Determination of interatomic coupling between two-dimensional crystals using angle-resolved photoemission spectroscopy

Following the isolation of graphene, many other atomically thin two-dimensional crystals have been produced and can even be stacked on top of each other in a desired order to form so called van der Waals heterostructures. In contrast to conventional heterostructures, in which chemical bonding at interfaces between two materials modifies their properties and requires lattice matching, thus restricting what materials can be placed next to each other, stacks of two-dimensional crystals are held together by weak forces without directional bonding. As a result, any two can be stacked together, providing extraordinary flexibility in the design of vertical structures. Moreover, subtle changes in the stacking, especially the angle between the crystallographic axes (special directions in the arrangement of atoms) of two adjacent layers, can have big impact on the properties of the whole heterostructure, with examples including appearance of superconductivity in twisted bilayer graphene with twist angle around 1.1 degrees (known as magic-angle twisted bilayer graphene).
In our work, we show that the coupling between atoms in two two-dimensional crystals, knowledge of which is necessary to describe the properties of the stack, can be determined by studying a structure made of three layers with two similar interfaces but one with crystallographic axes aligned and one twisted (see Figures 1 and 2). This is because each of the interfaces provides complementary information and together they enable self-consistent determination of the coupling. We use angle-resolved photoemission spectroscopy measurements carried out at the Spectromicroscopy beamline at Elettra and obtain interatomic coupling for carbon atoms by studying a three-layer stack of graphene. We also show that our result can be used to predict photoemission spectra of structures with different twist angles and number of layers. Our approach demonstrates how to extract fundamental information about interlayer coupling in a stack of two-dimensional crystals and can be applied to many other van der Waals interfaces.

Figure 1.    (a) Comparison of aligned and twisted interfaces for two-dimensional crystals and the corresponding descriptions in the real and reciprocal spaces. Blue and black balls indicate atoms in the top and bottom layer, respectively. (b) Schematic of twisted trilayer graphene with monolayer (blue) stacked at an angle on top of a Bernal bilayer (black). The red and purple arrows indicate the interlayer couplings for the Bernal and twisted interfaces. Inset shows photoemission intensity from copper substrate which is attenuated by graphene layers above, providing a measure of graphene layer number (the red arrows indicate each of the graphene layers and the cyan line corresponds to the distance of 10 μm).

Figure 2.    
(a) Left: Real-space interatomic coupling, t(∣r∣, c0), as a function of distance ∣r∣between carbon atoms. Right: Two-dimensional Fourier transform of the interatomic coupling, t(∣r∣, c0), as a function of wave vector ∣q∣. (b) Photoemission intensity for twisted trilayer with twist θ= 9.6°, measured along the direction connecting Brillouin zone corners K2 and K1 of the bilayer and monolayer crystals. The calculated miniband structure is shown with red dashed lines. White dashed and solid lines indicate the important energies used to fit the parameters of our theoretical model.


This research was conducted by the following research team:

J. J. P. Thompson1,2, D. Pei3, H. Peng3, H. Wang4, N. Channa1,5, H. L. Peng4, A. Barinov6, N. B. M. Schröter3,7, Y. Chen3, M. Mucha-Kruczyński1,8


Department of Physics, University of Bath, UK.
Chalmers University of Technology, Department of Physics, Sweden.
Clarendon Laboratory, Department of Physics, University of Oxford, UK.
Center for Nanochemistry, Beijing National Laboratory for Molecular Sciences, Peking University, China.
Department of Physics, University of Warwick, UK.
Elettra - Sincrotrone Trieste SCpA, Italy.
Swiss Light Source, Paul Scherrer Institute, Switzerland.
Centre for Nanoscience and Nanotechnology, University of Bath, UK.

Contact persons:

M. Mucha-Kruczyński, contact e-mail:



J. J. P. Thompson, D. Pei, H. Peng, H. Wang, N. Channa, H. L. Peng, A. Barinov, N. B. M. Schröter, Y. Chen, M. Mucha-Kruczyński,"Determination of interatomic coupling between two-dimensional crystals using angle-resolved photoemission spectroscopy", Nature Communications 11, 3582 (2020), DOI: 10.1038/s41467-020-17412-0,  
Last Updated on Thursday, 20 August 2020 11:46