Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Polarons in two-dimensional atomic crystals

Nature Physics volume 19, pages 629–636 (2023)Cite this article

Subjects

Abstract

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.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

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 https://doi.org/10.5281/zenodo.7496896. 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.

References

  1. Franchini, C., Reticcioli, M., Setvin, M. & Diebold, U. Polarons in materials. Nat. Rev. Mater. 6, 560 (2021).

    Article  ADS  Google Scholar 

  2. Emin, D. Polarons (Cambridge Univ. Press, 2012).

    Book  Google Scholar 

  3. Alexandrov, A. S. & Devreese, J. T. Advances in Polaron Physics Springer Series in Solid-State Sciences (Springer, 2012).

  4. Riley, J. M. et al. Crossover from lattice to plasmonic polarons of a spin-polarised electron gas in ferromagnetic EuO. Nat. Commun. 9, 20 (2018).

  5. Wang, Z. et al. Tailoring the nature and strength of electron–phonon interactions in the SrTiO3(001) 2D electron liquid. Nat. Mater. 15, 835 (2016).

    Article  ADS  Google Scholar 

  6. Chen, C., Avila, J., Frantzeskakis, E., Levy, A. & Asensio, M. C. Observation of a two-dimensional liquid of Fröhlich polarons at the bare SrTiO3 surface. Nat. Commun. 6, 8585 (2015).

    Article  ADS  Google Scholar 

  7. Cancellieri, C. et al. Polaronic metal state at the LaAlO3/SrTiO3 interface. Nat. Commun. 7, 10386 (2016).

    Article  ADS  Google Scholar 

  8. Pastor, E. et al. In situ observation of picosecond polaron self-localisation in α-Fe2O3 photoelectrochemical cells. Nat. Commun. 10, 3962 (2019).

    Article  ADS  Google Scholar 

  9. Moser, S. et al. Tunable polaronic conduction in anatase TiO2. Phys. Rev. Lett. 110, 196403 (2013).

    Article  ADS  Google Scholar 

  10. Baldini, E. et al. Discovery of the soft electronic modes of the trimeron order in magnetite. Nat. Phys. 16, 541 (2020).

    Article  Google Scholar 

  11. Miyata, K. & Zhu, X.-Y. Ferroelectric large polarons. Nat. Mater. 17, 379 (2018).

    Article  ADS  Google Scholar 

  12. Guzelturk, B. et al. Visualization of dynamic polaronic strain fields in hybrid lead halide perovskites. Nat. Mater. 20, 618 (2021).

    Article  ADS  Google Scholar 

  13. Miyata, K. et al. Large polarons in lead halide perovskites. Sci. Adv. 3, e1701217 (2017).

    Article  ADS  Google Scholar 

  14. Wu, B. et al. Strong self-trapping by deformation potential limits photovoltaic performance in bismuth double perovskite. Sci. Adv. 7, eabd3160 (2021).

    Article  ADS  Google Scholar 

  15. Novoselov, K. S. et al. Two-dimensional atomic crystals. Proc. Natl Acad. Sci. USA 102, 10451 (2005).

    Article  ADS  Google Scholar 

  16. Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).

    Article  Google Scholar 

  17. Chen, C. et al. Emergence of interfacial polarons from electron-phonon coupling in graphene/h-BN van der Waals heterostructures. Nano Lett. 18, 1082 (2018).

    Article  ADS  Google Scholar 

  18. Kang, M. et al. Holstein polaron in a valley-degenerate two-dimensional semiconductor. Nat. Mater. 17, 676 (2018).

    Article  ADS  Google Scholar 

  19. Sio, W. H., Verdi, C., Poncé, S. & Giustino, F. Polarons from first principles, without supercells. Phys. Rev. Lett. 122, 246403 (2019).

    Article  ADS  Google Scholar 

  20. Sio, W. H., Verdi, C., Poncé, S. & Giustino, F. Ab initio theory of polarons: formalism and applications. Phys. Rev. B. 99, 235139 (2019).

    Article  ADS  Google Scholar 

  21. Sio, W. H. & Giustino, F. Unified ab initio description of Fröhlich electron-phonon interactions in two-dimensional and three-dimensional materials. Phys. Rev. B. 105, 115414 (2022).

    Article  ADS  Google Scholar 

  22. Giustino, F. Electron-phonon interactions from first principles. Rev. Mod. Phys. 89, 015003 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  23. Giannozzi, P. et al. Advanced capabilities for materials modelling with quantum ESPRESSO. J. Phys. Condens. Matter 29, 465901 (2017).

