Abstract
Two dimensional (2D) materials consist of one to a few atomic layers, where the intralayer atoms are chemically bonded and the atomic layers are weakly bonded. The high bonding anisotropicity in 2D materials make their growth on a substrate substantially different from the conventional thin film growth. Here, we proposed a general theoretical framework for the epitaxial growth of a 2D material on an arbitrary substrate. Our extensive density functional theory (DFT) calculations show that the propagating edge of a 2D material tends to align along a high symmetry direction of the substrate and, as a conclusion, the interplay between the symmetries of the 2D material and the substrate plays a critical role in the epitaxial growth of the 2D material. Based on our results, we have outlined that orientational uniformity of 2D material islands on a substrate can be realized only if the symmetry group of the substrate is a subgroup of that of the 2D material. Our predictions are in perfect agreement with most experimental observations on 2D materials’ growth on various substrates known up to now. We believe that this general guideline will lead to the largescale synthesis of waferscale single crystals of various 2D materials in the near future.
Introduction
Twodimensional (2D) materials are potentially the most promising materials for future device applications but in practice, waferscale single crystals of various 2D materials are needed to realize these applications^{1,2,3,4}. Recently, the seamless coalescence of millions of wellaligned islands of a 2D material epitaxially grown on a substrate has been successfully used to synthesize waferscale single crystals of graphene^{5,6,7}, hexagonal boron nitride^{8,9}, and MoS_{2}^{10}. This strategy is expected to be generalized to grow various 2D single crystals in the near future. Nevertheless, the unique behavior of 2D materials growth, different from that predicted by classical theory of epitaxy, necessitates the development of a general theory for the epitaxial growth of 2D materials^{5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}.
In graphene CVD growth, the zigzag (ZZ) edge is generally the slowest propagating edge because of its highest barrier for edge propagation^{23,24} and the alignment of graphene on a substrate has been broadly observed to be dependent on the symmetry of the substrate. For example, graphene islands grown on a Cu(111) or Cu(110) surface are wellaligned but those grown on a Cu(100) surface are observed to be along two perpendicular directions^{5,11,12}. Among the three lowindex Cu surfaces, highly robust alignment of graphene can be obtained on the Cu(111) surface and currently, the principal method of graphene single crystal production is by epitaxially growing graphene single crystals on Cu(111) surface^{5,6,7}.
Because of the lower C_{3V} symmetry of hBN, the alignment of hBN on a substrate is different from that of graphene. Three of its six ZZ edges are nitrogen terminated and the other three are boron terminated (named as ZZN and ZZB edges hereafter). In most experiments, the ZZN edge has been proven to be the slowest propagating and kinetic Wulff construction leads to triangular hBN islands enclosed by three ZZN edges^{25,26,27}. In contrast to epitaxial graphene growth, wellaligned hBN islands have rarely been observed. When grown on Cu(111) or Cu(110) surfaces, triangular hBN islands aligned along two opposite directions were found^{13,14,15,28}, while those grown on Cu(100) surfaces had four different orientations^{14}. Recent works have shown that wellaligned hBN islands can be successfully achieved by using a Cu substrate with tailored step edges, thus enabling epitaxial growth of waferscale hBN single crystals^{8,9}.
Similar to hBN, most transition metal dichalcogenides (TMDCs) possess threefold symmetry and present very similar epitaxial behavior on substrates; for example, with two possible alignments on Au(111)^{16}, Al_{2}O_{3}(0001)^{17,18,19}, and GaN(0001)^{20} surfaces. Wellaligned WS_{2} islands have been grown on hBN surface^{21} and nearly wellaligned WSe_{2} islands have been grown on a vicinal Al_{2}O_{3}(0001) surface^{22}. Most recently, centimeter scale singlecrystalline MoS_{2} was obtained by the coalescence of wellaligned MoS_{2} grains on a vicinal Au(111) surface^{10}.
All these experimental observations strongly indicate that the alignment of a 2D material on a substrate depends on both its symmetry and that of the substrate and a general theory for 2D materials epitaxy that helps to predict the alignment of various 2D materials on different substrates is highly desirable to serve as a guideline for experimental design.
