Triangular nanoflake growth patterns have been commonly observed in synthesis of transition metal dichalcogenide sheets and their hybrid structures. Triangular nanoflakes not only show exceptional properties, but also can serve as building blocks for two or three dimensional structures. In this study, taking the MoS2 system as a test case, we propose a Matrix method to understand the mechanism of such unique growth pattern. Nanoflakes with different edge types are mathematically described with configuration matrices, and the total formation energy is calculated as the sum of the edge formation energies and the chemical potentials of sulfur and molybdenum. Based on energetics, we find that three triangular patterns with the different edge configurations are energetically more favorable in different ranges of the chemical potential of sulfur, which are in good agreement with experimental observations. Our algorithm has high efficiency and can deal with nanoflakes in microns which are beyond the ability of ab-initio method. This study not only elucidates the mechanism of triangular nanoflake growth patterns in experiment, but also provides a clue to control the geometric configurations in synthesis.

The novel physical and chemical properties of graphene, together with its technological applications1 have stimulated considerable interest in the search of other two-dimensional (2D) materials. Among them, monolayer transition metal dichalcogenides (TMDs) have currently been in the focus due to their various novel electronic2–8 and optical9–11 properties, and many promising applications in catalysis,12 hydrogen storage,13 transistors,14 photovoltaic,15 and Li-ion batteries.16 Despite the variety of compositions and properties, all monolayer TMDs (MX2, M = Mo, W, V; X = S, Se, Te) share an interesting common feature in structure, namely the nanoflakes of TMDs grow in triangular shapes varying in their edge compositions.17,18 For example, on Au substrate MoS2 sheets grow in triangular shape with the edge length up to 115 μm,19 which has also been observed on single-crystalline sapphire (0001) substrates in chemical vapor deposition (CVD),20 and these triangular nanoflakes can form three-dimensional spirals.21 Moreover, the intriguing triangular growth patterns have also been observed in MoS2-WS2 hybrid sheets.22,23 It has been found that the special triangular nanoflakes of TMDs exhibit many interesting physical and chemical properties.24 This unique growth pattern inspires people to investigate the underlying reasons. Although density functional theory (DFT) based calculations25 were carried out to understand the stability of triangular nanoflakes, the effect of edge configurations on the stability has not been studied yet. In addition, DFT calculation cannot deal with the nanoflakes in micron dimensions containing millions of atoms as observed in experiments.19 

In this work, we develop a simple but effective Matrix method to understand the mechanisms of triangular growth patterns of TMDS. Taking the MoS2 system as an example, we find that with different chemical potentials of sulfur, the triangular shaped MoS2 nanoflakes with different edge configurations have much lower formation energy among all the nanoflakes with other shapes. Our results not only explain the experimental observations of triangular growth patterns and various edge types of MoS2 nanoflakes, but also suggest an appropriate synthesis condition to control the edge configurations of nanoflakes.

Our DFT calculations were performed with the Vienna ab-initio simulation package (VASP)20 at the level of the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE)21 exchange and correlation functional. The wave functions are expanded with a plane-wave basis set having a cutoff energy of 400 eV. For geometry optimization and total energy calculations of 2D MoS2 nanosheets, a vacuum space of 20 Å is applied in the direction perpendicular to the sheet, and a k-mesh of 7 × 7 × 1 is used. While for MoS2 nanoribbons, a vacuum space of 25 Å × 20 Å along the y and z directions is employed with a k-mesh of 7 × 1 × 1. For MoS2 finite nanoflakes, a 20 Å × 20Å × 20 Å supercell is used with a k-mesh of 1 × 1 × 1.

We first conducted geometry optimization for a perfect single-layer MoS2 sheet, which is composed of the hexagonal honeycomb unit cells with one sulfur atomic layer on the top and bottom respectively, and one molybdenum atomic layer in the middle. The optimized structure has a lattice parameter of 3.184 Å, and the binding energy per MoS2 unit is calculated to be EMoS2 = − 21.797 eV, which agrees well with previous experimental and theoretical results.22,23

Four different types of edges that are commonly seen in single-layer MoS2 nanoflakes are taken into account: zigzag-Mo edge, zigzag-S edge, antenna-S edge, and antenna-Mo edge, labeled as I, II, III, and IV, respectively, as shown in Fig. 1.

