A simple method for understanding the triangular growth patterns of transition metal dichalcogenide sheets

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.

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 MoS 2 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.C 2015 Author(s).All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.[http://dx.doi.org/10.1063/1.4933021]The novel physical and chemical properties of graphene, together with its technological applications 1 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 electronic [2][3][4][5][6][7][8] and optical [9][10][11] properties, and many promising applications in catalysis, 12 hydrogen storage, 13 transistors, 14 photovoltaic, 15 and Li-ion batteries. 16espite the variety of compositions and properties, all monolayer TMDs (MX 2 , 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. 17,18For example, on Au substrate MoS 2 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. 21Moreover, the intriguing triangular growth patterns have also been observed in MoS 2 -WS 2 hybrid sheets. 22,23It has been found that the special triangular nanoflakes of TMDs exhibit many interesting physical and chemical properties. 24This unique growth pattern inspires people to investigate the underlying reasons.Although density functional theory (DFT) based calculations 25 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. 19n this work, we develop a simple but effective Matrix method to understand the mechanisms of triangular growth patterns of TMDS.Taking the MoS 2 system as an example, we find that with different chemical potentials of sulfur, the triangular shaped MoS 2 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 MoS 2 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 MoS 2 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 MoS 2 nanoribbons, a vacuum space of 25 Å × 20 Å along the y and z directions is employed with a k-mesh of 7 We first conducted geometry optimization for a perfect single-layer MoS 2 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 MoS 2 unit is calculated to be E MoS 2 = −21.797eV, which agrees well with previous experimental and theoretical results. 22,23our different types of edges that are commonly seen in single-layer MoS 2 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.
Based on the feature of honeycomb structural cell of 2D MoS 2 sheets and previous experimental findings of triangular and hexagonal MoS 2 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 MoS 2 nanoflake shown in Fig. 2(a), the configuration matrix X can be expressed as  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).
In an infinite MoS 2 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 γ I , γ II , γ III and γ IV , respectively, which is a function of the chemical potentials of sulfur and molybdenum.We define the edge formation energy vector Γ as Thus, the average edge formation energy can be expressed as Therefore, the formation energy computed by DFT method in previous work 24 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. 25,26or the MoS 2 nanoribbons with the different edge configurations constructed by the combination of the four edges, as shown in Fig. 3, we have γ where E is the total free energy of the nanoribbons, µ Mo and µ S are the chemical potentials of molybdenum and sulfur.
From Equation ( 3) to ( 6 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: With DFT calculations, we get the binding energy of the nanoflake: E 9II+3III = −245.931eV.
In an equilibrium growth and annealing condition of MoS 2 nanoflakes, we have which equals to the energy of a unit cell of MoS 2 nanosheet, namely -21.797 eV.With Equations ( 3)-( 8), the formation energy of each edge can be solved as following: In CVD process, the chemical potential of sulfur changes in the range of -5.4 eV ∼ -3.8 eV. 24To examine the accuracy of our method, the formation energy of four typical shaped MoS 2 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.
With the different chemical potential of sulfur µ S , 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 < µ S < −5.347 eV, −5.347 eV < µ S < −4.871 eV, and −4.871 eV < µ S < −3.8 eV, controlling the formation energy of nanoflakes with different edge configurations.(i) −5.4 eV < µ S < −5.347 eV Under this condition, the stability follows the order of zigzag-S edge > zigzag-Mo edge > antenna-S edge.However, a hexagonal MoS 2 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 µ S 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.
(ii) −5.347 eV < µ S < −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.
(iii) −4.871 eV < µ S < −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 µ S , 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. 9,17However, when µ S 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 MoS 2 has no priority, and orthohexagonal nanoflakes of MoS 2 can be synthesized in such special chemical condition.Recently, the Wang group reported the synthesis of such hexagonal MoS 2 nanoflake experimentally. 27o 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 µ S (the chemical potential of S).The calculated results for the triangular nanoflakes in shape A, B and C are plotted in Fig. 5 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 group 18 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.
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 MoS 2 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 WS 2 . 28

,FIG. 1 .
FIG. 1. Top and side views of the four different edge configurations of MoS 2 nanoflakes.Zigzag-Mo edge, zigzag-S edge, antenna-S edge, and antenna-Mo edge are labeled as I, II, III, and IV, respectively.

FIG. 2 .
FIG. 2. (a) A MoS 2 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.
), we obtain that E I +II + E III+IV equals E I +IV + E II+III .While from DFT calculations, we get E I +II = −126.373eV E II+III = −137.067eV E I +IV = −135.956eV E III+IV = −146.695eV Thus, E I +II + E III+IV = −273.068eV, and E I +IV + E II+III = −273.023eV, showing a good selfconsistency of the equations.Equations (

107105- 6 S
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 µ S for the triangular nanoflakes with different edge configurations.

FIG. 5 .
FIG. 5. Variation of the average edge formation energy with respect to the edge length N and chemical potential µ S for MoS 2 nanoflakes in shape A (a), B (b) and C (c).
(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 µ S < −4.871 eV, γ decreases with N; when µ S > −4.871 eV, γ increases with N.

TABLE I .
Total energy E total (in eV) and edge formation energy γ (in eV) of the four typical shaped MoS 2 nanoflakes calculated by DFT and our matrix method.