Here, based on extensive density functional theory (DFT) calculations, we present a general theory to explain how the alignment of a 2D material on a substrate is intimately related to its interaction with the substrate and how the epitaxial growth of the 2D material on a substrate is critically dependent on the interplay of the symmetries of the 2D material and the substrate. Our theory explains most known experimental observations on 2D material epitaxial growth and hence can serve as a guideline for the experimental synthesis of various 2D single crystals, as well as 2D polycrystalline materials with designed grain boundaries.
Results
2D material–substrate interaction and the alignment of a 2D material on a substrate
There are hundreds of important 2D materials and the possible substrate types are also of the same order of magnitude. So, it is impossible to calculate the interactions of all possible combinations of 2D materials and substrates, which is greater than 100,000. Without losing the generality, we can classify the interactions between 2D materials and various substrates into two sceneries:

(i) The edge of the 2D material is terminated by the substrate, such as graphene or hBN on an active metal substrate, where the strong interaction between the edge of the 2D material and the pristine substrate facet determines the alignment of the 2D material and its epitaxial growth behavior^{29,30};

(ii) The edge of the 2D material is selfpassivated or terminated by active atoms from the environment of its growth, such as H or OH groups^{31,32,33,34}, where the weak interaction between the bulk of the 2D material and the pristine substrate facet dominates the alignment of the 2D material.
To establish an epitaxial relationship between a 2D material and a substrate for scenery (i), we firstly explore the interaction between the edges of graphene or hBN with the three lowindex Cu surfaces, namely Cu(111), Cu(100), and Cu(110). The calculated binding energies between a graphene (hBN) ZZ(ZZN) edge on the three lowindex Cu surfaces as a function of the angle of edge alignment that is defined as the angle between the edge and a Cu〈110〉 direction of the substrate, are shown in Fig. 1 (please refer to method section and Supplementary Figs. 1–6 in Section 1 of supplementary information (SI) for more details on the calculation and modeling). We clearly see that on each of the three lowindex surfaces, the strongest binding energy appears when the ZZ(ZZN) edge of graphene (hBN) is along a Cu〈110〉 direction of the surface; the difference between the binding energy minimum and maximum is significant, >0.2 eV per edge atom. Hence, on a Cu surface, a wellaligned small graphene or hBN island of ~2 nm (which has only ~200 atoms of which ~40 are at the edge) has an energy advantage of >8 eV over misaligned ones. This binding energy difference is large enough to maintain a growing graphene or hBN island in a wellaligned configuration on a Cu surface.
To elucidate the reason behind the strongest binding of a ZZ(ZZN) edge along the 〈110〉 direction on a Cu surface, we have plotted the electron density distributions of Cu(111), Cu(100), and Cu(110) surfaces, respectively, in Fig. 2a–c. It can be seen that on all the three surfaces, the isosurface fluctuation in electron density is the lowest along the 〈110〉 direction, indicating that the closepacked 〈110〉 atomic rows form a pattern with alternative ridges and valleys of uniform height on the surface. In a straight ZZ(ZZN) edge of graphene(hBN), the less stable edge atoms form a straight line and this straight edge is preferentially passivated by either a ridge or a valley of the Cu surface instead of crossing over ridges and valleys on the surface, which results in distortion of the edge. To further illustrate preferential passivating of the graphene ZZ edge by a closepacked atomic row, we compare the atomic structures of the interfaces and the charge density differences of the graphene edge along both 〈110〉 and other directions on the three lowindex Cu surfaces (Fig. 2d–i). When the graphene ZZ edge is aligned along the Cu〈110〉 direction, all the edge atoms are well passivated by a Cu〈110〉 atomic row and the edge remains straight. In contrast, if the graphene edge is along another direction, some of the edge atoms are poorly passivated and the edge is no longer straight because of the fluctuating ridgevalley pattern of the surface. The above analysis clearly shows the superiority of the close packed direction of a substrate in passivating a highsymmetric edge in a 2D material.
Our analysis of a straight graphene/hBN edge preferring the direction of valley or ridge of the isosurface of Cu substrate can be generally applied to most combinations of 2D materials on various substrates. It is noted that the lattice constant of graphene/hBN ZZ edge matches that of the 〈110〉 direction of Cu substrate well. Thus, it is worth to consider a system without perfect latticematch. To address the effect of latticemismatch, we consider the interaction between a ZZ edge of graphene and the Pt(111) surface, where the lattice constant of graphene ZZ edge is about 12.6% smaller than that of Pt〈110〉 direction. The binding energies of a graphene ZZ edge on the Pt(111) surface as a function of the alignment angle of the graphene edge are shown in Supplementary Fig. 7. As expected and similar to that of a graphene ZZ edge on Cu(111) surface, the graphene ZZ edge prefers the alignment along a 〈110〉 direction of the Pt(111) surface. As an example, above calculation suggests that the conclusion that the slowest propagating (also highsymmetric) edge of a 2D material prefers to align along the highsymmetric direction of an active metal substrate, regardless of the latticematch between the 2D material and the substrate.
