Abstract
Lack of directional bonding between twodimensional crystals like graphene or monolayer transition metal dichalcogenides provides unusual freedom in the selection of components for vertical van der Waals heterostructures. However, even for identical layers, their stacking, in particular the relative angle between their crystallographic directions, modifies properties of the structure. We demonstrate that the interatomic coupling between two twodimensional crystals can be determined from angleresolved photoemission spectra of a trilayer structure with one aligned and one twisted interface. Each of the interfaces provides complementary information and together they enable selfconsistent determination of the coupling. We parametrise interatomic coupling for carbon atoms by studying twisted trilayer graphene and show that the result can be applied to structures with different twists and number of layers. Our approach demonstrates how to extract fundamental information about interlayer coupling in a stack of twodimensional crystals and can be applied to many other van der Waals interfaces.
Introduction
Following the isolation of graphene (a layer of carbon atoms arranged in regular hexagons) in 2004^{1}, many other atomically thin twodimensional crystals have been produced and can be stacked in a desired order on top of each other. In contrast to conventional heterostructures, in which chemical bonding at interfaces between two materials modifies their properties and requires lattice matching for stability, stacks of twodimensional crystals are held together by weak forces without directional bonding. As a result, any two of these materials can be placed on top of each other, providing extraordinary design flexibility^{2,3,4}. Moreover, subtle changes in atomic stacking, especially the angle between the crystallographic axes of two adjacent layers, can have big impact on the properties of the whole heterostructure, with examples including the observation of Hofstadter’s butterfly^{5,6} and interfacial polarons^{7} in graphene/hexagonal boron nitride heterostructures, interlayer excitons in transition metal dichalcogenide bilayers^{8,9}, appearance of superconductivity in magicangle twisted bilayer graphene^{10,11} and explicit twistdependence of transport measurements in rotatable heterostructures^{12,13,14}. Phenomena like these arise because the misalignment of two crystals changes the atomic registry at the interface and hence tunes the spatial modulation of interlayer interaction. Consequently, understanding the coupling between two twodimensional materials at a microscopic level is crucial for efficient design of van der Waals heterostructures.
The impacts of a twisted interface and modulated interlayer coupling on the electronic properties of twodimensional crystals include band hybridisation^{15,16,17}, band replicas and minigaps due to scattering on moiré potential^{15,18,19}, charge transfer and vertical shifting of bands^{17,20,21} as well as changes of the effective masses^{17,20}. Variations in the interlayer coupling as a function of the twist angle, θ, were probed for example using photoluminescence, Raman and angleresolved photoemission (ARPES) spectroscopies^{20,22,23,24}. Here, we use the last of those methods to image directly the electronic bands in trilayer graphene with one perfect and one twisted interface. From our data, we extract the interatomic coupling, t(r, z), describing coupling between two carbon atoms separated by a vector r_{3D} = (r, z) = (x, y, z). Such coupling functions, usually based on comparisons to ab initio calculations, can be used to determine electron hoppings in tightbinding^{25,26} and continuum^{27,28} models of corresponding van der Waals interfaces at any twist angle. We show that t(r, z) determined purely by measurements on one of the structures accurately describes electronic dispersions obtained for stacks with different θ and number of layers, providing an experimentally verified set of parameters to model twistronic graphene. Our approach makes use of the fact that a trilayer structure is the thinnest stack that can contain both a perfect and twisted interface. The former, due to translational symmetry, can be straightforwardly described in the real space using t(r, z). At the same time, the impact of the moiré pattern formed at the latter can be captured in the reciprocal space by considering scattering by moiré reciprocal vectors on the momentumdependent potential \(\tilde{t}({\bf{q}},z)\) which is a twodimensional Fourier transform \({\mathcal{F}}[t({\bf{r}},z)]\) of t(r, z) (see the comparison of the two cases in Fig. 1a). As a consequence, this method should enable determination of interatomic couplings for all van der Waals interfaces for which moiré effects were observed.
