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# Strain fields in twisted bilayer graphene

## Abstract

Van der Waals heteroepitaxy allows deterministic control over lattice mismatch or azimuthal orientation between atomic layers to produce long-wavelength superlattices. The resulting electronic phases depend critically on the superlattice periodicity and localized structural deformations that introduce disorder and strain. In this study we used Bragg interferometry to capture atomic displacement fields in twisted bilayer graphene with twist angles <2°. Nanoscale spatial fluctuations in twist angle and uniaxial heterostrain were statistically evaluated, revealing the prevalence of short-range disorder in moiré heterostructures. By quantitatively mapping strain tensor fields, we uncovered two regimes of structural relaxation and disentangled the electronic contributions of constituent rotation modes. Further, we found that applied heterostrain accumulates anisotropically in saddle-point regions, generating distinctive striped strain phases. Our results establish the reconstruction mechanics underpinning the twist-angle-dependent electronic behaviour of twisted bilayer graphene and provide a framework for directly visualizing structural relaxation, disorder and strain in moiré materials.

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## Data availability

The datasets generated and/or analysed during the current study have been made publicly available from the Zenodo repository. Digital object identifiers (DOIs) to these datasets are provided in refs. 12–30 of the Supplementary Information. Additional information on these datasets and the manuscript figures to which each dataset corresponds are detailed in Supplementary Table 4.

## Code availability

The computer code used for the dataset processing and strain analysis has been made publicly available at https://github.com/bediakolab/StrainFieldsInTwistedBilayerGraphene.

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## Acknowledgements

We thank P. Kim for discussions. The major experimental work was supported by the Office of Naval Research Young Investigator Program under award no. N00014-19-1-2199 (D.K.B.). M.V.W. acknowledges support from an NSF GRFP award and a UC Berkeley Chancellor’s Fellowship. Work at the Molecular Foundry was supported by the Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under contract no. DE-AC02-05CH1123. C.O. acknowledges the support of the Department of Energy Early Career Research Award programme. S.C. acknowledges support from the NSF through grant no. OIA-1921199. J.C. and H.G.B. acknowledge support from the Presidential Early Career Award for Scientists and Engineers (PECASE) through the US Department of Energy. D.K.B. acknowledges support from the Rose Hills Foundation through the Rose Hills Innovator Program. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant no. JPMXP0112101001, JSPS KAKENHI grant no. JP20H00354 and the CREST (JPMJCR15F3), JST.

## Author information

Authors

### Contributions

N.P.K., M.V.W., C.O., K.C.B., H.G.B. and D.K.B. conceived the study. M.V.W. designed and fabricated the samples. M.V.W., K.C.B. and J.C. acquired the 4D-STEM data. N.P.K., C.O. and H.G.B. created the data analysis code. S.C. carried out the band structure calculations and finite-element modelling. T.T. and K.W. provided the bulk hBN crystals. N.P.K. and M.V.W. processed the data. N.P.K., M.V.W. and D.K.B. analysed the data and wrote the manuscript. All the authors contributed to the overall scientific interpretation and edited the manuscript.

### Corresponding author

Correspondence to D. Kwabena Bediako.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Materials thanks Jeil Jung, Jannik Meyer and the other, anonymous, reviewer(s) 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

### Supplementary Information

Supplementary Sections 1–12, Figs. 1–28 and Tables 1–3.

### Supplementary Table 4

Details of all relevant 4D-STEM datasets (uploaded to Zenodo repository) corresponding to refs. 12–30 of the Supplementary Informaton.

## Rights and permissions

Reprints and Permissions

Kazmierczak, N.P., Van Winkle, M., Ophus, C. et al. Strain fields in twisted bilayer graphene. Nat. Mater. (2021). https://doi.org/10.1038/s41563-021-00973-w

• Accepted:

• Published:

• ### Nano-imaging photoresponse in a moiré unit cell of minimally twisted bilayer graphene

• Niels C. H. Hesp
• , Iacopo Torre
• , David Barcons-Ruiz
• , Hanan Herzig Sheinfux
• , Kenji Watanabe
• , Takashi Taniguchi
• , Roshan Krishna Kumar
•  & Frank H. L. Koppens

Nature Communications (2021)