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Polarons in two-dimensional atomic crystals


Polarons are quasiparticles that emerge from the interaction of fermionic particles with bosonic fields1. In crystalline solids, polarons form when electrons and holes become dressed by lattice vibrations. While experimental signatures of polarons in bulk three-dimensional materials abound4,5,6,7,8,9,10,11,12,13,14, only rarely have polarons been observed in two-dimensional atomic crystals. Here, we shed light on this asymmetry by developing a quantitative ab initio theory of polarons in atomically thin crystals. Using this conceptual framework, we determine the real-space structure of the recently observed hole polaron in hexagonal boron nitride, discover a critical condition for the existence of polarons in two-dimensional crystals and establish the key materials descriptors and the universal laws that underpin polaron physics in two dimensions.

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Fig. 1: Large hole polaron in bulk h-BN.
Fig. 2: Fröhlich polaron in monolayer h-BN.
Fig. 3: Continuum model of polarons in 2D atomic crystals.

Data availability

The datasets generated during and analysed during the current study are available in the Zenodo repository at Source data are provided with this paper.

Code availability

The codes used for the current study, namely Quantum ESPRESSO23, wannier90 (ref. 24) and EPW25, are all open-source software distributed via their corresponding websites. A modified version of EPW including the kernel of ref. 21 is available from the corresponding author on reasonable request.


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This research is primarily supported by the Computational Materials Sciences Program funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under award no. DE-SC0020129 (software development, theoretical model, ab initio calculations). This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract no. DE-AC02-05CH11231. We also acknowledge the Texas Advanced Computing Center at The University of Texas at Austin for providing additional high-performance computing resources, including the Frontera and Lonestar6 systems, that have contributed to the research results reported within this paper ( In the final stage, W.H.S. was supported by the Science and Technology Development Fund of Macau SAR under grant no. 0102/2019/A2 (ab initio calculations and data analysis). W.H.S. also acknowledges the Information and Communication Technology Office at the University of Macau and the LvLiang Cloud Computing Center of China for providing extra high-performance computing resources, including the High-Performance Computing Cluster and TianHe-2 systems.

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Authors and Affiliations



W.H.S. performed the calculations and data analysis. F.G. conceived and supervised the project. W.H.S. and F.G. developed the theoretical model and wrote the paper.

Corresponding author

Correspondence to Feliciano Giustino.

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Nature Physics thanks Karsten Jacobsen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Computational Methods, Notes 1–7, Tables 1–3, Figs. 1–12 and Refs.

Supplementary Data 1

Calculated polaron formation energies for Supplementary Fig. 1.

Supplementary Data 2

Calculated polaron formation energies for Supplementary Fig. 2.

Supplementary Data 3

Calculated polaron formation energies for Supplementary Fig. 3.

Supplementary Data 4

Calculated polaron formation energies for Supplementary Fig. 4.

Supplementary Data 5

Calculated polaron charge density for Supplementary Fig. 5.

Supplementary Data 6

Calculated atomic displacements for Supplementary Fig. 6.

Supplementary Data 7

Calculated polaron charge density for Supplementary Fig. 7.

Supplementary Data 8

Calculated potential function for Supplementary Fig. 8.

Supplementary Data 9

Calculated polaron charge density and polaron weights for Supplementary Fig. 9.

Supplementary Data 10

Calculated polaron charge density and polaron weights for Supplementary Fig. 10.

Supplementary Data 11

Calculated polaron weights for Supplementary Fig. 11.

Supplementary Data 12

Calculated polaron charge density and polaron weights for Supplementary Fig. 12.

Source data

Source Data Fig. 1

Source data for Fig. 1.

Source Data Fig. 2

Source data for Fig. 2.

Source Data Fig. 3

Source data for Fig. 3.

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Sio, W.H., Giustino, F. Polarons in two-dimensional atomic crystals. Nat. Phys. 19, 629–636 (2023).

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