    Article  Google Scholar 

  24. Mostofi, A. A. et al. An updated version of wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 185, 2309 (2014).

    Article  ADS  MATH  Google Scholar 

  25. Poncé, S., Margine, E. R., Verdi, C. & Giustino, F. EPW: electron–phonon coupling, transport and superconducting properties using maximally localized Wannier functions. Comput. Phys. Commun. 209, 116 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  26. Wickramaratne, D., Weston, L. & Van de Walle, C. G. Monolayer to bulk properties of hexagonal boron nitride. J. Phys. Chem. C 122, 25524 (2018).

    Article  Google Scholar 

  27. Arnaud, B., Lebégue, S., Rabiller, P. & Alouani, M. Huge excitonic effects in layered hexagonal boron nitride. Phys. Rev. Lett. 96, 026402 (2006).

    Article  ADS  Google Scholar 

  28. Wirtz, L. et al. Comment on “Huge excitonic effects in layered hexagonal boron nitride”. Phys. Rev. Lett. 100, 189701 (2008).

    Article  ADS  Google Scholar 

  29. Galvani, T. et al. Excitons in boron nitride single layer. Phys. Rev. B. 94, 125303 (2016).

    Article  ADS  Google Scholar 

  30. Zhang, F. et al. Intervalley excitonic hybridization, optical selection rules, and imperfect circular dichroism in monolayer h-BN. Phys. Rev. Lett. 128, 047402 (2022).

    Article  ADS  Google Scholar 

  31. Gil, B., Cassabois, G., Cusco, R., Fugallo, G. & Artus, L. Boron nitride for excitonics, nano photonics, and quantum technologies. Nanophotonics 9, 3483 (2020).

    Article  Google Scholar 

  32. Li, Z., Antonius, G., Wu, M., da Jornada, F. H. & Louie, S. G. Electron-phonon coupling from ab initio linear-response theory within the GW method: correlation-enhanced interactions and superconductivity in Ba1−xKxBiO3. Phys. Rev. Lett. 122, 186402 (2019).

    Article  ADS  Google Scholar 

  33. Dai, S. et al. Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride. Science 343, 1125 (2014).

    Article  ADS  Google Scholar 

  34. Tran, T. T., Bray, K., Ford, M. J., Toth, M. & Aharonovich, I. Quantum emission from hexagonal boron nitride monolayers. Nat. Nanotechnol. 11, 37 (2016).

    Article  ADS  Google Scholar 

  35. Cassabois, G., Valvin, P. & Gil, B. Hexagonal boron nitride is an indirect bandgap semiconductor. Nat. Photonics 10, 262 (2016).

    Article  ADS  Google Scholar 

  36. Elias, C. et al. Direct band-gap crossover in epitaxial monolayer boron nitride. Nat. Commun. 10, 2639 (2019).

    Article  ADS  Google Scholar 

  37. Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor. Nat. Mater. 13, 1091 (2014).

    Article  ADS  Google Scholar 

  38. Olsen, T., Latini, S., Rasmussen, F. & Thygesen, K. S. Simple screened hydrogen model of excitons in two-dimensional materials. Phys. Rev. Lett. 116, 056401 (2016).

    Article  ADS  Google Scholar 

  39. Paleari, F. et al. Excitons in few-layer hexagonal boron nitride: Davydov splitting and surface localization. 2D Mater. 5, 045017 (2018).

    Article  Google Scholar 

  40. Román, R. J. P. et al. Band gap measurements of monolayer h-BN and insights into carbon-related point defects. 2D Mater. 8, 044001 (2021).

    Article  Google Scholar 

  41. Hahn, T., Klimin, S., Tempere, J., Devreese, J. T. & Franchini, C. Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions. Phys. Rev. B. 97, 134305 (2018).

    Article  ADS  Google Scholar 

  42. Peeters, F. M., Xiaoguang, W. & Devreese, J. T. Ground-state energy of a polaron in n dimensions. Phys. Rev. B. 33, 3926 (1986).

    Article  ADS  Google Scholar 

  43. Ercelebi, A. & Süalp, G. A variational treatment of a two-dimensional polaron. J. Phys. Chem. Solids 48, 739 (1987).

    Article  ADS  Google Scholar 

  44. Jalabert, R. & Das Sarma, S. Quasiparticle properties of a coupled two-dimensional electron-phonon system. Phys. Rev. B. 40, 9723 (1989).

    Article  ADS  Google Scholar 

  45. Titantah, J. T., Pierleoni, C. & Ciuchi, S. Free energy of the Fröhlich polaron in two and three dimensions. Phys. Rev. Lett. 87, 206406 (2001).

    Article  ADS  Google Scholar 

  46. Kaasbjerg, K., Thygesen, K. S. & Jacobsen, K. W. Phonon-limited mobility in n-type single-layer MoS2 from first principles. Phys. Rev. B. 85, 115317 (2012).

