Dislocation-free two-dimensional concentric lateral heterostructures: MoS2–TaS2/Au(111)

We prepared two-dimensional concentric lateral heterostructures of the monolayer transition metal dichalcogenides MoS2 and TaS2 by reactive molecular beam epitaxy on chemically inert and weakly interacting Au(111). The heterostructures are in a size regime where quantum confinement can be expected. Despite large lattice mismatch a seamless interconnection of the two materials has been achieved, confirming that the semiconducting core is fully enclosed by a metallic border around its circumference. The resulting strain is analyzed on the atomic scale using scanning tunneling microscopy, corroborated by calculations based on empirical potentials and compared to results from finite elements simulations.


Introduction
The special properties of 2D-materials (2DMs) can be further modified by the introduction of lateral inplane confinement.The confinement in the resulting quasi-zero-dimensional quantum dots (QDs) leads to significant changes in the electronic band structure, ultimately resulting in atom-like states.In semiconducting QDs, such a boundary condition increases the band gap, which changes for example the optoelectronic properties: When the size is reduced down to the exciton Bohr radius, the exitonic binding energy increases, which manifests itself in a blueshift of the corresponding photoluminescence (PL) signals.
For the material of interest in this study, namely 2D transition metal dichalcogenides (TMDCs) with the general structure MX 2 (M = transition metal, X = chalcogen (S, Se, Te)), blue-shifted photoluminescence has been observed in QDs with diameters of just a few nm for MoS 2 (already in an early study) [1], MoSe 2 [2], and WS 2 .For the latter, also an enhanced quantum yield and giant spin-orbit coupling have been found and also attributed to confinement [3].These special properties make QDs also interesting for applications, as recently reviewed by Chiu et al [4].It can be expected that for small enough TMDC QDs also atom-like wave functions of intrinsic states as well as of image potential states could be observed as found in graphene QDs [5,6].
The edges of TMDCs are highly chemically active [7] and host a metallic edge state [8].While these edge-related properties deserve interest in their own right, they disturb the intrinsic properties of the QD [8].Enclosing the QD core with a protective border can passivate dangling bonds and quench the edge state.In consequence, the stability of the QD could be improved and the lifetime of excitons could be enhanced, leading to an increase in PL quantum yield.Such an enclosure is a well-established concept for three-dimensional core-shell QDs [9] and it is desirable to transfer it to the 2D world.Independent on dimensionality, the quality of the interface is important, as defects counteract the benefits of the enclosure.In the ideal case, the interface is atomically sharp (no intermixing), clean (no vacancies, interstitials, or impurities) and coherent (no dangling bonds or dislocations).While the first two conditions can in principle be met by improving preparation conditions, the presence of dislocations could be inherent to the system due to the lattice mismatch, which will be addressed in our study.
Pioneering studies reported the successful preparations of lateral heterostructures (LHs) of twodimensional TMDCs on the micrometer scale, but with significant intermixing, smearing out the interface over more than 10 nm [10,11].The first atomically sharp interface was found in WSe 2 -MoS 2 -LHs [12].Later, making elegant use of the magnification effect arising from the moiré superstructure between 2DM and substrate, Zhang et al could show that in this system roughly half of the lattice mismatch of 3.8% is compensated by strain, the other half by dislocations that they can directly observe using atomically resolved transmission electron microscopy [13].In our study, we scale these LHs down to the quantumdot size regime and focus on the compensation of the lattice mismatch.
We demonstrate the preparation of 2D concentric lateral heterostructures (CLHs) where the core is formed out of semiconducting MoS 2 and the border out of metallic TaS 2 .This combination is especially demanding as there is a considerable lattice mismatch.To give an example: When epitaxially grown on Au(111), the lattice constants are a MoS2 = 3.157 Å for MoS 2 [14] and a TaS2 = 3.300 Å for TaS 2 [15], leading to a mismatch of 4.5 %.