If the edge of a 2D material is selfterminated, such as the seleniumterminated edges of TMDC materials^{33}, or terminated by the H or OH functional groups, such as the edges of graphene or hBN grown on a less active metal substrate^{31,32,34}. In such a scenery, the interaction between the 2D material edge and the substrate is no longer very strong and the dominating interaction is the weak interaction between the bulk of the 2D material and the substrate. To elucidate the alignment of a 2D material on a substrate for scenery (ii), we calculate the interaction between hBN and Cu(111) (Fig. 3a, b, and Supplementary Fig. 8) and Au(111) (Supplementary Fig. 9) surfaces, respectively, by using the periodic boundary condition models and the interaction between a triangular WS_{2} cluster and the hBN surface (Fig. 3c, d and Supplementary Fig. 10). All these calculations, together with previous studies of graphene on Cu(111) surface^{35} and MoS_{2} on Al_{2}O_{3}(0001) surface^{36}, show that a highsymmetric direction of a 2D material (ZZ directions of graphene, hBN and TMDCs) prefers to align along the highsymmetric directions of a substrate, such as the 〈110〉 directions of Cu(111) and Au(111) surfaces, and 〈11\(\bar 20\)〉 direction of hBN and Al_{2}O_{3} (0001) surfaces.
The above results allow us to draw a conclusion of the alignment of a 2D material on an arbitrary surface, i.e., a highsymmetric direction of the 2D island prefers to align along a highsymmetric direction of the substrate. Although we just explored a very limited systems of 2D materials grown on various substrates, as will be seen later, this rule is in accordance with most experimental observations on the epitaxial growth of various 2D materials, such as the CVD synthesis of graphene, hBN, and TMDCs on various transition metals or nonmetallic substrates (please refer to Section 2 in SI). Thus, we believe that this rule can be applied for the epitaxial growth of various 2D materials on different substrates.
The alignment of 2D materials on various substrates
Having established the principle that determines the alignment of a 2D material on a substrate, we proceed to discuss the interplay between the symmetry of a 2D material and that of the substrate in epitaxial growth. Let us consider a 2D material with a G_{2D} symmetry group on a substrate with a symmetry group of G_{sub}. The symmetry group of the whole system, G_{2D@Sub}, must be a subgroup of either G_{2D} or G_{Sub} because any symmetry operation of G_{2D@Sub} will not change the alignment of the 2D material or the substrate. As shown in SI, we have proved that the number of equivalent but different directions of a 2D material on a substrate can be calculated by
where \(\left {{\mathrm{G}}_{{\mathrm{sub}}}} \right\) and \(\left {{\mathrm{G}}_{2{\mathrm{D}}@{\mathrm{sub}}}} \right\) are the orders, or numbers of different symmetry operations, of G_{sub} and G_{2D@sub}, respectively.
According to the principle of 2D materials alignment discussed in the previous paragraph, the presence of a highsymmetry edge of a 2D material along a highsymmetry direction of the substrate ensures that the symmetry group of the whole system, G_{2D@Sub}, is the largest subgroup of both G_{sub} and G_{2D}. We have considered various combinations of the symmetries of the 2D material and the substrate and the numbers of equivalent but different alignments of various 2D materials are shown in Table 1.
Without loss of generality, we have used fcc(111), fcc(100), fcc(110), and hBN(0001) surfaces as different examples of a substrate with 6, 4, 2 and 3fold symmetries to illustrate the alignment of various 2D materials on them. Figure 4 presents the various ways in which 2D materials with 2, 3, 4 and 6fold symmetries are aligned on these substrates. From the figure, we can deduce that in order to keep the whole system with the highest symmetry, there are:

1, 2, 1 and 1 equivalent but different alignments for a 6fold symmetric 2D material on 6, 4, 2 and 3fold symmetric substrates;

2, 4, 2, and 1 equivalent but different alignments for a 3fold symmetric 2D material on 6, 4, 2 and 3fold symmetric substrates;

3, 1, 1 and 3 equivalent but different alignments for a 4fold symmetric 2D material on 6, 4, 2 and 3fold symmetric substrates;

3, 2, 1 and 3 equivalent but different alignments for a 2fold symmetric 2D material on 6, 4, 2 and 3fold symmetric substrates, respectively.
