Suk Hyun Sung1,Yin Min Goh1,Hyobin Yoo2,Rebecca Engelke3,Hongchao Xie1,Zidong Li1,Andrew Ye4,Parag Deotare1,Andrew Mannix5,Jiwoong Park4,Liuyan Zhao1,Philip Kim3,Robert Hovden1
University of Michigan1,Sogang University2,Harvard University3,The University of Chicago4,Stanford University5
Suk Hyun Sung1,Yin Min Goh1,Hyobin Yoo2,Rebecca Engelke3,Hongchao Xie1,Zidong Li1,Andrew Ye4,Parag Deotare1,Andrew Mannix5,Jiwoong Park4,Liuyan Zhao1,Philip Kim3,Robert Hovden1
University of Michigan1,Sogang University2,Harvard University3,The University of Chicago4,Stanford University5
At low twist angles, moiré heterostructures periodically restructure and spontaneously break symmetry to form complex superstructures [1,2,3]. Periodic restructuring of twisted 2D materials is a direct consequence of competition between interlayer van der Waals registry energy and intralayer elastic energy cost. Here, we use a torsional periodic lattice distortion (PLD) model [4] to concisely describe the relaxed superstructure and electron diffraction patterns across a variety of twisted 2D materials. The moiré of twisted bilayer graphene (TBG) periodically unwinds energetically favorable AB/BA stacked regions to increase AB/BA area, and unfavorable AA stacked regions are further twisted to decrease their area. We use a torsional PLD model to describe the in-plane distortion of twisted 2D materials. The torsional PLD (<b>Δ</b><sub>N</sub>) is made-up of three non-orthogonal, transverse distortion waves of equal amplitudes and harmonics thereof.<br/><br/>Torsional PLDs in twisted 2D materials are a universal phenomenon at low twist angles and not limited to TBG [2,3,5,6]. We found evidence of torsional PLD existing in four distinct twisted 2D systems: a) low twist angle TBG, b) 4-layer (4L) of WS<sub>2</sub>, c) twisted double bilayer CrI<sub>3</sub> and d) twisted WS<sub>2</sub>/MoSe<sub>2</sub> heterostructure. Quantum mechanical multislice simulation [7] of electron diffraction patterns with torsional PLDs applied show good agreement with the experimental diffraction patterns. Notably, for near magic-angle TBG, single-harmonic torsional PLD (<b>Δ</b><sub>1</sub>) matches quantitatively with the experimental SAED patterns [4].<br/><br/>A torsional PLD model reduces the complexity of low-twist angle moiré crystals to a single order parameter across a variety of 2D materials ranging from graphene, metal dichalcogenides, metal trihalides homostructures to heterostructures of 2D materials.<br/><br/><br/><br/><br/><br/>References:<br/><br/>[1] H Yoo et al., Nat. Mater. <b>18</b> (2019), p. 448. doi: 10.1038/s41563-019-0346-z<br/>[2] AJ Mannix et al., Nat. Nanotechnol. (2022). doi: 10.1038/s41565-021-01061-5<br/>[3] H Xie et al., Nat. Phys. <b>18</b> (2022), p. 30. doi: 10.1038/s41567-021-01408-8<br/>[4] SH Sung et al., arXiv:2203.06510 (2022). doi: 10.48550/arXiv.2203.06510<br/>[5] K Yasuda et al., Science <b>372</b> (2021) p. 1458 doi: 10.1126/science.abd3230<br/>[6] A Weston et al., Nat. Nanotechnol. <b>15</b> (2020) p. 592 doi: 10.1038/s41565-020-0682-9<br/>[7] EJ Kirkland, “Advanced Computing in Electron Microscopy 2<sup>nd</sup> edition” (2010)