Beyond mere enclosure, a lateral 2D junction between a semiconductor and a metal has special properties as recently reviewed by Tan et al [16]: When the work function of the metal matches the electron affinity of the semiconductor, the metal side can be used as an electrode to address the semiconducting side with a low contact resistance [17].High-quality lateral epitaxy ensures the absence of defects that could lead to Fermi level pinning.For mismatched electronic levels, a Schottky contact forms with its characteristic electronic and optoelectronic properties [18].The combination of different materials in the form of 2D lateral heterostructures is especially fascinating as it opens the door to device manufacturing at the smallest possible level.
We prepare our MoS 2 -TaS 2 CLHs by epitaxial growth under well-controlled conditions (single crystal substrates, ultra-high vacuum), which is a versatile method to prepare 2DMs in high quality.Especially for transition metal disulfides, flat, clean, chemically inert, and weakly interacting Au(111) has emerged as a suitable substrate for reactive molecular beam epitaxy where evaporated transition metal atoms react with a suitable sulfur source to form the ultrathin layer [19,20].The weak interaction of our TMDCs with the substrate manifests itself by only weak hybridisation between the electronic states [20,21], a rather large vertical separation [14,15], and a longranged moiré superstructure [14,15].We modify our established growth process [14] to yield MoS 2 nanoislands in the QD size regime and enclose this core by an epitaxial border of TaS 2 [15].The structure of the CLHs is determined on the atomic scale using scanning tunneling microscopy (STM), corroborated by calculations based on empirical potentials.

Methods
Sample preparation and analysis were carried out in a home-built ultra-high vacuum (UHV) chamber with a base pressure of 3 × 10 −11 mbar equipped with a variable temperature STM.A clean Au(111) substrate was prepared by sputtering with Ar + at 1.5 keV, first at room temperature (RT) for 30 min, then at 900 K for 30 min, followed by annealing at 900 K for 10 min.Cleanliness was verified by STM.We evaporated Mo and Ta at room temperature via an e-beam evaporator (EGCO4 Oxford Applied Research).The evaporation rates are given in monolayers (ML) per min.In this context, we define 1 ML as a full pseudomorphic layer of the transition metal atoms on Au(111), i.e. 1.39 × 10 19 atoms m −2 .The rates were determined using the coverage of pristine TMDC islands after a full preparation assuming an areal density of Mo in MoS 2 of 1.16 × 10 19 atoms m −2 [14] and Ta in TaS 2 of 1.06 × 10 19 atoms m −2 [15].To prepare the heterostructures, first Mo was evaporated with a rate of 0.007 ML min −1 for 15-30 min at RT, followed by heating for 20 min at 900 K in an H 2 S atmosphere characterized by a chamber pressure of 1 × 10 −5 mbar measured far away from the sample, provided by a stainless steel tube with an opening diameter of ≈1 cm at a distance of ≈3 cm from the sample surface.The sample normal and the tube have an angle of 50 • to each other.The sulfur atmosphere is kept until the sample cools down to 450 K. Then Ta is evaporated at RT for 40-130 s with a rate of 0.115 ML min −1 .This is followed by the same process as previously described with a decreased H 2 S pressure of 1 × 10 −6 mbar.
All STM measurements were conducted at room temperature in constant current mode.Tunneling parameters are given in each image caption as: (bias voltage, tunneling current, image size).Labeled arrows indicate crystal directions.Images were analyzed with WSxM [22] and Gwyddion [23], using the known periodicity of Au(111) [24], the moiré structure of MoS 2 /Au(111) [14] or the moiré structure of TaS 2 /Au(111) [15] for calibration.Techniques to normalise the brightness of outlier scan-lines have been applied on a case by case basis.
We validated the ability of the EP-calculation to describe the TaS 2 -MoS 2 interface energy in the heterostructure by comparing the energy penalty versus the interface width to those obtained using density functional theory (DFT) calculations [27] and found a good agreement.The structures are fully optimized using a nonlinear conjugate gradient energy minimization method.To mimic the effect of the substrate, the z component of the force on all the atoms is set to zero during energy minimization while in-plane (x, y) force components are allowed to relax.The atomistic model of core-shell heterostructures was constructed using a triangular supercell containing 3577 atoms with dimensions of about 15.7 nm × 15.7 nm.An initial strain is applied to the outer TaS 2 envelope.The bond length and displacement analyses were performed using the Open Visualization Tool (OVITO) [28].