The number of equivalent but different alignments of various 2D materials on substrates of different symmetries are shown in Fig. 4 and these numbers are in perfect agreement with the symmetry analysis shown in Table 1. Besides the number of equivalent but different alignments of a 2D material on a substrate, Fig. 4 also gives the misalignment angles of equivalent islands of 2D materials. On the substrates with 6, 4, 3, 2fold symmetries, the misalignment angles are \(\frac{i}{3}\pi ,\frac{i}{2}\pi ,\frac{{2i}}{3}\pi ,i\pi ,i = 1,2,3,\)… respectively.
It is important to note that a highsymmetric edge of a 2D material along a highsymmetric direction of a substrate is critical for the above analysis. If the highsymmetric edge of the 2D material is along a lowsymmetric direction of the substrate with mirror symmetry, the symmetry group G_{2D@Sub} will have no mirror symmetry and the least number of equivalent but different alignments of the 2D material will be 2, which makes orientational uniformity impossible.
Comparison with experimental observations
We have summarized most of known experimental obversions on 2D materials epitaxial growth, including graphene growth on lowindex Cu surfaces (Supplementary Table 1), hBN growth on lowindex Cu surfaces (Supplementary Table 2), TMDCs growth on various lowindex substrates, including Al_{2}O_{3}, Au, GaN, hBN (Supplementary Table 3), and various 2D material grown on different highindex substrates (Supplementary Table 4). It is interesting to note that there is a perfect agreement between the experimentally observed numbers of alignments and misalignment angles of 2D materials on these substrates with those predicted by our theoretical analysis. Hence, we believe that the epitaxial relationship is valid for the epitaxial growth of most 2D materials grown on different substrates.
Strategies toward the epitaxial growth of various 2D single crystals
From Eq. (1), we can see that once the symmetry group of the substrate, G_{Sub}, is a subgroup of the 2D material, G_{2D}, or in other words
the symmetry group of the system G_{2D@Sub} could be same as G_{Sub}. This means that there is only one most preferential alignment of the 2D material on the substrate and the orientational uniformity of a large number of islands of the 2D material is possible. As shown in Table 1 and Fig. 4, this can be applied to 2D materials with 6fold symmetry, such as graphene on a 6, 3 or 2fold symmetric substrate, 2D materials with 4fold symmetric on a 4 or 2fold symmetric substrate, 3fold symmetricy 2D materials on a substrate with 3fold symmetry, and a 2fold symmetric 2D material on a 2fold symmetric substrate. Up to now, the seamless stitching of wellaligned graphene islands have been realized on Cu(111) and Ge(110) surfaces, both of which are in accordance with the above described analysis^{5,6,7,37,38}.