Results
ARPES of twisted trilayer graphene
We grew our graphene trilayers on copper foil using chemical vapour deposition^{29,30}. The inset of Fig. 1b shows the intensity map of copper dband photoelectrons which are attenuated differently by the overlying graphene layers depending on their number. This provides means to identify all of the layers in our stack, shown in the inset with different shades of grey and indicated with the red arrows. As depicted schematically in the main panel of Fig. 1b, the bottom two layers form a Bernal bilayer (2L) while the crystallographic axes of the top monolayer (1L) are rotated by an angle θ with respect to those of the layer underneath. As a result, the Brillouin zones corresponding to the bilayer and monolayer are also rotated with respect to each other, Fig. 1c. We focus here on the vicinity of one set of the corners of the two Brillouin zones, which we denote K_{2} and K_{1}, for the bilayer and monolayer, respectively. The separation between these two points, dependent on the twist angle, defines an effective superlattice Brillouin zone, indicated in orange in the inset of Fig. 1c.
In Fig. 2a, we present ARPES intensity along a cut in the kspace connecting K_{2} and K_{1}, with the energy reference point set to the linear crossing (Dirac point) at K_{1}. Close to each corner, the intensity reflects the lowenergy band structures of unperturbed 2L and 1L. Because the bilayer flake is below the monolayer, signal from the former is attenuated due to the electron escape depth effect. In between the two spectra, coupling of the two crystals leads to anticrossings of the bands and opening of minigaps (marked as \({\varepsilon }_{{\rm{I}}}^{{\rm{g}}}\) and \({\varepsilon }_{{\rm{II}}}^{{\rm{g}}}\) in the figure). As the size of the superlattice Brillouin zone depends on the twist angle, the energy positions of the minigaps also depend on θ. Moreover, the magnitudes of the minigaps depend on the interlayer coupling between the bilayer and monolayer and also, in principle, vary with θ. However, fundamentally, all of the features in our spectrum originate in interactions between carbon atoms, be it in the same or different layers, at the twisted or aligned interface. This provides us with an opportunity to study the interatomic coupling t(r, z) in carbon materials.
Parametrising carboncarbon interaction potential
In order to understand our data, we use a generic Hamiltonian for a van der Waals heterostructure comprised of three layers of the same twodimensional crystal
In this Hamiltonian, the diagonal block, \({\hat{H}}_{0}\left({\theta }_{i},{\varepsilon }_{i}\right)\) describes the ith layer at a twist angle θ_{i}, with onsite energies of atomic sites in this layer, ε_{i}. Here, because only the relative twist between any two adjacent layers is important, we have θ_{1} = θ_{2} = 0 and θ_{3} = θ. Also, our choice of energy reference point is equivalent to ε_{3} = 0 and we introduce potential energy difference, 2u = ε_{1} − ε_{2}, as well as average energy, Δ = (ε_{1} + ε_{2})/2, of layers 1 and 2 (the charge transfer between the copper foil and the graphene layers giving rise to u ≠ Δ ≠ 0 is discussed in more detail in ref. ^{29}). For graphene, the intralayer blocks \({\hat{H}}_{0}\) can be straightforwardly described using a tightbinding model^{31} for a triangular lattice with two inequivalent atomic sites, A and B, per unit cell and nearest neighbour coupling between them γ_{0} ≡ −t(r_{AB}, 0), where r_{AB} is a vector connecting neighbouring A and B atoms with the carboncarbon bond length ∣r_{AB}∣ = 1.46 Å.
Of more importance for us, however, are the offdiagonal blocks \(\hat{T}({\theta }_{i}{\theta }_{i1})\) which capture the twistdependent interlayer interactions between adjacent layers (we neglect the interaction between the bottom and the top layers which is at least an order of magnitude weaker^{32}). As the bottom two layers are stacked according to the Bernal stacking, a realspace description of the interlayer interaction block \(\hat{T}(0)\) is possible with the leading coupling t(0, c_{0}) ≡ γ_{1}, with interlayer distance c_{0} = 3.35 Å, due to atoms with neighbours directly above or below them, as shown in Fig. 1a^{33}. In contrast, we describe the coupling between the twisted layers, i = 2, 3, in the reciprocal space based on electron tunnelling from a state with wave vector k in layer 2 to a state with wave vector \({\bf{k}}^{\prime}\) in layer 3 with the requirement that crystal momentum is conserved^{34,35}, \({\bf{k}}+{\bf{G}}={\bf{k}}^{\prime} +{\bf{G}}^{\prime}\), where G and \({\bf{G}}^{\prime}\) are the reciprocal vectors of layers 2 and 3, respectively. The strength of a given tunnelling process is set by the twodimensional Fourier transform, \({\mathcal{F}}[t({\bf{r}},z)]=\tilde{t}({\bf{q}},z)\), of the realspace coupling t(r, z) so that
where τ = (−∣r_{AB}∣, 0) and \({\hat{R}}_{\theta }\) is a matrix of clockwise rotation by angle θ (see Supplementary Note 1 for more details on the construction of the Hamiltonian \(\hat{H}\)).