FIG. 1.

Top and side views of the four different edge configurations of MoS2 nanoflakes. Zigzag-Mo edge, zigzag-S edge, antenna-S edge, and antenna-Mo edge are labeled as I, II, III, and IV, respectively.

FIG. 1.

Top and side views of the four different edge configurations of MoS2 nanoflakes. Zigzag-Mo edge, zigzag-S edge, antenna-S edge, and antenna-Mo edge are labeled as I, II, III, and IV, respectively.

Close modal

Based on the feature of honeycomb structural cell of 2D MoS2 sheets and previous experimental findings of triangular and hexagonal MoS2 nanoflakes, we consider the convex hexagonal nanoflakes with all angles of 120°, and the six edges in the different configurations shown in Fig. 1. We use a matrix to describe such a nanoflake. For instance, for the MoS2 nanoflake shown in Fig. 2(a), the configuration matrix X can be expressed as

X = 0 0 0 2 4 0 0 0 0 2 0 0 0 0 4 0 0 2 0 0 4 0 0 0 ,

where the four columns represent the four types of edges, and the six rows are for the edge length of the six edges characterized by the number of the honeycomb structural unit cells, starting from the top-left edge, as labeled in Figure 2(a).

FIG. 2.

(a) A MoS2 nanoflake with convex hexagonal shape. Edges are numbered I to VI from the top-left edge, corresponding to the row in matrix. (b) Edge length of the hexagonal nanoflake.

FIG. 2.

(a) A MoS2 nanoflake with convex hexagonal shape. Edges are numbered I to VI from the top-left edge, corresponding to the row in matrix. (b) Edge length of the hexagonal nanoflake.

Close modal

In an infinite MoS2 monolayer, one Mo atom is coordinated to six S atoms, and one S atom is coordinated to three Mo atoms, forming a stable structure. However, in a finite MoS2 nanoflake, Mo atoms on the edges no longer have perfect triangular prismatic coordination, and S atoms on the edges no longer have perfect triangular pyramid coordination. Therefore, such unsaturated atoms on the edges would increase the free energy, which is termed as edge formation energy, acting as a controlling thermodynamical parameter in formation and growth of nanoflakes. The average edge formation energy per unit length of an edge (γ) is used to measure the stability of a nanoflake with different edge configurations, namely the lower γ, the lower total free energy, and more stable the nanoflake is.

To calculate parameter γ, we label the formation energy of the four types of the edges in Fig. 1 as γI, γII, γIII and γIV, respectively, which is a function of the chemical potentials of sulfur and molybdenum. We define the edge formation energy vector Γ as

Γ = γ I γ II γ III γ IV .
(1)

Thus, the average edge formation energy can be expressed as

γ = X Γ X
(2)

Therefore, the formation energy computed by DFT method in previous work24 can be calculated now by the simple algebra equations in our Matrix method.

In order to find the formation energy of different edges, a common method is to calculate the total free energy of the nanoribbons with different edge configurations, and subtract the chemical potentials of the species in their corresponding 2D crystalline nanolayers.25,26

For the MoS2 nanoribbons with the different edge configurations constructed by the combination of the four edges, as shown in Fig. 3, we have

γ I + γ II = E I + II 6 μ Mo 12 μ S
(3)
γ II + γ III = E II + III 6 μ Mo 14 μ S
(4)
γ I + γ IV = E I + IV 7 μ Mo 12 μ S
(5)
γ III + γ IV = E III + IV 7 μ Mo 14 μ S
(6)

where E is the total free energy of the nanoribbons, μMo and μS are the chemical potentials of molybdenum and sulfur.

FIG. 3.

MoS2 nanoribbons with four different combinations of the typical edge configurations: (a) I + II, (b) I + IV, (c) II + III, and (d) III + IV.

FIG. 3.

MoS2 nanoribbons with four different combinations of the typical edge configurations: (a) I + II, (b) I + IV, (c) II + III, and (d) III + IV.

Close modal

From Equation (3) to (6), we obtain that EI+II + EIII+IV equals EI+IV + EII+III.