Among the thousands of known 2D materials, most of them have the 3fold symmetry, such as the most explored hBN and TMDCs. As shown above, a substrate with 3fold symmetry is expected to be suitable for the epitaxial growth of 2D materials with a threefold symmetry, but it is difficult to find proper lowindex substrates with 3fold symmetry. Although some lowindex substrates has the 3fold symmetry, such as the Cu(111) surface and the Al_{2}O_{3}(0001) surface, the atoms of the top layer of the substrate generally have a higher symmetry, such as the atoms of the top Cu(111) layer have the 6fold symmetry. So, for a C_{3V} 2D material on a C_{3V} substrate, there are generally two deep local minima in the formation profile which corresponds two high symmetric configurations, such as the 0˚ and 60˚ configurations of a WS_{2} on the hBN surface shown in Fig. 3d and Supplementary Fig. 10, and the 0˚ and 60˚ configurations of MoS_{2} on the Al_{2}O_{3} (0001) surface in Fig. 3h of ref. ^{34} (please refer to Supplementary Fig. 11 for the configuration difference). Experimentally, antiparallel TMDC islands on 3fold symmetric substrates are generally seen and parallel aligned TMDC islands could be realized by precise control of the experimental condition (Supplementary Table 3)^{17,18,19,20,21}. From Eq. (2), we can see that if the substrate has the C_{1} symmetry group, the condition for epitaxial growth can be satisfied for any 2D materials, implying that on a substrate with no symmetry, we may be able to achieve orientational uniformity for any 2D materials. In practice, substrates with very low symmetry could be a highindex surface or a vicinal surface of a lowindex surface. As illustrated in Fig. 5, a highindex surface has a large number of lowindex terraces connected by parallel step edges. These step edges of the substrate can serve as nucleation sites to initiate the growth of the 2D material. Furthermore, these step edges interact preferentially with an edge of the 2D material to promote the orientational uniformity of the 2D material. In this manner, the orientational uniformity of various 2D materials has been widely observed. As listed in Supplementary Table 4, epitaxial growth of wellaligned hBN islands have been observed on Cu(102), Cu(103), and vicinal Cu(110) surfaces^{39,40}, where one of the three edges of the triangular hBN island is aligned along the step edge of the substrate. In addition, wellaligned WSe_{2} islands were also observed on Al_{2}O_{3}(0001) surface with step edges^{22}. Recently, such a strategy has been used to grow wafer scale singlecrystalline hBN on vicinal Cu(110) surface and Cu(111) surface with step egdes^{8,9}, and centimeter scale singlecrystalline MoS_{2} on an Au(111) surface with aligned step edges^{10}. DFT calculations in these studies have shown that the weak interaction between the bulk of the 2D material and the substrate and/or the strong interaction between the edge of 2D material and step edge of the substrate lead to the favorable alignment of the 2D islands along the step edges of the substrate^{8,9,10,39,40}. Since highindex surfaces can be easily obtained by miscutting a single crystal, we believe that this could be a general strategy for the synthesis of various 2D materials in the future.
Discussion
We would like to note that the fact that a high symmetric direction of a 2D material prefers to align along a high symmetric direction of a substrate, which is revealed by our extensive DFT calculations in this study, is the foundation of our main conclusion. Otherwise, orientational uniformity of a 2D material on a substrate, even the symmetry group of the substrate is a subgroup of that of the 2D material, cannot be realized. Currently, most previous studies on the synthesis of largesized single crystalline hBN and MoS_{2} employed vicinal (111) or (110) surfaces with parallel step edges^{8,9,10}, where 2D materials tend to align along these step edges and the general study on the epitaxial 2D materials growth on various high index surface is very rare. Besides the vicinal surfaces which are close to one of the lowindex surfaces, our study also predicts that the highindex surfaces that are largely deviated from all the lowindex surfaces, such as the (123) surface of an fcc material, are also ideal for the epitaxial growth of largescale singlecrystalline 2D materials. Furthermore, our theory provides a principle to determine the alignment of a 2D material on any given substrates and, thus, it offers a strategy of synthesizing 2D materials with welldefined grain boundaries, for instance, polycrystalline graphene with grain boundaries of a 30^{o} misorientation angle can be synthesized on an fcc(100) surface, similar to the case of graphene growth on a liquid Cu surface^{41}.
In conclusion, our study clearly demonstrates that the interplay of the symmetries of the 2D material and the substrate is critical for the epitaxial growth of 2D materials. Both theoretical analysis and experimental observations show that a highsymmetric direction of a 2D material tends to be aligned along a highsymmetric direction of a substrate, so that the 2D material–substrate system has the highest possible symmetry. Based on the symmetry analysis and the rules for the preferential alignment of a 2D material on a substrate, we established a library of the different possible alignments of various 2D materials on different substrates to serve as a guideline for experimental design. Furthermore, we theoretically proved that the epitaxial growth of a 2D single crystal can be realized only if the symmetry group of the substrate is a subgroup of that of the 2D material. To meet the requirement for singlecrystalline 2D material growth, we propose using substrates with highindex surfaces which have lower symmetry to template the epitaxial growth of various 2D materials; this strategy has been successfully demonstrated (Nature 570, 91 (2019); Nature 579, 219 (2020); ACS nano 14, 5036 (2020)) and is in agreement with experimental observations. After submission of this manuscript, we have noticed that same strategy has been employed for the epitaxial synthesis of singlecrystalline nanoribbons of TMDCs (Nat. Mater. DOI 10.1038/s4156302007954 (2020)), which further validated the proposed approach of synthesizing large area singlecrystal 2D materials. Our study thus provides a theoretical foundation for the synthesis of waferscale 2D single crystals.