The uniqueness of a trilayer with one perfect and one twisted interface (as exemplified in Fig. 1a for the case of graphene) lies in the fact that the Hamiltonian \(\hat{H}\) contains interlayer blocks based on both the realspace (\(\hat{T}(0)\)) and reciprocalspace (\(\hat{T}(\theta )\)) descriptions which provide complementary information and at the same time are related to each other because of the Fourier transform connection between t(r, z) and \(\tilde{t}({\bf{q}},z)\). Because of this, comparison of the photoemission data with the spectrum calculated based on Eq. (1) provides more information about the interatomic coupling t(r, z) than structures with one type of interface only. For our graphene trilayer, we compute the miniband spectrum of \(\hat{H}\) (see Methods for more details) assuming a SlaterKosterlike twocentre ansatz for t(r, z)^{25},
where V_{π} and V_{σ} represent the strength of the π and σ bonding^{36}, respectively, and α_{π} and α_{σ} their decay with increasing interatomic distance.
In fitting our numerical results to the experimental data in Fig. 2a, we first determine the position of 1L Dirac point what sets the ε = 0 reference point. We then use the electronic band gap at K_{2} to fix the electrostatic potential 2u and position the bilayer neutrality point halfway in the gap, establishing the potential energy shift Δ. We obtain the inplane nearest neighbour hopping γ_{0} from the slope of the 1L linear dispersion close to the Dirac point at K_{1} while the direct interlayer coupling γ_{1} is set by the splitting of the 2L lower valence band from the neutrality point at K_{2}. Finally, the decay constants α_{π} and α_{σ} are found numerically using the constraints that (i) the magnitudes of the gaps \({\varepsilon }_{{\rm{I}}}^{{\rm{g}}}\) and \({\varepsilon }_{{\rm{II}}}^{{\rm{g}}}\) in Fig. 2a match the experimental data and (ii) in the limit of θ = 0, \(\hat{T}(\theta )\) from Eq. (2) converges to the realspace form of \(\hat{T}(0)\) as used for coupling between the Bernal stacked layers (see Supplementary Note 2 for further discussion).
The miniband spectrum resulting from our model is shown in red dashed lines in Fig. 2a, the functions t(∣r∣, c_{0}) and \(\tilde{t}( {\bf{q}} ,{c}_{0})\) are plotted in Fig. 2b and the corresponding values of the parameters γ_{0}, γ_{1}, α_{π} and α_{σ} are summarised in Table 1. The interatomic potential we obtain decays more rapidly in the real space (and hence slower in the reciprocal space) than suggested by computational results^{25}. Importantly, parametrization of t(r, z) does not depend on the twist angle and so should be applicable to other graphene stacks with twisted interfaces. It also does not depend on the doping level because, for the relevant range of electric fields, the electrostatic energies Δ and u do not modify the electron hoppings. At the same time, once these energies are determined for a particular stack, their influence on the band structure (shifting of the positions and magnitudes of anticrossings) is captured through the Hamiltonian \(\hat{H}\). To confirm applicability of a single parametrization of t(r, z) to different graphene stacks, we compare in Fig. 3 the miniband spectra computed using the parameters from Table 1 to ARPES intensities measured along a similar K_{2}K_{1}kspace cut for, in Fig. 3a, a trilayer with θ = 9^{∘} and, in Fig. 3b, twisted bilayer with θ = 19.1^{∘}. Our model describes the bands of both of the structures well, despite changes in the twist angle, number of layers, potentials u and Δ (which vary with growth conditions and thickness of the stack^{29} and are determined for each structure individually) and the magnitudes of minigaps.
