Mechanism of substrate-induced anisotropic growth of monolayer WS2 by kinetic Monte Carlo simulations

Controlled anisotropic growth of two-dimensional materials provides an approach for the synthesis of large single crystals and nanoribbons, which are promising for applications as low-dimensional semiconductors and in next-generation optoelectronic devices. In particular, the anisotropic growth of transition metal dichalcogenides induced by the substrate is of great interest due to its operability. To date, however, their substrate-induced anisotropic growth is typically driven by the optimization of experimental parameters without uncovering the fundamental mechanism. Here, the anisotropic growth of monolayer tungsten disulfide on an ST-X quartz substrate is achieved by chemical vapor deposition, and the mechanism of substrate-induced anisotropic growth is examined by kinetic Monte Carlo simulations. Results show that, besides the variation of substrate adsorption, the chalcogen to metal (C/M) ratio is a major contributor to the large growth anisotropy and the polarization of undergrowth and overgrowth; either perfect isotropy or high anisotropy can be expected when the C/M ratio equals 2.0 by properly controlling the linear relationship between gas flux and temperature. The anisotropic growth of WS2 is governed by the chalcogen to metal ratio. A team led by Gaofeng Wang at Hangzhou Dianzi University established a substrate-sensitive kinetic Monte Carlo (kMC) model that accounts for the local substrate effects on adsorption, desorption, and diffusion processes, to study the growth anisotropy in atomically thin transition metal dichalcogenides. Using the representative case of chemical vapor deposition growth of monolayer WS2 on ST-X quartz, the physical mechanism of substrate-induced anisotropic growth was investigated using first-principles calculations to obtain the energy parameters for modeling the kinetic events, followed by kinetic quantum Monte Carlo simulations within the kMC framework. The driving force of the anisotropic growth behavior was found to be the chalcogen to metal ratio, which could be controlled by tailoring the gas flux and temperature.


Introduction
Transition metal dichalcogenides (TMDs) have been a star family of 2D materials [1], which is attributed to their excellent elctronic and optical properties.In particular, monolayer TMDs [2,3] typically including MoS 2 , MoSe 2 , WS 2 , and WSe 2 , with remarkable advantages over the few-layers or blocks of such materials, have more potential applications in low-dimensional semiconductors and next-generation optoelectronic devices.[4][5][6][7] Methods including mechanical exfoliation, liquid exfoliation, physical vapor deposition, and CVD have been developed to prepare monolayer or fewlayers TMD materials.[8,9] Among those methods, vapor-phase-based growth approaches like CVD are more desirable due to the potential to scale up and obtain wafer-scale 2D TMDs.
To enable the direct growth of large single crystals and nanoribbons by CVD [10], the flake alignment of 2D materials can be one of key factors and the growth anisotropy should be intensively investigated, since it has been shown that the wafer-scale singlecrystalline graphene can be grown if the initial graphene nuclei have the same orientation [11] and the alignment of monolayer TMD nanoribbons is largely determined by the orientation of the crystal substrate.[12,13] Polycrystals instead of single crystals are produced as randomly nucleated and orientated flakes are more likely to form grain boundaries when adjacent flakes merge together, which will lead to the formation of defects.Currently [14], in many synthesis approaches, 2D TMD materials nucleate randomly on substrates, and their orientation cannot be well controlled.Controlled anisotropic growth of large-area or high-aspect-ratio single crystals is still in the early stage of development, and the detailed mechanism remains unclear, though there have been efforts and attempts on the location-and orientation-controlled growth of monolayer TMDs.[15][16][17] Recently, van der Waals (vdW) substrates such as sappire, mica, graphite [18][19][20][21] have been used in the alignment of as-grown TMD flakes, which is attributed to the vdW epitaxial interaction between the crystal substrate and the monolayer TMDs.The substrateinduced growth anisotropy of 2D TMD flakes has been found in the experiments using vdW substrates.For example, Chen et al. reported a step-edgeguided approach for the aligned or oriented growth of 2D WSe 2 on the C-plane sapphire substrate by CVD and found that at a high temperature (>950 °C), the growth is strongly guided by the atomic steps on the substrate surface; [18] using graphene on Ir(111) as substrates, Hall et al. grew well-oriented monolayer flakes of TMDs by a two-step molecular beam epitaxy (MBE) synthesis.[22] However, the substrate-induced anisotropic growth of 2D TMD samples is almost driven by experimental parameter optimization without understanding the fundamental mechanisms dictating the 2D domain morphology under diverse growth conditions.
Systematic understanding of the anisotropic growth of monolayer TMDs is a theoritical challenge due to the diversity of the involved kinetic mechanisms and the wide range of growth conditions (e.g., atomic flux, C/M ratio, temperature, and substrate conditions).The kMC simulation [23,24], as an excellent tool to investigate the cumulative statistical effects of the kinetic processes at the atomic level, can help quantify these diverse experimental conditions and significantly reduce the number of variables, so that one can develop a unified conceptual framework on the deposition mechanism of 2D compound crystals.Rajan et al. proposed a generalized kMC model with special consideration to CVD reactor parameters for the growth of 2D TMD monolayers, which is predictive of mophological evolution with variations of growth conditions [25].Nie et al. introduced a full-diffusion kMC model coupled with first-principles calculations to study the deposition process of WSe 2 monolayers on graphene, [26] which can reproduce different morphologies such as compact, fractal, and dendrite.Nevertheless, substrate effects [27,28] that can largely determine the growth anisotropy are not introduced to those kMC models.
Study on the substrate-induced anisotropic growth of monolayer WS 2 provides a complete perspective on the growth of 2D materials, especially for large single crystals and nanoribbons, which can significantly advance the progress of next-generation electronics.In this work, considering local substrate effects on the adsorption, desorption, and diffusion processes, a substrate-sensitive kMC model is established to study the growth anisotropy using the representative case of the CVD growth of monolayer WS2 on ST-X quartz.Also, for the quantitative analysis on the growth morphological anisotropy, an equation is formulated to evaluate the AGR of monolayer TMD flakes grown in different conditions.Finally, a proof-of- concept experiment is conducted to validate the kMC simulation.