Methods
DFT calculations of edge binding energies
All DFT calculations were carried out via the Vienna ab initio simulation Package (VASP)^{42,43}. The exchangecorrelation effect was treated by the PerdewBurkeErnzerhof generalized gradient approximation (GGA)^{44}. The interaction between valence electrons and ion cores was described by the projected augmented wave (PAW) method^{45} and the kpoint mesh was sampled by a separation of 0.03 Å^{−1}.
To calculate the binding energy of a graphene ZZ edge to Cu (111), (100), and (110) surfaces, Cu slabs consisting of three (111), (100), or (110) atomic layers were constructed to mimic the Cu substrates. Graphene nanoribbons along the ZZ direction, which were three hexagons wide and one of the two ZZ edges passivated by hydrogen, were constructed. Because of the incommensurate lattice constants between the graphene ZZ nanoribbon and the Cu substrates, only a small number of periodic structures can be constructed with a graphene ZZ nanoribbon adsorbed on lowindex Cu surfaces along different directions, of which the number of atoms is not too large and can be handled by DFT calculations, as shown in Supplementary Figs. 1–3. The initial distance between the graphene ZZ nanoribbon and the Cu substrate is set to 3.06 Å, which was estimated from DFTD2 calculations^{46}, and it is the typical equilibrium distance between a graphene layer and the underlying Cu substrate surface. Structure optimization was conducted with the atomic positions of the lowest Cu atomic layer fixed. In addition, the vertical positions of carbon atoms that are passivated by hydrogen were also fixed during structure optimization. To eliminate the imaginary interaction between periodic images along the vertical direction, a vacuum layer with a 15 Å thickness was used to separate the Cu slabs. All the structures were relaxed until the force on each unfixed atom was <0.01 eV/Å, with an energy convergence of 10^{−4} eV.
In a similar manner, hBN nanoribbons along the ZZ direction were also constructed with their ZZB edges passivated by hydrogen. The structures with hBN ZZ ribbons adsorbed on lowindex Cu substrate surfaces are shown in Supplementary Figs. 4 and 6. The distance between hBN nanoribbon and the Cu substrate is set to be 3.10 Å, which was obtained by optimizing a hBN sheet on Cu(111) surface by DFTD2 calculations^{46}. During structure optimization, the vertical positions of boron atoms passivated by hydrogen were fixed. In addition, the atomic positions of the lowest Cu atomic layer were also fixed.
The method similar to above calculations was also employed to calculate the binding energies between the graphene ZZ edge and Pt(111) surface. The calculated structures are shown in Supplementary Fig. 7.
The binding energy of a ZZ edge of graphene or hBN to the substrate is defined as
where E_{ribbon}, E_{sub}, and E_{total} are, respectively, the energies of the nanoribbon, the substrate and the total energy of the nanoribbon adsorbed on the substrate; L is the edge length of the nanoribbon.
To calculate the weak interaction between hBN wall and Cu(111) or Au(111) surface, a hBN layer was stacked to a Cu(111) or Au(111) slab consisting of three atomic layers under periodic boundary condition and with different alignment angles. The calculated structures are provided in Supplementary Figs. 8 and 9. During structure optimization, the bottom atomic layer of the metal slab was fixed. The binding energy between the hBN wall and the metal substrate is defined as
where E_{total}, E_{hBN}, and E_{sub} are the energies of the whole system, the hBN layer and the substrate, respectively. N_{BN} is the number of BN pairs of the hBN layer in the unit cell of the whole system.
To calculate the weak interaction between WS_{2} and hBN layer, a triangular WS_{2} cluster consisting of 60 S atoms and 27 W atoms was placed on a hBN layer with different orientations (Supplementary Fig. 10). Because the edges of the WS_{2} cluster are passivated by S and the hBN layer is chemically inert, the interaction between the WS_{2} cluster and the hBN layer should be dominated by the WS_{2} wall and the hBN layer. The binding energy between the WS_{2} cluster and the hBN layer is defined as
where E_{total}, E_{hBN}, and E_{WS2} are the energies of the whole system, the hBN layer and the WS_{2} cluster, respectively. N_{W} is the number of W atoms in the WS_{2} cluster.
Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.
