The term heterostrain was proposed in 2018 in the context of materials science to simplify the designation of possible strain situations in van der Waals heterostructures where two (or more) two-dimensional materials are stacked on top of each other.[1] These layers can experience the same deformation (homostrain) or different deformations (heterostrain). In addition to twist, heterostrain can have important consequences on the electronic[2][3] and optical[4] properties of the resulting structure. As such, the control of heterostrain[5][6] is emerging as a sub-field of straintronics in which the properties of 2D materials are controlled by strain. Recent works have reported a deterministic control of heterostrain by sample processing[7] or with the tip of an AFM[8] of particular interest in twisted heterostructures. Heterostrain alone (without twist) has also been identified as a parameter to tune the electronic properties of van der Waals structures as for example in twisted graphene layers with biaxial heterostrain.[9]

Etymology

Heterostrain is constructed from the Greek prefix hetero- (different) and the noun strain. It means that the two layers constituting the structure are subject to different strains.[1] This is in contrast with homostrain in which the two layers as subject to the same strain.[1] Heterostrain is designated as "relative strain" by some authors.[10]

Manifestation and measurement of heterostrain

For simplicity, the case of two graphene layers is considered. The description can be generalized for the case of different 2D materials forming an heterostructure.

In nature, the two graphene layers usually stack with a shift of half a unit cell. This configuration is the most energetically favorable and is found in graphite. If one layer is strained while the other is left intact, a moiré pattern signaling the regions where the atomic lattices of the two layers are in or out of registry. The shape of the moiré pattern depends on the type of strain.

  • If the layer is deformed along one direction (uniaxial heterostrain), the moiré is one dimensional.
  • If the layer is strained in the same way along two directions (biaxial heterostrain), the moiré is a two-dimensional superstructure.
Bernal stacking
Graphene layers in Bernal stacking
Biaxial heterostrain
Graphene layers with biaxial heterostrain
Uniaxial heterostrain
Graphene layers with uniaxial heterostrain

In General, a layer can be deformed by an arbitrary combination of both types of heterostrain.

Heterostrain can be measured by scanning tunneling microscope which provides images showing both the atomic lattice of the first layer and the moiré superlattice. Relating the atomic lattice to the moiré lattice allows to determine entirely the relative arrangement of the layers (biaxial, uniaxial heterostrain and twist).[11] The method is immune to calibration artifacts which affect the image of the two layers identically which cancels out in the relative measurement. Alternatively, with a well calibrated microscope and if biaxial heterostrain is low enough, it is possible to determine twist and uniaxial heterostrain from the knowledge of the moiré period in all directions.[12] On the contrary it is much more difficult to determine homostrain which necessitates a calibration sample.

Origin and impact of heterostrain

Heterostrain is generated during the fabrication of the 2D materials stack. It can result from a meta-stable configuration during bottom up assembly[1] or from the layer manipulation in the tear and stack technique.[13] It has been shown to be ubiquitous in twisted graphene layers near the magic twist angle and to be the main factor in the flat band width of those systems.[2][3] Heterostrain has a much larger impact on electronic properties than homostrain.[1] It explains some of the sample variability which had previously been puzzeling.[3][14] Research is now moving towards understanding the impact of spatial fluctuations of heterostrain.[15]