Finite elements simulations were made using COMSOL Multiphysics ® ver.6.0 [29].We use an unspecified isotropic reference material (Poisson ratio ν = 0.3, thermal expansion coefficient α = 6 × 10 −3 1/K).With exception of ν, we consider the exact values of the parameters unimportant for the conclusions we draw from the simulation.To account for the strain in coherent heterostructures due to the mismatch between the constituencies, we increased the thermal energy of the part representing TaS 2 to an amount that leads to an expansion of 4.5% for an equivalent homogeneous structure.We defined the cells in a way that we can identify each one with a unit cell of the TMDC based on our experimentally determined lattice parameters of the extended individual TMDCs.

Results and discussion
After the first preparation step, hexagonal MoS 2islands in the size range of 5-50 nm are present on Au(111), see figure 1(a).The islands are located on different Au(111) terraces separated by monoatomic steps.The edges of the islands are parallel to the dense-packed atomic rows of the substrate.The enhanced brightness of the edges is the fingerprint of the metallic edge state [8].The hexagonal pattern on the surface of the islands is the moiré superstructure due to the difference in lattice constants of the material and Au(111) [14,30].The moiré lattice is aligned with Au(111), directly indicating that the dense-packed rows of MoS 2 follow the dense-packed rows of Au(111).On the uncovered substrate one can discern the Au(111) herringbone reconstruction [31].
After the second preparation step triangular CLHs of MoS 2 and TaS 2 are formed, see figure 1(b) for an overview and c) for a close-up of an individual CLH.
A typical CLH has a side length of around 14 nm and a core diameter of around 8 nm.The CLHs are also strictly aligned with Au(111).We identify the inner, darker, hexagonal core with a smaller atomic periodicity as MoS 2 and the outer, brighter, triangular border with a larger atomic periodicity as TaS 2 .We consistently find a sharp line interface between core and border.We never observed a special electronic signature of the interface under a wide range of tunneling voltages.
Present in the background of all images is the Au(111) herringbone reconstruction (see also figure 3), recognizable by its wave like double lines that separate the crystal surface in two different stacking orders: In the larger spaces between two sets of double lines, there is the regular fcc stacking, whereas in the small spaces between one set of double lines a stacking fault is present between the topmost surface layer and the deeper ones, leading to hcp stacking [31].When only the first two Au(111) layers are taken into account, the fcc-areas and the hcp-areas are related by a rotation of 180 • .The islands avoid growing across the double lines to such an extend that they locally deform the herringbone pattern.
Figure 2 shows a model of the CLH.For simplicity, we assume commensurability between CLS and Au(111) and focus on the fcc-stacking.Note that the lateral heterostructures (LHs) forming the interface between the MoS 2 -core and the TaS 2 -border can be of two different kinds: Either the bonds across the interface are between a metal atom of the core (M 1 ) and a chalcogen atom (X), called M 1 -X-LH in the following (in our specific case Mo-S), or between a chalcogen and a metal atom of the border (M 2 ), dubbed X-M 2 -LH (here S-Ta).Similarly, the edges bounding the CLH can either be metal terminated (M-edge) or chalcogen terminated (X-edge).In figure 2 we tentatively assumed that one type of edge dominates.Note that our designation of the edges is purely geometrical (i.e.referring to the ideal truncation of the 2D layer) and not chemical.The well-known tendency of the edges of 2D sulfides to adsorption and reconstruction [32] is not relevant for our discussion here.
The MoS 2 -islands have alternating short and long edges, making them slightly asymmetric hexagons.This indicates that there is only a weak preference for one edge type.In contrast, for the CLHs only one type of edge is observed, leading to a fully formed triangular envelope.This shows that for the TaS 2 -border one type of edge is energetically favored.The MoS 2core retains its hexagonal shape, but the asymmetry between the long and short LHs appears to increase in comparison to the edges of pristine MoS 2 .
