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Nanoscale Turing patterns in a bismuth monolayer


Turing’s reaction–diffusion theory of morphogenesis has been very successful for understanding macroscopic patterns within complex objects ranging from biological systems to sand dunes. However, Turing patterns on microscopic length scales are extremely rare. Here we show that a strained atomic bismuth monolayer assembled on the surface of NbSe2—and subject to interatomic interactions and kinetics—displays Turing patterns. Our reaction–diffusion model produces stripe patterns with a period of five atoms (approximately 2 nm) and domain walls with Y-shaped junctions that bear a striking resemblance to what has been experimentally observed. Our work establishes that Turing patterns can occur at the atomic scale in a hard condensed-matter setting.

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Fig. 1: Crystal structure of monolayer Bi on the top Se layer.
Fig. 2: Emergence of nanoscale patterns in an atomic monolayer.
Fig. 3: Changes in wavenumber in Turing patterns.
Fig. 4: Dynamic properties of Turing patterns.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The codes that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.


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We thank A. Fang for providing the raw STM data discussed in this manuscript, and N. Sasaki and K. Izawa for fruitful discussions. Funding: this work was initiated through a ‘QuantEmX’ Exchange Awards (A.K.) and the Promotion of Joint International Research (Y.F.) at ESPCI. Work at UEC (Japan) was supported by JSPS grant nos. 15KK0155, 18KK0132 and 19K21844. Work at Stanford University was supported by the US Department of Energy (DOE) Office of Basic Energy Science, Division of Materials Science and Engineering, Stanford, under contract no. DE-AC02- 76SF00515.

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Y.F., K.B. and A.K. initiated this work. Y.F. and H.K. constructed the theory and carried out the calculations. Y.F., K.B. and A.K. wrote the paper.

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Correspondence to Yuki Fuseya or Aharon Kapitulnik.

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Peer review informationNature Physics thanks Denise Gabuzda, Talvikki Hovatta and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Fuseya, Y., Katsuno, H., Behnia, K. et al. Nanoscale Turing patterns in a bismuth monolayer. Nat. Phys. 17, 1031–1036 (2021).

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