dislocation structures in greater detail since they play a vital role in determining the physical properties of GB systems including stability. The distorted dislocations in the S4 model (Fig. 1) have two antisymmetric structures in the two GB walls: an S–S 5|7 ring in an S-rich environment and an element-inverted W–W 5|7 ring in a W-rich environment. One may argue that the element-inverted motifs are an artificial effect when enforcing the periodic-boundary conditions along the perpendicular direction. However, the antisymmetric patterns are absent in the S2 model, where the lateral GB wall has a 6|8 ring but the central GB wall has a 4|6 ring. To understand the differences of S2 and S4 models, the dislocations before structural optimization are shown in the left panels of Fig. 2. The 5|7 rings [Fig. 2(a and d)] can keep their overall appearance in the structural relaxation. By contrast, the W-rich 6|8 ring [Fig. 2(b)] can easily transform to an S-poor 4|6 ring [Fig. 2(c)] via dislocation climbing, indicating that the former configuration is an energetic saddle-point. Similar instability is also found for the W-poor 4|6 ring [Fig. 2(f)]. Therefore, we identify four low-energy dislocation motifs, i.e., S–S 5|7 ring, W–W 5|7 ring, S-poor 4|6 ring, and S-rich 6|8 ring, confirmed experimentally (see the right panels of Fig. 2). It is worth noting that most of these non-hexagonal rings have been theoretically reported earlier.5,9,13,17,18 Our construction approach provides an intuitive understanding of the translational degree of freedom in determining the low-energy structures.
Fig. 2 Energetically favorable dislocation rings in WS2 GB models. The left plots (with a white background) are theoretical structures before optimization. W atoms with dangling bonds in subplot (b) and 4-fold coordinates in subplot (f) are energetic saddle-points, and the structural transformations are denoted by pink arrows. The experimental HAADF-STEM images (with a black background) are shown on the right for subplots (a, d, and e), cited from ref. 15. Note: on the right of subplot (c) is the STEM-ADF image of MoS2.13 |
Our identified structures are the ideal least-distorted low-energy configurations of WS2 GBs, especially with low misorientation angles. In addition, stoichiometry is assumed for simplicity. Since real GBs are complicated nonequilibrium defects, energetics is not the only decisive factor for their formation. Nevertheless, our finding provides valuable insights for understanding the considerable inconsistency. For example, it was reported that 6|8 rings are the dominant type over 5|7 cores in WS2, especially under S-rich conditions.9 In contrast, a study of MoS2 GBs only observed 5|7 dislocation cores.11 Third, all the 4|6, 5|7, and 6|8 rings were found in ref. 13.
The ring types may have a direct consequence on the electronic properties, and one notable example is magnetic instability. Among the four rings, we find that the W–W 5|7 ring (specifically, the W atoms in the W–W homoelemental bond) is most likely to develop magnetic moment, as shown in the low-angle system θ = 7.341° in Fig. S4.† However, the magnetic moment is quite small. We carry our further calculation in a more defective high-angle GB with θ = 42.103°. In this case, the highest magnetic moment is about 0.5 μB. A previous report on the significant magnetization is in the presence of anti-site defects.12 We conclude that the magnetic instability is quite weak in low-energy GBs.
(2) |
E(θ, nWS2) is the internal energy of the WS2 GB model with a misorientation angle θ and a formula number nWS2. μWS2 is the internal energy of pristine WS2. The parameter d in the denominator is equivalent to the lattice constant b. Note that 3D bulk materials' GB formation energy is averaged onto the grain interfacing area. By contrast, the atomic layer thickness of WS2 is not well defined, and the formation energy is the energy density over the boundary length. The factor of 2 in the denominator is due to two parallel dislocation walls in each model. Ef(θ) can have two representations, i.e., measured with respect to the GB length (eV nm−1) or dislocation core numbers (eV per core). Empirically, the Ef(θ) (in the unit of eV nm−1) of low-angle GBs follows the Read–Shockley relationship,1,2
Ef(θ) = θE0[A − ln(θ)]. | (3) |
E 0 depends only on the GB orientation and the macroscopic elastic constants. The parameter A depends upon both the misorientation angle and the atomic energy at the dislocation core. Equivalently, the formation energy per dislocation core is,4
Ef(θ) = C − Dln(θ), | (4) |
The first-principles results of 23 WS2-GB models are shown in Fig. 3(a), compared with that of graphene reproduced from ref. 4. Compared with graphene, WS2 GBs not only have higher formation energies at each angle but also lack a V-shaped energy dip at the high-angle end. This can be qualitatively understood, bearing in mind that GB formation energy is composed of (1) local atomistic distortions around the dislocation cores and (2) elastic strain associated with the long-range lattice deformation.4 Graphene has only one atomic layer and is mechanically flexible. GBs in graphene release strains through the local dislocation reconstruction and the prominent out-of-plane buckling (with a height of ∼5 Å). Moreover, dislocation cores of graphene combine into pairs because of their mutual attractions, leading to an energy dip at θ ≈ 32.2°.4 The three-atom-thick WS2 is more rigid than graphene and resembles 3D materials to a large extent. WS2, in the presence of distorted dislocations, has a high resistance to being buckled at the nanoscale. Strains of dislocations are primarily released from the stretch or compression of W–S bonds [Fig. 3(b)]. Due to the out-of-plane freedom, the bond variations are more effective than 3D bulk materials. This finding agrees well with the experimental observation of slight warping in WS2.9
Fig. 3 Electronic properties of symmetrical [0001] tilt grain boundaries. (a) Formation energies as a function of misorientation angles. The energy densities are represented along the dislocation-core-line (eV nm−1, the left y-axis) or averaged to the dislocation core (eV per core, the right y-axis). The solid symbols are DFT results for WS2 systems, and the open marks are for graphene calculated using the force-field approach.4 The solid lines for the low-angles are fitted to eqn (3) or (4). The dotted lines are the direct connection of the data points. (b) Changes of the vertical S–S distances in the system with θ = 7.34°. The inset shows the cut view of the crystal structure, and the green lines mark the out-of-plane warping around the dislocation core. |
The critical value for differentiating low and high-angle GBs is defined as the position, where formation energy deviates from the empirical Read–Shockley relationship (eqn (3) and (4)). This predicts θc ≈ 14° for WS2 and θc ≈ 20° for graphene [Fig. 3(a) and S5†]. Above these angles, the Read–Shockley relationship ceases to work because other energies (due to dislocation coupling) besides the linear-elastic energy come into play. Interestingly, the critical angle can also be used for organizing the electronic properties of WS2 GBs. Fig. 4(c–f) shows the electronic density-of-states of a few systems. Localized in-gap states enhance with respect to the misorientation angles. A quasi-linear relationship holds up to the critical angle [Fig. 4(a)]. Above the critical angle, the massive in-gap states become delocalized, resulting in a notable metallic behavior18 along the GB directions [e.g., Fig. 4(f) for θc ≈ 32.204°].
Footnote |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2na00709f |
This journal is © The Royal Society of Chemistry 2023 |