In view of these significant differences it is desirable to unambiguously identify the specific kind of edge and LH.However, this turned out to be notoriously difficult.Our solution is based on a detailed analysis of the orientation and the registry of the islands taking the herringbone reconstruction into account, see figure 3.
We find two orientations of the triangular CLHs.Following our earlier work [33] we denote them (+) and (−), which differ by a rotation of 180 • .For (+), one tip of the triangular shaped island points along the   shows the results of a statistical analysis where the CLHs are sorted with respect to their orientation (+/−) and the stacking of their Au(111) substrate (fcc/hcp).To illustrate the statistic, figure 3(a) shows CLHs with '+' and '−' placed on them.The colour of the sign indicates the stacking of the substrate (white: fcc, blue: hcp).Occasionally we find pure TaS 2 -islands among the CLHs (see e.g. the unmarked islands in figure 3(a)).These were not included in the statistics.The majority of CLHs lie in the fcc-stacking of the surface.Within the fcc-stacking the (+)-orientation strongly dominates.For the hcpstacking, the preferred orientation flips.This is to be expected as the hcp-regions are well approximated by a 180 • -rotated fcc-region.
The orientation alone is not sufficient to distinguish between the two kinds of LHs and TaS 2 edges.Typically, the three-fold symmetric TMDCs can have two different registries with the sixfold-symmetric topmost layer of an fcc(111) surface [33][34][35].This is illustrated in figure 2. As introduced previously [33] the two possible registries are labelled according to the relative position of the transition metal M (Mo or Ta) and the chalcogen X (S) with respect to the Au substrate as M fcc X top and M hcp X top .In explicit, top means that the atoms sit directly on top of an Au in the topmost substrate layer, fcc indicates that the atoms sit at threefold-hollow sites of fcc-type, and correspondingly hcp indicates that the atoms sit at threefold-hollow sites of hcp-type.Unfortunately, as illustrated in figure 2, the same orientation can be explained by either an M fcc X top registry and a preference of Ta-edges or a M hcp X top -registry and a preference of S-edges.It is therefore mandatory that we know the registry as well as the orientation.
For pristine MoS 2 /Au(111), Krane et al [36] determined the registry using deviations from sixfold symmetry both in the island shape and in lowtemperature STM images of the moiré structure.They found a strong preference for M fcc X top (called 'regular stacking' in [36]).
We assume that once the registry of the MoS 2core is fixed in the growth process, it does not change upon addition of the TaS 2 -border (this would necessitate a rotation of the core by 180 • ).Therefore, we conclude that the preferred registry of the CLHs is M fcc X top as well.In consequence, the CLHs are bound by Ta-edges.
To check the self-consistency of our analysis, we also determined registry and substrate stacking of the MoS 2 -cores after the first preparation step, see figure 3 (b) and table 1(b).As the islands are not fully hexagonal, but have the shape of truncated triangles, we can continue to use the (+/−)-naming scheme.We find the same pattern as for the CLHs, confirming our assumption that the registry is kept during the growth process.We corroborate the preference for M fcc X top found in [36].
To summarize: We can confirm that the pristine MoS 2 -islands slightly prefer the metal edge, as found in previous studies [8] and predicted in theoretical calculations [37].For the CLHs, the TaS 2 -border strongly favours the metal edge.The shape of the core shows a modest preference for the M 1 -X LH, where a Mo atom from the core is bound to an S atom from the border.
Regarding the shape of the TaS 2 -border, we propose that this is also mainly governed by the TaS 2 edge energies which dominate the effects of the MoS 2 -core.This assumption is supported by the fact that we find the identical shape for pristine TaS 2 -islands (see 3(a)).Martincová et al [38] found a strong preference for the Ta-edge, which is especially pronounced for Spoor conditions but prevails over a large range of the S chemical potential.This explains our experimental observation.As an aside, our direct identification of the preferred edge for TaS 2 also validates the more indirect determination in [33].