While from DFT calculations, we get

E I + II = 126 . 373 eV
E II + III = 137 . 067 eV
E I + IV = 135 . 956 eV
E III + IV = 146 . 695 eV

Thus, EI+II + EIII+IV = − 273.068 eV, and EI+IV + EII+III = − 273.023 eV, showing a good self-consistency of the equations.

Equations (3)-(6) derived from the nanoribbons are not enough to solve the formation energy of each edge, so that a triangular nanoflake with the shape characterized by the configuration matrix:

X = 0 3 0 0 0 0 1 0 0 3 0 0 0 0 1 0 0 3 0 0 0 0 1 0

is computed to provide another equation:

0 3 0 0 0 0 1 0 0 3 0 0 0 0 1 0 0 3 0 0 0 0 1 0 γ I γ II γ III γ IV = 9 γ II + 3 γ III = E 9 II + 3 III 10 μ Mo 30 μ S
(7)

With DFT calculations, we get the binding energy of the nanoflake: E9II+3III = − 245.931 eV.

In an equilibrium growth and annealing condition of MoS2 nanoflakes, we have

μ MoS 2 = μ Mo + 2 μ S
(8)

which equals to the energy of a unit cell of MoS2 nanosheet, namely -21.797 eV.

With Equations (3)-(8), the formation energy of each edge can be solved as following:

γ I = 5 . 929 eV + 2 3 μ S γ II = 1 . 518 eV 2 3 μ S γ III = 4 . 765 eV 4 3 μ S γ IV = 10 . 696 eV + 4 3 μ S .

In CVD process, the chemical potential of sulfur changes in the range of -5.4 eV ∼ -3.8 eV.24 To examine the accuracy of our method, the formation energy of four typical shaped MoS2 nanoflakes (triangle, parallelogram, hexagon, and pear) were calculated by both DFT and our Matrix method. The calculated results are given in Table I, which shows that the results provided by Matrix method are in good agreement with those obtained by DFT, confirming the accuracy of our method.

TABLE I.

Total energy Etotal (in eV) and edge formation energy γ (in eV) of the four typical shaped MoS2 nanoflakes calculated by DFT and our matrix method.

DFT computationMatrix method
ShapeX matrixEtotalγEtotalγ
 040000100400001004000010 -359.288 2.13645μS -359.460 2.16745μS 
 001002002000010020000200 -162.730 1.16413μS -163.015 1.13813μS 
 020020000200200002002000 -234.789 2.231 -235.098 2.201 
 030000100300200002002000 -277.803 0.428413μS -276.554 0.524413μS 
DFT computationMatrix method
ShapeX matrixEtotalγEtotalγ
 040000100400001004000010 -359.288 2.13645μS -359.460 2.16745μS 
 001002002000010020000200 -162.730 1.16413μS -163.015 1.13813μS 
 020020000200200002002000 -234.789 2.231 -235.098 2.201 
 030000100300200002002000 -277.803 0.428413μS -276.554 0.524413μS 

With the different chemical potential of sulfur μS, the formation energies of the edges change gradually, as shown in Figure 4(d). Hence, different shapes of nanoflakes are determined by the energetics. Based on Figure 4(d), we can clearly see that antenna-Mo edge configuration is energetically most unfavorable, while the relative stability of other three edge configurations depends on the chemical potential of S, which can be discussed in the three ranges: −5.4 eV < μS < − 5.347 eV, −5.347 eV < μS < − 4.871 eV, and −4.871 eV < μS < − 3.8 eV, controlling the formation energy of nanoflakes with different edge configurations.

  • −5.4 eV < μS < − 5.347 eV

FIG. 4.

(a), (b), and (c) Geometries of the three thermodynamically stable triangular nanoflakes. (d) Variation of the edge formation energy with respect to the chemical potential of sulfur μS for the triangular nanoflakes with different edge configurations.

FIG. 4.

(a), (b), and (c) Geometries of the three thermodynamically stable triangular nanoflakes. (d) Variation of the edge formation energy with respect to the chemical potential of sulfur μS for the triangular nanoflakes with different edge configurations.

Close modal

Under this condition, the stability follows the order of zigzag-S edge > zigzag-Mo edge > antenna-S edge. However, a hexagonal MoS2 nanoflake cannot be formed with zigzag-S edges merely. In fact, at least three alternated edges should be either zigzag-Mo edges or antenna-S edges. Therefore, the thermodynamically most stable structure within such μS range should be in a triangular nanoflake, with three zigzag-S edges as long as possible, and three zigzag-Mo edges as short as possible, as shown in Fig. 4(a), named shape A.