Probing electron wave function
We assess the accuracy of our parametrization of the interatomic potential, t(r, z), further by modelling directly the ARPES intensity data (we use approach developed in ref. ^{37} and applied to the graphene/hexagonal boron nitride heterostructure in ref. ^{38}; see Methods and Supplementary Note 3 for further details). In graphene materials, interference of electrons emitted from different atomic sites within the unit cell provides additional information about the electronic wave function^{37}. This is best visualised by ARPES intensity patterns at constant electron energy, which we present, both as obtained experimentally (top row) and simulated theoretically (bottom row), in Fig. 4 for the trilayer sample with θ = 9^{∘} and energies indicated with grey dashed lines in Fig. 3. For the map at the energy ε = 0, the two spots of high intensity indicate the positions of the valleys K_{1} and K_{2}. For energies 0 < ε < −0.6 eV, the bilayer and monolayer dispersions are effectively uncoupled. The crescentlike intensity pattern in the vicinity of K_{1} reflects the pseudospin of n = 1 (evidence of Berry phase of π^{39}) of electrons in monolayer graphene. In contrast, in bilayer graphene, the lowenergy band hosts massive chiral fermions^{40} with pseudospin n = 2 so that the outer ring pattern in the vicinity of K_{2} displays two intensity maxima, feature best visible in panel (II). Because in our model all electron hoppings are generated naturally by t(r, z), agreement of our ARPES simulation with experimental data provides confirmation that our model and parametrization of the interatomic coupling t(r, z) leads to the correct band structure. Finally, panels (III)(V) in Fig. 4 show the constantenergy maps in the vicinity of the minigaps which open due to hybridisation of the bilayer and monolayer bands. The merging of 1L and 2L contours in panel (III) leads to a van Hove singularity and an associated peak in the electronic density of states, similarly to the case of twisted bilayer graphene^{15} and discussed also for twisted trilayer graphene^{29} (in the latter, the position of the van Hove singularity is established by tracking the minigap; the former is caused by saddle points in the electronic dispersion as the bands flatten at the anticrossings and so every minigap is accompanied by a van Hove singularity). Overall, our simulated patterns correctly reflect the evolution of the minigap as a function of energy and wave vector as well as the measured photocurrent intensity.
Discussion
Our parametrization of t(r, z) is applicable to a wide range of twist angles, including the magicangle regime^{10,34} as well as the 30^{∘}twisted bilayer graphene quasicrystal^{41,42}. To mention, it yields the kspace interlayer coupling at the graphene Brillouin zone corner K, \(\tilde{t}( {\bf{K}} ,{c}_{0})=0.11\) eV. This agrees with the values used in effective models of the lowtwist limit of twisted bilayer graphene^{27,34,35,43} which require \(\tilde{t}( {\bf{K}} ,{c}_{0})\) as the only parameter. Overall, our form of t(r, z) decays more rapidly in the real space (and hence slower in the reciprocal space) than usually assumed. This might explain the discrepancy between theory and experimental ARPES intensities of Dirac cone replicas observed for the case of 30^{∘}twisted bilayer graphene in ref. ^{41}.