References
Geim, A. K. & Novoselov, K. S. The rise of graphene. Nat. Mater. 6, 183–191 (2007).
Watanabe, K., Taniguchi, T. & Kanda, H. Directbandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal. Nat. Mater. 3, 404–409 (2004).
Zhou, J. et al. A library of atomically thin metal chalcogenides. Nature 556, 355–359 (2018).
Song, C.L. et al. Molecularbeam epitaxy and robust superconductivity of stoichiometric FeSe crystalline films on bilayer graphene. Phys. Rev. B 84, 020503 (2011).
Xu, X. et al. Ultrafast epitaxial growth of metresized singlecrystal graphene on industrial Cu foil. Sci. Bull. 62, 1074–1080 (2017).
Deng, B. et al. Wrinklefree singlecrystal graphene wafer grown on strainengineered substrates. ACS Nano 11, 12337–12345 (2017).
Nguyen, V. L. et al. Seamless stitching of graphene domains on polished copper (111) foil. Adv. Mater. 27, 1376–1382 (2015).
Wang, L. et al. Epitaxial growth of a 100squarecentimetre singlecrystal hexagonal boron nitride monolayer on copper. Nature 570, 91–95 (2019).
Chen, T.A. et al. Waferscale singlecrystal hexagonal boron nitride monolayers on Cu (111). Nature 579, 219–223 (2020).
Yang, P. et al. Epitaxial growth of centimeterscale singlecrystal MoS2 monolayer on Au (111). ACS Nano 14, 5036–5045 (2020).
Murdock, A. T. et al. Controlling the orientation, edge geometry, and thickness of chemical vapor deposition graphene. ACS Nano 7, 1351–1359 (2013).
Ogawa, Y. et al. Domain structure and boundary in singlelayer graphene grown on Cu(111) and Cu(100) films. J. Phys. Chem. Lett. 3, 219–226 (2012).
Wood, G. E. et al. van der Waals epitaxy of monolayer hexagonal boron nitride on copper foil: growth, crystallography and electronic band structure. 2D Mater. 2, 025003 (2015).
Song, X. et al. Chemical vapor deposition growth of largescale hexagonal boron nitride with controllable orientation. Nano Res. 8, 3164–3176 (2015).
Uchida, Y., Iwaizako, T., Mizuno, S., Tsuji, M. & Ago, H. Epitaxial chemical vapour deposition growth of monolayer hexagonal boron nitride on a Cu(111)/sapphire substrate. Phys. Chem. Chem. Phys. 19, 8230–8235 (2017).
Grønborg, S. S. et al. Synthesis of epitaxial singlelayer MoS2 on Au(111). Langmuir 31, 9700–9706 (2015).
Zhang, X. et al. Diffusioncontrolled epitaxy of large area coalesced WSe2 monolayers on sapphire. Nano Lett. 18, 1049–1056 (2018).
Aljarb, A. et al. Substrate latticeguided seed formation controls the orientation of 2D transitionmetal dichalcogenides. ACS Nano 11, 9215–9222 (2017).
Dumcenco, D. et al. Largearea epitaxial monolayer MoS2. ACS Nano 9, 4611–4620 (2015).
Ruzmetov, D. et al. Vertical 2D/3D semiconductor heterostructures based on epitaxial molybdenum disulfide and gallium nitride. ACS Nano 10, 3580–3588 (2016).
Lee, J. S. et al. Waferscale singlecrystal hexagonal boron nitride film via selfcollimated grain formation. Science 362, 817–821 (2018).
Chen, L. et al. Stepedgeguided nucleation and growth of aligned WSe2 on sapphire via a layeroverlayer growth mode. ACS Nano 9, 8368–8375 (2015).
Dong, J., Zhang, L. & Ding, F. Kinetics of graphene and 2D materials growth. Adv. Mater. 31, 1801583 (2019).
Ma, T. et al. Edgecontrolled growth and kinetics of singlecrystal graphene domains by chemical vapor deposition. Proc. Natl Acad. Sci. USA 110, 20386–20391 (2013).
Sekerka, R. F. Equilibrium and growth shapes of crystals: how do they differ and why should we care? Cryst. Res. Technol. 40, 291–306 (2005).
Zhang, Z., Liu, Y., Yang, Y. & Yakobson, B. I. Growth mechanism and morphology of hexagonal boron nitride. Nano Lett. 16, 1398–1403 (2016).