References

  1. 1 2 3 4 5 Huder, Loïc; Artaud, Alexandre; Le Quang, Toai; de Laissardière, Guy Trambly; Jansen, Aloysius G. M.; Lapertot, Gérard; Chapelier, Claude; Renard, Vincent T. (2018-04-12). "Electronic Spectrum of Twisted Graphene Layers under Heterostrain". Physical Review Letters. 120 (15): 156405. arXiv:1803.03505. Bibcode:2018PhRvL.120o6405H. doi:10.1103/PhysRevLett.120.156405. PMID 29756887. S2CID 21734003.
  2. 1 2 Bi, Zhen; Yuan, Noah F. Q.; Fu, Liang (2019-07-31). "Designing flat bands by strain". Physical Review B. 100 (3): 035448. arXiv:1902.10146. Bibcode:2019PhRvB.100c5448B. doi:10.1103/PhysRevB.100.035448. ISSN 2469-9950. S2CID 118982311.
  3. 1 2 3 Mesple, Florie; Missaoui, Ahmed; Cea, Tommaso; Huder, Loic; Guinea, Francisco; Trambly de Laissardière, Guy; Chapelier, Claude; Renard, Vincent T. (2021-09-17). "Heterostrain Determines Flat Bands in Magic-Angle Twisted Graphene Layers". Physical Review Letters. 127 (12): 126405. arXiv:2012.02475. Bibcode:2021PhRvL.127l6405M. doi:10.1103/PhysRevLett.127.126405. ISSN 0031-9007. PMID 34597066. S2CID 227305789.
  4. ↑ Zheng, Huiyuan; Zhai, Dawei; Yao, Wang (2021-10-01). "Twist versus heterostrain control of optical properties of moiré exciton minibands". 2D Materials. 8 (4): 044016. arXiv:2110.01487. Bibcode:2021TDM.....8d4016Z. doi:10.1088/2053-1583/ac2d16. ISSN 2053-1583. S2CID 238259645.
  5. ↑ Kapfer, Mäelle; Jessen, Bjarke S.; Eisele, Megan E.; Fu, Matthew; Danielsen, Dorte R.; Darlington, Thomas P.; Moore, Samuel L.; Finney, Nathan R.; Marchese, Ariane; Hsieh, Valerie; Majchrzak, Paulina; Jiang, Zhihao; Biswas, Deepnarayan; Dudin, Pavel; Avila, José (2023). "Programming twist angle and strain profiles in 2D materials". Science. 381 (6658): 677–681. arXiv:2209.10696. Bibcode:2023Sci...381..677K. doi:10.1126/science.ade9995.
  6. ↑ Peña, Tara; Dey, Aditya; Chowdhury, Shoieb A.; Azizimanesh, Ahmad; Hou, Wenhui; Sewaket, Arfan; Watson, Carla L.; Askari, Hesam; Wu, Stephen M. (2023). "Moiré engineering in 2D heterostructures with process-induced strain". Applied Physics Letters. 122 (14): 143101. arXiv:2210.03480. Bibcode:2023ApPhL.122n3101P. doi:10.1063/5.0142406. S2CID 252762502.
  7. ↑ Tara, Pena (2023). "Moiré engineering in 2D heterostructures with process-induced strain". Appl. Phys. Lett. 122 (14): 143101. arXiv:2210.03480. Bibcode:2023ApPhL.122n3101P. doi:10.1063/5.0142406. S2CID 252762502 – via AIP Publishing.
  8. ↑ Kapfer, Maëlle; Jessen, Bjarke S.; Eisele, Megan E.; Fu, Matthew; Danielsen, Dorte R.; Darlington, Thomas P.; Moore, Samuel L.; Finney, Nathan R.; Marchese, Ariane; Hsieh, Valerie; Majchrzak, Paulina; Jiang, Zhihao; Biswas, Deepnarayan; Dudin, Pavel; Avila, José (2023-08-11). "Programming twist angle and strain profiles in 2D materials". Science. 381 (6658): 677–681. arXiv:2209.10696. Bibcode:2023Sci...381..677K. doi:10.1126/science.ade9995. ISSN 0036-8075. PMID 37561852. S2CID 252438585.
  9. ↑ Mesple, Florie; Walet, Niels R.; Trambly de Laissardière, Guy; Guinea, Francisco; Došenović, Djordje; Okuno, Hanako; Paillet, Colin; Michon, Adrien; Chapelier, Claude; Renard, Vincent T. (September 2023). "Giant Atomic Swirl in Graphene Bilayers with Biaxial Heterostrain". Advanced Materials. 35 (41). arXiv:2308.13230. Bibcode:2023AdM....3506312M. doi:10.1002/adma.202306312. ISSN 0935-9648. PMID 37615204.
  10. ↑ Engelke, Rebecca; Yoo, Hyobin; Carr, Stephen; Xu, Kevin; Cazeaux, Paul; Allen, Richard; Valdivia, Andres Mier; Luskin, Mitchell; Kaxiras, Efthimios; Kim, Minhyong; Han, Jung Hoon; Kim, Philip (2022-07-16). "Non-Abelian topological defects and strain mapping in 2D moiré materials". arXiv:2207.05276 [cond-mat.mes-hall].
  11. ↑ Artaud, A.; Magaud, L.; Le Quang, T.; Guisset, V.; David, P.; Chapelier, C.; Coraux, J. (2016). "Universal classification of twisted, strained and sheared graphene moiré superlattices". Scientific Reports. 6 (1): 25670. arXiv:1605.07356. Bibcode:2016NatSR...625670A. doi:10.1038/srep25670. ISSN 2045-2322. PMC 4867435. PMID 27181495.
  12. ↑ Kerelsky, Alexander; McGilly, Leo J.; Kennes, Dante M.; Xian, Lede; Yankowitz, Matthew; Chen, Shaowen; Watanabe, K.; Taniguchi, T.; Hone, James; Dean, Cory; Rubio, Angel (2019). "Maximized electron interactions at the magic angle in twisted bilayer graphene". Nature. 572 (7767): 95–100. arXiv:1812.08776. Bibcode:2019Natur.572...95K. doi:10.1038/s41586-019-1431-9. ISSN 0028-0836. PMID 31367030. S2CID 199057045.
  13. ↑ Kim, Kyounghwan; Yankowitz, Matthew; Fallahazad, Babak; Kang, Sangwoo; Movva, Hema C. P.; Huang, Shengqiang; Larentis, Stefano; Corbet, Chris M.; Taniguchi, Takashi; Watanabe, Kenji; Banerjee, Sanjay K. (2016-03-09). "van der Waals Heterostructures with High Accuracy Rotational Alignment". Nano Letters. 16 (3): 1989–1995. Bibcode:2016NanoL..16.1989K. doi:10.1021/acs.nanolett.5b05263. ISSN 1530-6984. PMID 26859527.
  14. ↑ Parker, Daniel E.; Soejima, Tomohiro; Hauschild, Johannes; Zaletel, Michael P.; Bultinck, Nick (2021-07-06). "Strain-Induced Quantum Phase Transitions in Magic-Angle Graphene". Physical Review Letters. 127 (2): 027601. arXiv:2012.09885. Bibcode:2021PhRvL.127b7601P. doi:10.1103/PhysRevLett.127.027601. hdl:1854/LU-01GQF91F1WRRP5WQDFR3XBHF2J. ISSN 0031-9007. OSTI 1821673. PMID 34296891. S2CID 229331622.
  15. ↑ Nakatsuji, Naoto; Koshino, Mikito (2022-06-14). "Moiré disorder effect in twisted bilayer graphene". Physical Review B. 105 (24): 245408. arXiv:2204.06177. Bibcode:2022PhRvB.105x5408N. doi:10.1103/PhysRevB.105.245408. ISSN 2469-9950. S2CID 248157360.
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