As we have shown, the same interatomic coupling t(r, z) can be used in graphene structures with different number of layers as, similarly to the case of perfect graphite and other layered materials, coupling to the nearest layer dominates the interlayer couplings. The continuum approach has been applied extensively to model the graphene/graphene interface, including to predict the existence of the magic angle^{34}. Hence, in Supplementary Figure 1, we use our results to simulate ARPES spectra for twist angles in the vicinity of the magic angle, θ ≈ 1.1^{∘}, and show qualitative agreement with the recent experimental data^{44,45}. The continuum model was also used successfully to interpret experimental observations in graphene on hexagonal boron nitride^{5} as well as homo and heterobilayers of transition metal dichalcogenides^{46,47}. Our approach allows for experimental parametrization of the interatomic coupling t(r, z) for each of these interfaces as well as for others for which influence of neighbouring crystals can be approximated by considering the harmonics of the moiré potential^{43,48,49,50,51,52}. To comment, previous studies suggest that adapting our model to stacks of transition metal dichalcogenides requires taking into account changes in the interlayer distance as a function of the twist angle^{20}. Moreover, in contrast to graphene, for which the part of \(\tilde{t}({\bf{q}},z)\) most relevant to modelling twisted interfaces is that for q pointing to the Brillouin zone corner, q ≈ K, for transition metal dichalcogenides more significant changes due to interlayer coupling occur in the vicinity of the Γ point. In multilayers of 2H semiconducting dichalcogenides MX_{2} (M = Mo, W, and X = S, Se), coupling of the degenerate states at the Γ point built of transition metal \({d}_{{z}^{2}}\) and chalcogen p_{z} orbitals leads to their hybridisation and splitting which drives the directtoindirect band gap transition^{53,54}. Using the form of t(r, z) suggested in ref. ^{26} for chalcogen p_{z}top_{z} hopping (which dominates the interlayer coupling) in transition metal disulfides and diselenides, we computed the corresponding \(\tilde{t}({\bf{q}},z)\) and obtained an estimate of \(\tilde{t}(\Gamma ,{c}_{{\rm{X}}{\rm{X}}}) \sim 1.2\) eV for interlayer nearest neighbour distance between chalcogen sites, c_{X−X} ≈ 3 Å. Taking into account the fractional contribution of the p_{z} orbitals to the top valence band states at Γ in a monolayer^{26}, we obtain coupling between two such states in bilayer ~0.4 eV. This, in turn, suggests band splitting of ~0.8 eV, in qualitative agreement with observations^{53,54,55}. This supports the idea that our model can accurately describe and parametrise interatomic coupling between materials other than graphene.
Experimentally, our approach requires fabrication of trilayer (or thicker) stacks with one twisted and one perfect interface in order to benefit from the complementarity of the information obtained from selfconsistent real and momentumspace description of the interfaces. However, to note, building on the observations of superconductivity in magicangle twisted bilayer graphene^{10,11}, structures containing both a twisted and a perfect interface like twisted trilayer graphene^{56,57}, double bilayer graphene^{58,59,60,61,62,63} or double bilayer WSe_{2}^{64} recently attracted attention on its own due to observation of correlated electronic behaviour. Our approach provides one of the avenues to build an experimentally validated singleparticle base to study such effects. It could be, in principle, also applied to stacks of different materials, as long as one of the interfaces is commensurate and can be described in the real space in a tightbindinglike fashion. Finally, apart from continuum models, the interatomic coupling t(r, z) can also be used directly in large scale tightbinding calculations for commensurate twist angles^{25,26,65,66,67}.
Methods
ARPES measurements
The ARPES measurements were performed at the Spectromicroscopy beamline at the Elettra synchrotron (Trieste, Italy). Before measurements, the samples were annealed at 350^{∘} for 30 minutes. The experiment was then performed at a base pressure of 10^{−10} mbar in ultrahigh vacuum and at the temperature of 110 K. We used photons with energy of 74 eV and estimate our energy and angular resolution as 50 meV and 0.5^{∘}, respectively. For each sample, we determined the twist angle θ by measuring the distance between the Brillouin zone corners K_{2} and K_{1} which depends on the twist angle, \( {{\bf{K}}}_{2}{{\bf{K}}}_{1} =\frac{8\pi }{3\sqrt{3} {{\bf{r}}}_{AB} }\sin \frac{\theta }{2}\). Further comments on experimental analysis of ARPES intensity are provided in Supplementary Note 4.
Theoretical calculations
We write the Hamiltonian \(\hat{H}\) in Eq. (1) in the basis of sublattice Bloch states constructed of carbon p_{z} orbitals ϕ(r_{3D})^{31},
where k is electron wave vector, X = A, B is the sublattice, R_{l} are the lattice vectors of layer l and τ_{X,l} points to the site X in layer l within the unit cell selected by R_{l}. We include in the basis all states coupled to k through \(\hat{T}(\theta )\) which are less than a distance \(\frac{28\pi }{3\sqrt{3}{r}_{AB}}\sin \frac{\theta }{2}\) away from it, compute the matrix elements of \(\hat{H}\) in this truncated basis and diagonalize the resulting matrix numerically. In order to simulate the ARPES intensity, we project the eigenstates of the moiré Hamiltonian, \(\hat{H}\), on a planewavelike final state (see Supplementary Note 3 for more details and ref. ^{38} for a detailed discussion of this approach for the case of graphene on hexagonal boron nitride). We determine the broadening of the ARPES signal as well as the decay constant for the intensity of Bernal bilayer signal by fitting to the experimental data.