Substrate-sensitive kMC model
To study the anisotropic growth of 2D TMDs, we propose a full-diffusion kMC model coupling with local substrate effects including substrate adsorption and surface diffusion, which are considered as two key factors that influence the topological evolution of TMD monolayers and would result in the anisotropy of film growth.Compared with the bond-counting model of first order approximation, as shown in Figure 1a, our model additionally does the on-the-fly counting of interactions between source atoms (W or S) and substrate atoms (Si or O).Therefore, substrateinduced variations of adsorption energy and diffusion energy are calculated in real-time and then, specifically, our model can simulate the anisotropic growth of WS 2 on the ST-X quartz substrate.
For the growth of monolayer TMDs, the kinetic events consist of adsorption (on the substrate or TMD flakes), desorption, and diffusion, where diffusion events (see Figure 1b-d) include diffusion of an adatom on the surface of substrate (i.e., surface diffusion), diffusion of atoms between two neighbor atomic layers (i.e., interlayer diffusion), and diffusion of a bonded atom along the flake edges (i.e., edge diffusion).Among those events, substrate adsorption and surface diffusion would be strongly influenced by the specific substrate.In theory, TMD monolyers tend to stay in several predicted structures of minimum potential energy such as triangle and hexagon; while, in practice, it is difficult to predict the morphology of monolayer flakes under the strong influence of local substrate effects of a specific substrate.
First-principles calculations should be conducted before the kMC simulation to obtain energy parameters for modeling the above-mentioned kinetic events.Without loss of generality, the growth of WS 2 monolayers on a ST-X quartz substrate is investigated.ST-X quartz, also called 42.75°Y-X quartz, has been widely used as the substrate of surface acoustic wave (SAW) resonators.[29] Because the normal to ST-X quartz surface is 42.75°rotated with respect to axis Y, there is a periodical sawtooth profile with a slightly tilt on the ST-X quartz surface (see Figure 2a) and the vdW interaction between the WS 2 and the substrate varies peridically along the sawtooth profile.It should be mentioned that the temperature in the CVD process of 2D WS 2 is much higher than the phase-transition temperature of quartz crystals (around 550°C), so the ST-X quartz is in β phase during the growth process.For simplicity, as illustrated in Figure 2b, the ST-X quartz substrate is reduced into a binary model that only has two types of substrate adsorption domain: strong adsorption domain marked by Si and ordinary adsorption domain marked by O. Also, the atomic structure of WS 2 is reduced into a simplified hexagonal nuclei (W 3 S 6 ). Figure 2c displays the top view of WS 2 nuclei and SiO 2 substrate for the following kMC simulations, where the substrate is a 30×30 lattice with a 2×30 'belt' marked by Si of strong substrate adsorption compared to other areas marked by O of ordinary substrate adsorption.