Stehle, Y. Y. et al. Anisotropic etching of hexagonal boron nitride and graphene: question of edge terminations. Nano Lett. 17, 7306–7314 (2017).
Tay, R. Y. et al. Synthesis of aligned symmetrical multifaceted monolayer hexagonal boron nitride single crystals on resolidified copper. Nanoscale 8, 2434–2444 (2016).
Yuan, Q., Yakobson, B. I. & Ding, F. Edgecatalyst wetting and orientation control of graphene growth by chemical vapor deposition growth. J. Phys. Chem. Lett. 5, 3093–3099 (2014).
Zhang, X., Xu, Z., Hui, L., Xin, J. & Ding, F. How the orientation of graphene is determined during chemical vapor deposition growth. J. Phys. Chem. Lett. 3, 2822–2827 (2012).
Dong, J. et al. Formation mechanism of overlapping grain boundaries in graphene chemical vapor deposition growth. Chem. Sci. 8, 2209–2214 (2017).
Ren, X. et al. Grain boundaries in chemicalvapordeposited atomically thin hexagonal boron nitride. Phys. Rev. Mater. 3, 014004 (2019).
Sang, X. et al. In situ edge engineering in twodimensional transition metal dichalcogenides. Nat. Commun. 9, 2051 (2018).
Shu, H., Chen, X. & Ding, F. The edge termination controlled kinetics in graphene chemical vapor deposition growth. Chem. Sci. 5, 4639–4645 (2014).
Dong, J., Zhang, L., Zhang, K. & Ding, F. How graphene crosses a grain boundary on the catalyst surface during chemical vapour deposition growth. Nanoscale 10, 6878–6883 (2018).
Ji, Q. et al. Unravelling orientation distribution and merging behavior of monolayer MoS2 domains on sapphire. Nano Lett. 15, 198–205 (2015).
Jin, S. et al. Colossal grain growth yields singlecrystal metal foils by contactfree annealing. Science 362, 1021–1025 (2018).
Lee, J.H. et al. Waferscale growth of singlecrystal monolayer graphene on reusable hydrogenterminated germanium. Science 344, 286–289 (2014).
Li, J. et al. Growth of polar hexagonal boron nitride monolayer on nonpolar copper with unique orientation. Small 12, 3645–3650 (2016).
Wang, S. et al. Catalystselective growth of singleorientation hexagonal boron nitride toward highperformance atomically thin electric barriers. Adv. Mater. 31, 1900880 (2019).
Dong, J., Geng, D., Liu, F. & Ding, F. Formation of twinned graphene polycrystals. Angew. Chem. Int. Ed. 58, 7723–7727 (2019).
Kresse, G. & Hafner, J. Ab initio molecular dynamics for openshell transition metals. Phys. Rev. B 48, 13115–13118 (1993).
Kresse, G. & Furthmüller, J. Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758–1775 (1999).
Grimme, S. Semiempirical GGAtype density functional constructed with a longrange dispersion correction. J. Comput. Chem. 27, 1787–1799 (2006).
Acknowledgements
Authors acknowledge support from the Institute for Basic Science (IBSR019D1), South Korea. Computational resources from Cimulator, CMCM, IBS are also acknowledged.
Author information
Authors and Affiliations
Contributions
F. Ding led the development of the concept and supervised the research. J. Dong, L. Zhang and X. Dai performed calculations and data analysis. J. Dong and F. Ding wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks Yunqi Liu and the other, anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Dong, J., Zhang, L., Dai, X. et al. The epitaxy of 2D materials growth. Nat Commun 11, 5862 (2020). https://doi.org/10.1038/s41467020197523
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467020197523
This article is cited by

Twodimensional single crystal monoclinic gallium telluride on silicon substrate via transformation of epitaxial hexagonal phase
npj 2D Materials and Applications (2023)

Mechanisms of the epitaxial growth of twodimensional polycrystals
npj Computational Materials (2022)

Singlecrystal twodimensional material epitaxy on tailored nonsinglecrystal substrates
Nature Communications (2022)

Epitaxial singlecrystal hexagonal boron nitride multilayers on Ni (111)
Nature (2022)

Dualcouplingguided epitaxial growth of waferscale singlecrystal WS2 monolayer on vicinal aplane sapphire
Nature Nanotechnology (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.