In the following, we will focus on the interface between the core and the border and its influence on the morphology of the CLHs, especially its strain state.Therefore, we combine the identification of the type of junction established above with a characterization on the atomic scale.As an example, figure 4(a) shows an atomically resolved image of a CLH where the dark MoS 2 -core is enclosed in a bright TaS 2border.All visible atomic rows are continuous across the lateral MoS 2 -TaS 2 -heterojunctions, indicating a coherent interface.In fact, we have not observed dislocations for any CLH prepared under this or similar conditions.To become quantitative: The total length of the interface in figure 4(a) is 71 MoS 2 unit cells.Given the lattice mismatch of 4.5%, this length corresponds to 68 unstrained unit cells of Ta 2 , so one would expect three dislocations in case of full compensation of the mismatch by defects, but we observe none.
The absence of dislocations is in line with our recent theoretical analysis applied to the specific combination of MoS 2 and TaS 2 .In short, for the elementary case of thermal equilibrium, we compared the two competing mechanisms for the compensation of lattice mismatch between various combinations of TMDCs, namely strain and dislocations.The formation energy for an interface increases with the width of the border for the case of strain, but is independent of width for the case of dislocations.Therefore, we expect a critical width of the border for any given combination of TMDCs where the preferred relaxation mechanism changes from strain to dislocations.The TaS 2 -borders in the present study fall below our theoretical limit.In addition, we determined that the formation energy of the Mo-S-LH is lower than the one of the S-Ta-LH.This explains the modest preference of the core shape for the former type observed in our experiment.In view of the width-dependence of the interface formation energy discussed above, we expect to find dislocations for significantly thicker borders, in line with the findings in [13].That said, growth processes as used here typically take place at a certain distance from thermal equilibrium, so one could also find systems trapped in a metastable state, where our theory is not fully applicable.
The absence of dislocations implies that there are severe elastic deformations in the CLH that we will analyze in the following.In principle, the strain state of our structures could be determined on the atomic scale from highly resolved STM images like the one shown in figure 4(a).In practice, noise, drift and creep of the microscope do not allow this.A sophisticated alternative is to exploit the magnifying effect of the moiré pattern [13][14][15].However, for the size of the CLHs studied here, the core contains just a few moiré supercells, and the border oftentimes even less than one (see for example figure 4(a)).This method therefore has a reduced precision when applied to the core and cannot be applied at all to the border.Our solution is to use appropriately averaged atomic parameters in the following to describe the strain state.
In the first step, we quantify the deformation of the MoS 2 -core.By averaging over all three high symmetry directions in several LHs we obtain (9.7 ± 0.5) MoS 2 unit cells in the moiré periodicity (error is the standard deviation).This value is significantly smaller than 10.6 ± 0.1 determined previously for pristine MoS 2 /Au(111) [14].A moiré analysis yields an average core lattice constant ãMoS2 = (0.318 ± 0.002) nm, meaning that the core expands by ≈0.7% compared to pristine MoS 2 /Au(111), it is under tensile stress.This has to be expected as it is coherently bound to the TaS 2 border with a larger equilibrium lattice constant.The small size of the core does not allow us to go beyond averaging and determine the core lattice constant for individual CLHs or even locally resolved within the core.
To quantify strain in the TaS 2 -border, we determined the average atomic distance a along lines that are parallel to the junction (exemplified by the white dashed lines in figure 4(a)).We distinguish between MoS 2 (red boxes) and TaS 2 (blue boxes) and label each value of a by the index of the corresponding atomic row i.The junction is located where row zero and row one meet (boundary between colored boxes).Thus, we obtain a set of values a MoS2,i and a TaS2,i .This type of measurement is done for 22 junctions in 6 individual CLHs.We distinguish between Mo-S-heterojunctions and S-Ta-heterojunctions.