Data availability
The data used in this study are available from the University of Bath data archive at https://doi.org/10.15125/BATH00864^{68}.
References
 1.
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
 2.
Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013).
 3.
Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).
 4.
Liu, Y. et al. Van der Waals heterostructures and devices. Nat. Rev. Mater. 1, 16042 (2016).
 5.
Ponomarenko, L. A. et al. Cloning of dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).
 6.
Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum hall effect in moiré superlattices. Nature 497, 598–602 (2013).
 7.
Chen, C. et al. Emergence of interfacial polarons from electronphonon coupling in graphene/hBN van der Waals heterostructures. Nano Lett. 18, 1082–1087 (2018).
 8.
Fang, H. et al. Strong interlayer coupling in van der waals heterostructures built from singlelayer chalcogenides. Proc. Natl Acad. Sci. USA 111, 6198–6202 (2014).
 9.
Rivera, P. et al. Interlayer valley excitons in heterobilayers of transition metal dichalcogenides. Nat. Nanotechnol. 13, 1004–1015 (2018).
 10.
Cao, Y. et al. Correlated insulator behaviour at halffilling in magicangle graphene superlattices. Nature 556, 80–84 (2018).
 11.
Cao, Y. et al. Unconventional superconductivity in magicangle graphene superlattices. Nature 556, 43–50 (2018).
 12.
Chari, T., RibeiroPalau, R., Dean, C. R. & Shepard, K. Resistivity of rotated graphitegraphene contacts. Nano Lett. 16, 4477–4482 (2016).
 13.
RibeiroPalau, R. et al. Twistable electronics with dynamically rotatable heterostructures. Science 361, 690–693 (2018).
 14.
Finney, N. R. et al. Tunable crystal symmetry in grapheneboron nitride heterostructures with coexisting moiré superlattices. Nat. Nanotechnol. 14, 1029–1034 (2019).
 15.
Ohta, T. et al. Evidence for interlayer coupling and Moiré periodic potentials in twisted bilayer graphene. Phys. Rev. Lett. 109, 186807 (2012).
 16.
Diaz, H. C. et al. Direct observation of interlayer hybridization and dirac relativistic carriers in graphene/MoS_{2} van der Waals heterostructures. Nano Lett. 15, 1135–1140 (2015).
 17.
Wilson, N. R. et al. Determination of band offsets, hybridization, and exciton binding in 2D semiconductor heterostructures. Sci. Adv. 3, e1601832 (2017).
 18.
Pierucci, D. et al. Band alignment and minigaps in monolayer MoS_{2}graphene van der Waals heterostructures. Nano Lett. 16, 4054–4061 (2016).
 19.
Ulstrup, S. et al. Direct observation of minibands in a twisted graphene/WS_{2} bilayer. Sci. Adv. 6, eaay6104 (2020).
 20.
Yeh, P.C. et al. Direct measurement of the tunable electronic structure of bilayer MoS_{2} by interlayer twist. Nano Lett. 16, 953–959 (2016).
 21.
Zribi, J. et al. Strong interlayer hybridization in the aligned SnS_{2}/WSe_{2} heterobilayer structure. npj 2D Mater. Appl. 3, 27 (2019).
 22.
van der Zande, A. M. et al. Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist. Nano Lett. 14, 3869–3875 (2014).
 23.
Huang, S. et al. Probing the interlayer coupling of twisted bilayer MoS_{2} using photoluminescence spectroscopy. Nano Lett. 14, 5500–5508 (2014).
 24.
Liu, K. et al. Evolution of interlayer coupling in twisted molybdenum disulfide bilayers. Nat. Commun. 5, 4966 (2014).
 25.
de Laissardiere, G. T., Mayou, D. & Magaud, L. Localization of dirac electrons in rotated graphene bilayers. Nano Lett. 10, 804–808 (2010).
 26.
Fang, S. et al. Ab initio tightbinding Hamiltonian for transition metal dichalcogenides. Phys. Rev. B 92, 205108 (2015).
 27.
Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene bilayer with a twist: electronic structure. Phys. Rev. Lett. 99, 256802 (2007).
 28.
Wallbank, J. R., Patel, A. A., MuchaKruczynski, M., Geim, A. K. & Fal’ko, V. I. Generic miniband structure of graphene on a hexagonal substrate. Phys. Rev. B 87, 245408 (2013).
 29.
Peng, H. et al. Substrate doping effect and unusually large angle van Hove singularity evolution in twisted bi and multilayer graphene. Adv. Mater. 29, 1606741 (2017).
 30.
Mattevi, C., Kim, H. & Chhowalla, M. A review of chemical vapour deposition of graphene on copper. J. Mater. Chem. 21, 3324–3334 (2011).
 31.
Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).
 32.
Koshino, M. & McCann, E. Gateinduced interlayer asymmetry in ABAstacked trilayer graphene. Phys. Rev. B 79, 125443 (2009).
 33.
McCann, E. & Fal’ko, V. I. Landaulevel degeneracy and quantum hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006).
 34.
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted doublelayer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).
 35.
Koshino, M. Interlayer interaction in general incommensurate atomic layers. N. J. Phys. 17, 015014 (2015).
 36.
Slater, J. C. & Koster, G. F. Simplified LCAO method for the periodic potential problem. Phys. Rev. 94, 1498–1524 (1954).
 37.
MuchaKruczynski, M. et al. Characterization of graphene through anisotropy of constantenergy maps in angleresolved photoemission. Phys. Rev. B 77, 195403 (2008).
 38.
MuchaKruczynski, M., Wallbank, J. R. & Fal’ko, V. I. Moiré miniband features in the angleresolved photoemission spectra of graphene/hBN heterostructures. Phys. Rev. B 93, 085409 (2016).
 39.
Zhang, Y., Tan, Y.W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).
 40.
Novoselov, K. S. et al. Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene. Nat. Phys. 2, 177–180 (2006).
 41.
Ahn, S. J. et al. Dirac electrons in a dodecagonal graphene quasicrystal. Science 361, 782–786 (2018).
 42.
Yao, W. et al. Quasicrystalline 30^{∘} twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling. Proc. Natl Acad. Sci. USA 115, 6928–6933 (2018).
 43.
Jung, J., Raoux, A., Qiao, Z. & MacDonald, A. H. Ab initio theory of moiré superlattice bands in layered twodimensional materials. Phys. Rev. B 89, 205414 (2014).
 44.
Utama, M. I. B. et al.Visualization of the flat electronic band in twisted bilayer graphene near the magic angle twist. Preprint at https://arxiv.org/abs/1912.00587 (2019).
 45.
Lisi, S. et al. Direct evidence for flat bands in twisted bilayer graphene from nanoARPES. Preprint at https://arxiv.org/abs/2002.02289 (2020).
 46.
Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures. Nature 567, 81–86 (2019).
 47.
Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).
 48.
Wallbank, J. R., MuchaKruczynski, M. & Fal’ko, V. I. Moiré minibands in graphene heterostructures with almost commensurate \(\sqrt{3}\times \sqrt{3}\) hexagonal crystals. Phys. Rev. B 88, 155415 (2013).
 49.
Yu, H., Wang, Y., Tong, Q., Xu, X. & Yao, W. Anomalous light cones and valley optical selection rules of interlayer excitons in twisted heterobilayers. Phys. Rev. Lett. 115, 187002 (2015).
 50.
Tong, Q. et al. Topological mosaics in moiré superlattices of van der Waals heterobilayers. Nat. Phys. 13, 356–362 (2017).
 51.
RuizTijerina, D. A. & Fal’ko, V. I. Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transitionmetal dichalcogenides. Phys. Rev. B 99, 125424 (2019).
 52.
Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019).
 53.
Jin, W. et al. Direct measurement of the thicknessdependent electronic band structure of MoS_{2} using angleresolved photoemission spectroscopy. Phys. Rev. Lett. 111, 106801 (2013).
 54.
Zhang, Y. et al. Direct observation of the transition from indirect to direct bandgap in atomically thin epitaxial MoSe_{2}. Nat. Nanotechnol. 9, 111–115 (2014).
 55.