Simulation of substrate-induced anisotropic growth
To perform in-depth analysis on the anisotropic growth, a dynamic analysis framework is first proposed for classifying and measuring the morphology of 2D flakes.To make the anisotropic growth measurable, anisotropic growth ratio or AGR is defined to describe the extent of growth anisotropy and the formula for calculating AGRs is as follows: Also, to make the high-dimensional simulation data readable, the morphological classification of TMD flakes is conducted to extract the growth trend.It has been found that the morphology of TMD flakes ranges from upward triangle to downward triangle and to hexagon due to the variation of conditions such as temperature (T), gas flux (Ra, defined as adsorption rate of W in the kMC model), the ratio of calcogen to metal (C/M ratio), and substrate adsorption contrast (xEads, the ratio of the highest adsorption energy to the lowest one on the same substrate).Figure 3a/b illustrates how to calculate AGR of a specific flake and what exactly the AGR measures (see also Equation ( 1)) when the flake is shaped as triangle or hexagon.Figure 3c-e are selected as three representative morphologies of monolayer WS 2 flakes having AGRs of about 1.0, which are grown with uniform substrate adsorption under different growth conditions.If the strong adsorption belt is enabled, the flake is likely to grow anisotropically along the belt and correpondingly the AGR would increase.For the morphology classification, the flake with the area ratio (to the defined substrate lattice) greater than 20% is set as overgrowth while the one with the area ratio less than 3% as undergrowth according to rules of thumb.Similarly, the flake with AGR > 4.0 is considered as (extremely) anisotropic growth whereas the flake with AGR around 1.0 as isotropic growth.
To explore the relationship between the flake morphology and the growth condition, three groups of kMC simulations are designed, which are presented in Table 2.The anisotropic growth and isotropic growth co-occur when the substrate adsorption is neither large nor small (xEads = 1.25 or 1.5), which agrees with the observations in the early proofof-concept experiment, so the substrate adsorption contrast (xEads) is estimated to be 1.4.As is shown in Figure 4, one can found that the C/M ratio is a major contribution to the surge of growth anisotropy and the polarization of undergrowth and overgrowth; the specific linear relationship between gas flux and temperature can help growth isotrophy or anisotropy maintain in an expected manner.The AGR surges at the vicinity of the C/M ratio of 2.0, as is shown in Figure 4a/c, and too high or too low temperature (or too small or too large Ra) can weaken the dramatic increase, which is observed at the AGR curves of T = 1123, 1173 and Ra = 3, 9, 12. Figure 4b/d show that anisotropic growth is observed as the C/M ratio increases and reaches over 2.0, while at the same time very low temperature or very large Ra would result in overgrowth.At the dashed line of the C/M ratio of 2.0, the morphological classification is diverse and the morphology varies with the change of either temperature or Ra.Moreover, Figure 4c/e provide more details for the calculated results when  the C/M ratio equals to 2.0.It is clearly shown that WS 2 monolayers grow isotropically only in the center area where both the temperature and Ra are not too low or too high, and the anisotropic growth of WS 2 monolayers also has a similar "safe zone".After linear regression calculations, it is found that the domains for perfect isotropic growth and extremely anisotropic growth are near two regression lines (see Figure 4f), i.e., R a = 0.045T − 41.535 and R a = 0.054T − 56.442, respectively.That would be the key to the synthesis of large-area or extremely-anisotropic WS 2 monolayers.In practice, AGR measurement curves and morphological classification maps provide a whole picture and detailed guide for operation.The steep slope of AGR curves displayed in Figure 4a/c should be avoided to improve the controllability and reliability of experiments.Because the gas flow is usually unidirectional during CVD process, the C/M ratio on the same substrate may fluctuate with the position.The temperature and Ra can be optimized according to the morphological classification maps shown in Figure 4b/d.When the C/M ratio is significantly greater than 2.0, the varied range of C/M ratio is still away from the steep slope, therefore, 2D TMD flakes of uniform morphology can be obtained.In addition, although it is challenging to finely control the C/M ratio, the well control of the C/M ratio benefits much more than others.Those two regression lines indicate that, when the C/M ratio is well controlled, the transition between anisotropy and isotropy can be easily made by adjusting either temperature or Ra or both.The above gives a whole picture of the dynamical morphologies in the anisotropic growth of monolayer WS 2 on ST-X quartz substrate.Generally, those results and discussions can also apply to other TMDs and even other 2D materials.