To compensate for measurement errors, we normalized both a MoS2,i and a TaS2,i by the average value āMoS2 = ∑ for the core and analog- for the border is shown in figure 4(b).Red data points were obtained in the MoS 2 -core, blue data points in the TaS 2 -border.With caution, the relative lattice constants δ i can be converted to absolute values under the assumption that the average determined for each CLH corresponds to the average lattice constant from the moiré analysis, i.e. āMoS2 = ãMoS2 (right axis in figure 4(b)).
The relative lattice constant for the MoS 2 core clusters around 1 (green dashed line) and does not fluctuate much.As implied in our moiré analysis, this is slightly expanded with respect to the equilibrium lattice constant of MoS 2 (red dashed line).
In contrast to MoS 2 , TaS 2 changes significantly with respect to its equilibrium structure, additionally, there are striking differences between the two kinds of junctions.For the Mo-S-junction, TaS 2 reduces its lattice constant to fully match MoS 2 .This corresponds to a contraction of 3.6% with respect to the equilibrium lattice constant of TaS 2 (blue dashed line).The contraction is present until the edge of the island is reached.
At the S-Ta-junction TaS 2 behaves differently.Close to the border with MoS 2 the lattice constant is significantly lower so that it matches MoS 2 , but with increasing distance to the border, it expands.Surprisingly, the TaS 2 border does not only expand until its equilibrium lattice constant is reached (dashed blue line) but significantly beyond that.We find a dilation of up to 10% compared to the lattice constant of freestanding TaS 2 .Note that we did not find an expansion of the tips in triangular pristine TaS 2 -islands.
It is intriguing that the strain in TaS 2 shows such a strong variation in our CLHs, ranging from 3.6% compression to ≈10% expansion, especially since optical, vibrational and electronic properties of 2DMs are highly dependent on strain [39].Since the strain in CLHs heterostructures is junction dependent, a manipulation of the junction length and border width could be a versatile way of strain-tuning a system to achieve desired properties.
To reach a deeper understanding of the strain pattern in our heterostructures, we modelled them using empirical potentials (see Methods for details).We build a CLH where the hexagonal MoS 2 core has a side length of 11 unit cells and the triangular TaS 2 envelope has a side length of 47 unit cells.This approximates the island shown in figure 4(a).Mimicking the experimental situation, the atomistic model was created by using MoS 2 lattice parameters while applying a compressive strain on the border lattice of TaS 2 .Upon relaxation, the Ta-S bonds in the border starts to expand to compensate the induced lattice strain in TaS 2 (see figure 5(a) and movie 1 (supplementary material)).The extension of the bonds is stronger perpendicular to the junction than parallel to it.The longest bonds are found in the tips.
We analyze the change in lattice parameter parallel to the boundary analogously to the experimental procedure.As the equilibrium lattice constants of the extended individual TMDCs are different, it is a bit tedious to relate the experiment and EP calculation.To facilitate the comparison analysis, we normalized the parameters from the EP calculation in a way that the equilibrium lattice constants match the experimental ones.Explicitly, we set a TMDC = 0.722 × a EP TMDC + 0.0906 nm and obtained the green symbols in figure 4(b).The calculation corroborates the experimental findings that (i) the MoS 2 core expands slightly, (ii) the TaS 2 border at the Mo-S-junction is compressed parallel to the junction over its full width to match the MoS 2 core, and (iii) the TaS 2 lattice at the S-Ta-junction gradually relaxes with increasing distance from the junction.This in turn verifies our experimental analysis.After relaxation, the atoms have shifted from their starting positions as shown in figure 5(b) and movie 2 (supplementary material).
In the empirical potential simulation, the overextension of the TaS 2 lattice at the tips of the triangle is much less pronounced (1%) than in our experiment (10%).We rule out that the discrepancy is due to disregarding the substrate as the overextension is not found in the experiment for pristine TaS 2 triangles.A likely candidate are electronic effects which are not part of an empirical potential calculation In principle, one could investigate the system deeper using DFT calculations, which is, however, computationally too expensive given the large size of our system.
