Strain fields in twisted bilayer graphene

Abstract

Twisted bilayer graphene (TBG) displays a host of correlated electronic phases associated with the formation of flat electronic bands near an interlayer 'magic angle' (MA) of 1.1 degrees 1-9 . Intralayer lattice reconstruction 10-13 , which involves local rotations with consequent localized strain 14,15 , and symmetry breaking due to extrinsic heterostrain have significant implications for electronic behavior at the MA 9,16,17 . Although reconstruction and strain are therefore fundamental to the properties of TBG, directly mapping the reconstruction mechanics in the MA regime has been elusive and the strain tensor fields of TBG have not been measured. Here, we introduce Bragg interferometry, based on four-dimensional scanning transmission electron microscopy (4D-STEM) 18-21 , to capture the atomic displacement fields of TBG with twist angles ranging from 0.1 to 1.6 degrees. Sub-nanometer resolution allows us to image atomic reconstruction in MA-TBG and resolve twist angle disorder at the level of individual moiré domains. We quantitatively map the strain tensor fields and uncover that reconstruction proceeds in two distinct regimes depending on the twist angle-in contrast to previous models depicting a single continuous process-and we distinguish the contributions of these regimes to the band structure. Further, we find that over a twist angle range encompassing the MA, applied heterostrain accumulates anisotropically in saddle point (SP) regions to generate distinctive striped strain phases. Our results thus establish the reconstruction mechanics underpinning the twist angle dependent electronic behavior of TBG, and provide a new framework for directly visualizing strain and reconstruction in other moiré materials.

Imaging moiré deformation and dynamics in twisted bilayer graphene

“…1 e. The observed twist angles on this sample range from <0.1° to 0.7°. For smaller angles, we observe local reconstruction towards Bernal stacking within the moiré lattice, consistent with literature 12 , 15 . The best resolution was reached on another sample with a twist angle of 1.3° (See Supplementary Fig.…”

“…The twist angle variation within each domain is much smaller than the variation in twist angle between the separate, fold-bounded areas. Within domains, standard deviations range from 0.005° to 0.015°, i.e., significantly smaller (by a factor 3–10) than previously reported 13 , 15 , 33 . The strain observed is around a few tenths of a percent, which is considerable.…”

“…Our data suggest that domain boundary displacement follows a random pattern of forward and backward steps. This indicates a possible source for the twist angle disorder observed in low(er) temperature experiments 10 , 13 , 15 , 33 : frozen-in thermal fluctuations of the moiré lattice. The thermal fluctuations found, corresponding to ±0.005° for twist angle and ±0.02% for strain, are smaller than the extracted static deformations, though not negligible.…”

Interferometric 4D‐STEM for Lattice Distortion and Interlayer Spacing Measurements of Bilayer and Trilayer 2D Materials

“…Very recently, a 4D-STEM technique based on Bragg interferometry was demonstrated that allowed in-plane structure distortions to be measured in bilayer graphene. [29] While this provides the valuable ability to resolve local structural distortions, it was applied only to low-twist angle materials, up to 1.6°, and again cannot provide information about out-of-plane structure, such as the interlayer spacings. Finally, none of these projection techniques can provide information about the direction of twist of one layer with respect to the other, which could be useful for studying chiral materials.…”

A scalable network model for electrically tunable ferroelectric domain structure in twistronic bilayers of two-dimensional semiconductors

“…Two-dimensional material twistronics, fueled by discoveries of new phenomena in twisted graphene bilayers [1][2][3][4][5][6][7][8][9][10] and trilayers [11][12][13][14][15][16], has recently expanded onto a broader range of van der Waals systems [17][18][19][20][21][22][23]. In general, twistronic structures are associated with geometrical moiré patterns: a periodic variation local stacking of the two layers.…”