  • −5.347 eV < μS < − 4.871 eV

In this range, Fig. 4(d) shows that the zigzag-S edge configuration is more stable than the antenna-S edge one, which is in turn energetically preferable than the zigzag-Mo edge configuration. Therefore the most thermodynamically stable structure should also be in a triangular shape, with three zigzag-S edges as long as possible, and three antenna-S edges as short as possible, as shown in Fig. 4(b), named shape B.

  • −4.871 eV < μS < − 3.8 eV

In this case, the antenna-S edges have the lowest formation energy, and then the zigzag-S edges. Therefore the most thermodynamically stable structure should still be in a triangular shape, with three antenna-S edges as long as possible, and three zigzag-S edges as short as possible, as shown in Figure 4(c), named shape C.

Based on above discussions, three thermodynamically stable configurations are identified for different μS, and all of them have triangular shape with favorable formation energy. This is the reason why triangular growth patterns have been observed in many experiments.9,17 However, when μS is about -4.871 eV, the formation energies of the zigzag-S edges and antenna-S edges are comparable. Therefore, growth of six edges of the hexagonal MoS2 has no priority, and orthohexagonal nanoflakes of MoS2 can be synthesized in such special chemical condition. Recently, the Wang group reported the synthesis of such hexagonal MoS2 nanoflake experimentally.27 

To simulate the configuration evolution during the growth from small to large size in microns for these three stable triangular structures, we study the relationship among γ (average edge formation energy per unit length of the edge), N (the edge length, characterizing by the number of hexagonal unit cells in the edge), and μS (the chemical potential of S). The calculated results for the triangular nanoflakes in shape A, B and C are plotted in Fig. 5(a), 5(b) and 5(c), respectively, where the color with scale bar represents edge formation energy. The Figure shows that the average edge formation energies of shape A and C become smaller when the size N is getting larger. For shape B, when μS < − 4.871 eV, γ decreases with N; when μS > − 4.871 eV, γ increases with N. In other words, in a Mo-rich environment, all three shapes have tendencies to grow thermodynamically, while in a S-rich environment, the growth of shape A and C to large size is preferable but not for shape B. The Lauritsen group18 reported that shape B only exists in the nanoflakes with edge length less than 5, and big nanoflakes are practically in shape C. Based on above discussions, we conclude that the synthesis of nanoflakes in shape A and C should be in a S-rich condition, where the growth of nanoflakes in shape B is unfavorable, while shape C is thermodynamically favorable.

FIG. 5.

Variation of the average edge formation energy with respect to the edge length N and chemical potential μS for MoS2 nanoflakes in shape A (a), B (b) and C (c).

FIG. 5.

Variation of the average edge formation energy with respect to the edge length N and chemical potential μS for MoS2 nanoflakes in shape A (a), B (b) and C (c).

Close modal

In summary, we propose a Matrix method to calculate the edge formation energy of nanoflakes where the configuration of nanoflakes is described with a matrix, and the total formation energy is approximated to be the sum of edge formation energies and the chemical potentials of sulfur and molybdenum. We have identified three thermodynamically stable triangular shapes with different edge configurations for MoS2 nanoflakes which are controlled by the chemical potentials of sulfur and molybdenum. In a Mo-rich synthesis condition, shape A and B are preferable, while in a S-rich synthesis environment, shape C is more favorable. Our Matrix method not only provides the results that agree well with those obtained from DFT calculations, but also it is beyond DFT as it can deal with large nanoflakes in microns containing millions of atoms as observed in experimental synthesis. Furthermore, we can extend the matrix method to other transition metal dichalcogenide nanoflakes like WS2.28 

This work is partially supported by grants from the National Natural Science Foundation of China (NSFC-11174014, NSFC-51471004), the National Grand Fundamental Research 973 Program of China (Grant No. 2012CB921404), and the Doctoral Program of Higher Education of China (20130001110033).

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See supplementary material at http://dx.doi.org/10.1063/1.4933021 for WS2 growth pattern calculated with matrix method.

Supplementary Material