Roldan, R. et al. Electronic properties of singlelayer and multilayer transition metal dichalcogenides MX_{2} (M = Mo, W and X = S, Se). Ann. der Phys. 526, 347–357 (2014).
 56.
Chen, S. et al. Electrically tunable correlated and topological states in twisted monolayerbilayer graphene. Preprint at https://arxiv.org/abs/2004.11340 (2020).
 57.
Shi, Y. et al. Tunable van Hove singularities and correlated states in twisted trilayer graphene. Preprint at https://arxiv.org/abs/2004.12414 (2020).
 58.
Liu, X. et al. Spinpolarized correlated insulator and superconductor in twisted double bilayer graphene. Preprint at https://arxiv.org/abs/1903.08130 (2019).
 59.
Burg, G. W. et al. Correlated insulating states in twisted double bilayer graphene. Phys. Rev. Lett. 123, 197702 (2019).
 60.
Shen, C. et al. Correlated states in twisted double bilayer graphene. Nat. Phys. 16, 520–525 (2020).
 61.
Cao, Y. et al. Tunable correlated states and spinpolarized phases in twisted bilayerbilayer graphene. Nature 583, 215–220 (2020).
 62.
He, M. et al. Tunable correlationdriven symmetry breaking in twisted double bilayer graphene. Preprint at https://arxiv.org/abs/2002.08904 (2020).
 63.
Rickhaus, P. et al. Densitywave states in twisted doublebilayer graphene. Preprint at https://arxiv.org/abs/2005.05373 (2020).
 64.
An, L. et al. Interaction effects and superconductivity signatures in twisted doublebilayer WSe2. Preprint at https://arxiv.org/abs/1907.03966 (2019).
 65.
Nam, N. N. T. & Koshino, M. Lattice relaxation and energy band modulation in twisted bilayer graphene. Phys. Rev. B 96, 075311 (2017).
 66.
Lin, X. & Tomanek, D. Minimum model for the electronic structure of twisted bilayer graphene and related structures. Phys. Rev. B 98, 081410 (2019).
 67.
Zhao, X.J., Yang, Y., Zhang, D.B. & Wei, S.H. Formation of bloch flat bands in polar twisted bilayers without magic angles. Phys. Rev. Lett. 124, 086401 (2020).
 68.
Thompson, J. J. P. Dataset for article determination of interatomic coupling between twodimensional crystals using angleresolved photoemission spectroscopy by Thompson Bath: University of Bath Research Data Archive. https://doi.org/10.15125/BATH00864.
Acknowledgements
J.J.P.T. was supported by EPSRC through the University of Bath Doctoral Training Partnership, EPSRC Grant No. EP/M507982/1. N.C. was supported by the Institute for Mathematical Innovation at the University of Bath. M.M.K. acknowledges funding from the University of Bath International Research Funding Scheme. D.P. and H.P. acknowledge support from the China Scholarship Council.
Author information
Affiliations
Contributions
H.P. and N.S. carried out the ARPES measurements with the assistance of A.B. and Y.C. D.P. analysed the ARPES data. H.W. and H.L.P. grew the samples. J.J.P.T. and M.M.K. built the theoretical model. J.J.P.T. and N.C. performed the miniband and ARPES simulations. M.M.K. conceived the project and supervised the theoretical analysis. J.J.P.T. and M.M.K. wrote the manuscript with input from D.P., A.B. and Y.C.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks the anonymous reviewers 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
Thompson, J.J.P., Pei, D., Peng, H. et al. Determination of interatomic coupling between twodimensional crystals using angleresolved photoemission spectroscopy. Nat Commun 11, 3582 (2020). https://doi.org/10.1038/s41467020174120
Received:
Accepted:
Published:
Further reading

In Operando Angle‐Resolved Photoemission Spectroscopy with Nanoscale Spatial Resolution: Spatial Mapping of the Electronic Structure of Twisted Bilayer Graphene
Small Science (2021)

Molecular switching operation in gate constricted interface of MoS2 and hBN heterostructure
Applied Materials Today (2021)

Interfacial Electronic Properties and Adjustable Schottky Barrier at Graphene/CsPbI 3 van der Waals Heterostructures
physica status solidi (RRL) – Rapid Research Letters (2021)
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.