Proof-of-concept experiment
A proof-of-concept experiment is conducted and here a preliminary comparison of the kMC simulation result and the experiment result is presented.As a proofof-concept experiment of the anisotropic growth of monolayer WS 2 , it is conducted independent on the above-mentioned kMC simulations.The experimental is detailed in Appendix B. S source is placed at the entry of gas flow, and more than 30 cm away at the downstream, the W source and substrate are arranged face to face.Specificially, sample #1 is placed at the downstream of the S source and sample #2 is arranged at the further downstream, which is 1.5 cm away from sample #1. Figure 5 shows the growth of monolayer WS 2 on the ST-X quartz substrate at specific locations, where   The AFM measurement results show that most of flakes are monolayers and the monolayer thickness is 0.75 nm (refer to the SI).There is a high degree of consistency in orientation of WS 2 flakes shown in Figure 5.However, the flakes located at the upstream of gas flow have a higher AGR than those located at the downstream, which is due to the variation of the C/M ratio.The temperature variation at the sample substrate can be ignored while the C/M ratio may vary significantly.Usually, the C/M ratio goes down slowly along the direction of gas flow.[33] On the whole, the aspect ratio of flakes is positively proportional to the C/M ratio, which agrees well with Figure 4a/c.Particularly, as is shown in Figure 5f, there is a morphological evolution of WS 2 flakes from trapzoid to triangle.The similar evolution is reproduced in the kMC simulation result, and comparisons of the first, third, and fifth rows in Figure 6 show that the growth morphology evolves from anisotropy to isotropy as the C/M ratio slightly decreases.So the transition from trapzoid to triangle is more likely to be observed at the downstream of gas flow.Moreover, comparisons between the second, third, and fourth rows in Figure 6 indicate that the increase of temperature also drives the morphological evolution from high anisotropy to almost perfect isotropy.The proof-of-concept experiment and its results provide a short-cut yet confident verification of the kMC simulations.

Conclusions
The anisotropic growth of monolayer TMDs such as WS 2 is a critical step to reach the controlled growth, which would be a practical and effective approach to produce large single crystals and nanoribbons.As an initial attempt, we studied the extremely anisotropic growth induced by the specific substrate using kMC simulations, proposed a quantitative model to measure the growth anisotropy and determine whether it is overgrown or undergrown, and provided a phenomenological picture of the morphological evolution of TMD monolayers.
The proposed substrate-sensitive kMC model introduces the on-the-fly calculation of substrateinduced variation of atomic potential energy.With the reduced configuration of the ST-X quartz substrate, it can simplify the kMC computations without loss of physical content.It has been found that both the isotropic growth and the anisotropic growth can be achieved by controlling the temperature and gas flux in a specific linear relationship within a moderate range, while the high polarity of parameters are more likely to result in the overgrowth and undergrowth.Also, the proof-of-concept experiment has been conducted and the comparisons showed that the kMC simulation and the proof-of-concept experiment are well consistent with each other.
In future studies, we will not only construct more complex substrate that is close to real and refine the optimization of parameters for the growth of monolayer TMDs in a specific CVD reactor, but also choose different crystals as substrates for different purposes and change cut types to adjust the contrast ratio of substrate adsorption.That will advance the development of 2D materials, especially for the direct growth of large single crystals and nanoribbons.

Figure 2 .
Figure 2. (a) displays the original atomic structure of WS 2 on the ST-X quartz substrate, (b) and (c) show the simplified configuration of (a) from different views.
(a) and (b) are photographed with amplification of 1000 at two upstream neighbor spots of sample #1; (c) and (d) are photographed with amplification of 1000 at two downstream neighbor spots of sample #1; (e) and (f) are photographed with amplification of 1000 at two separated spots downstream from sample #2.

Figure 5 .
Figure 5. Growth of monolayer WS 2 on the ST-X quartz substrate; (a) and (b) are two neighbor spots at the front side of sample #1; (c) and (d) are two neighbor spots at the rear side of sample #1; (e) and (f) are two separated spots at the rear side of sample #2.

Table 1 .
Energy parameters calculated by QE (unit: eV) † † The adsorption energy varies periodically in a specific range.

Table 2 .
Growth conditions for kMC simulations