To check to what extend the deformation of the CLHs is already accessible in a simplistic model based on continuum mechanics, we performed finite elements simulations using an unspecified isotropic reference material (see Methods).We use a thin triangle with a side length of 15.15 nm containing a hexagonal core with a side length of 3.47 nm.This corresponds to the structure used in the EP calculation.We mechanically strained the border with respect to the core to represent the lattice mismatch between MoS 2 and TaS 2 .As we are only interested in the resulting shifts of the atomic positions, we deem the exact input parameters for the simulation irrelevant.The one exception is the Poisson ration ν, which we put at ν = 0.3, reasonably close to both MoS 2 (ν = 0.3 [40]) and TaS 2 (ν = 0.26 [41]).The resulting displacement after relaxation is shown in figure 5(c).The simulation has been evaluated in the same way as the experiment and is also plotted in figure 4(b) using purple circles.The results from the simple finite element simulation are almost indistinguishable from the more sophisticated approach based on empirical potentials.Also here the key experimental findings are reproduced.This shows that for quick screening or an approximate treatment of TMDC heterostructures such finite element calculations can be a helpful tool.Naturally, also this calculation does not reproduce the overextension of the tip.

Conclusion
We successfully transferred the concept of encapsulation of a semiconducting core in the QD by a protective shell into the 2D world: Epitaxial growth on Au(111) enables enclosure of MoS 2 islands by an outer border of TaS 2 .The resulting CLHs inherit their preference for the fcc-stacking and the M fcc X topregistry from their MoS 2 -nucleus.As pristine TaS 2 they prefer to be bound by Ta-edges.Despite the significant lattice mismatch on 4.5%, we do not observe dislocation at the interface (lateral heterostructures) between the two materials, in line with recent theoretical results [27].In consequence, the mismatch is compensated by strain.We find a small expansion of the core, but the main part of the strain is present in the TaS 2 -border.This strain is junction-dependent.The Mo-S-junctions show a uniform contraction of TaS 2 to fit to MoS 2 .For S-Ta-junctions on the other hand, the TaS 2 contracts its lattice constant to fit the MoS 2 directly at the junction, but relaxes with increasing distance.This effect goes beyond what would be expected for pure strain relaxation however, as up to 10% dilation in comparison to freestanding TaS 2 has been measured.The key findings are corroborated by empirical potential simulations using a sophisticated description of the atomic interactions.We show that the mechanical deformations can already be well approximated by simplistic finite element calculations based on continuum mechanics.

Figure 2 .
Figure 2. Models in top view of the possible stacking variations of concentric lateral heterostructures on Au(111): M fcc Xtop (left) and M hcp Xtop (right).

Figure 4 .
Figure 4. (a) Atomically resolved STM image of a triangular concentric lateral MoS2-TaS2-heterostructure on Au(111) (−0.9 V, 1 pA, 12.5 nm × 15.8 nm); (b) Graph with relative (left scale) and absolute (right scale) lattice constants measured parallel to the heterojunction as indicated by white dashed lines in panel (a) of several Mo-S-junctions (top) and S-Ta-junctions (bottom), on several different islands (two examples labelled in panel (a)).Red symbols denote MoS2, blue symbols denote TaS2, see also corresponding boxes in panel (a).Blue/red dashed lines represent the equilibrium lattice constant of the constituents, see text.Green circles: empirical potential calculations (normalized, see text).Purple circles: finite element simulations.

i a MoS 2
,i i from each set.The resulting relative lattice constant δ i = a MoS 2 ,i āMoS 2

Figure 5 .
Figure 5. Structure of a relaxed concentric lateral heterostructure after relaxation using empirical potential calculations (a), (b) and finite element simulations (c).(a) Metal-sulfur bond lengths, the entirety of the inner blue region aligns precisely with the MoS2 core.(b) Displacement of the atoms during relaxation in the EP calculation.(c) Displacement of the unit cells (represented by the metal atoms) during relaxation in the FE simulation.The hexagon indicates the MoS2 core.

Table 1 .
Probability (normalized to 1) to find a structure in a specific combination of their orientation (+/−) and stacking of their Au(111) substrate (fcc/hcp).Sums are indicated in bold.