    Article  ADS  Google Scholar 

  47. Sohier, T., Calandra, M. & Mauri, F. Two-dimensional Fröhlich interaction in transition-metal dichalcogenide monolayers: theoretical modeling and first-principles calculations. Phys. Rev. B. 94, 085415 (2016).

    Article  ADS  Google Scholar 

  48. Deng, T. et al. Ab initio dipolar electron-phonon interactions in two-dimensional materials. Phys. Rev. B 103, 075410 (2021).

    Article  ADS  Google Scholar 

  49. Landau, L. D. Electron motion in crystal lattices. Phys. Z. Sowjet. 3, 664 (1933).

    Google Scholar 

  50. Pekar, S. Local quantum states of electrons in an ideal ion crystal. Zh. Eksp. Teor. Fiz. 16, 341 (1946).

    Google Scholar 

  51. Devreese, J. T. Fröhlich polarons. lecture course including detailed theoretical derivations. Preprint at arXiv https://arxiv.org/abs/1012.4576 (2020).

  52. Devreese, J. T. & Alexandrov, A. S. Fröhlich polaron and bipolaron: recent developments. Rep. Prog. Phys. 72, 066501 (2009).

    Article  ADS  Google Scholar 

  53. Spanier, J. & Oldham, K. B. An Atlas of Functions (Taylor & Francis/Hemisphere, 1987).

    MATH  Google Scholar 

  54. Rytova, N. S. Screened potential of a point charge in a thin film. Proc. MSU, Phys. Astron. 3, 30 (1967).

    Google Scholar 

  55. Keldysh, L. V. Coulomb interaction in thin semiconductor and semimetal films. JETP Lett. 29, 658 (1979).

    ADS  Google Scholar 

  56. Cudazzo, P., Tokatly, I. V. & Rubio, A. Dielectric screening in two-dimensional insulators: implications for excitonic and impurity states in graphane. Phys. Rev. B. 84, 085406 (2011).

    Article  ADS  Google Scholar 

  57. Feynman, R. P. Slow electrons in a polar crystal. Phys. Rev. 97, 660 (1955).

    Article  ADS  MATH  Google Scholar 

  58. Prokof’ev, N. V. & Svistunov, B. V. Polaron problem by diagrammatic quantum Monte Carlo. Phys. Rev. Lett. 81, 2514 (1998).

    Article  ADS  Google Scholar 

  59. Hahn, T., Klimin, S., Tempere, J., Devreese, J. T. & Franchini, C. Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions. Phys. Rev. B. 97, 134305 (2018).

    Article  ADS  Google Scholar 

  60. Nair, R. R. et al. Fluorographene: a two-dimensional counterpart of Teflon. Small 6, 2877 (2010).

    Article  Google Scholar 

  61. Chang, S.-J. et al. van der Waals epitaxy of 2D h-ALN on TMDS by atomic layer deposition at 250 °C. Appl. Phys. Lett. 120, 162102 (2022).

    Article  ADS  Google Scholar 

  62. Al Balushi, Z. Y. et al. Two-dimensional gallium nitride realized via graphene encapsulation. Nat. Mater. 15, 1166 (2016).

    Article  ADS  Google Scholar 

  63. Zhang, H. et al. Epitaxial growth of two-dimensional insulator monolayer honeycomb BeO. ACS Nano 15, 2497 (2021).

    Article  Google Scholar 

  64. Wang, H. et al. Improved thermoelectric performance of monolayer HfS2 by strain engineering. ACS Omega 6, 29820 (2021).

    Article  Google Scholar 

  65. Lv, H. Y., Lu, W. J., Shao, D. F., Lu, H. Y. & Sun, Y. P. Strain-induced enhancement in the thermoelectric performance of a ZrS2 monolayer. J. Mater. Chem. C. 4, 4538 (2016).

    Article  Google Scholar 

  66. Li, W., Poncé, S. & Giustino, F. Dimensional crossover in the carrier mobility of two-dimensional semiconductors: the case of InSe. Nano Lett. 19, 1774 (2019).

    Article  ADS  Google Scholar 

  67. Ferreira, F., Chaves, A. J., Peres, N. M. R. & Ribeiro, R. M. Excitons in hexagonal boron nitride single-layer: a new platform for polaritonics in the ultraviolet. J. Opt. Soc. Am. B. 36, 674 (2019).

    Article  ADS  Google Scholar 

  68. Zhang, W., Huang, Z., Zhang, W. & Li, Y. Two-dimensional semiconductors with possible high room temperature mobility. Nano Res. 7, 1731 (2014).

    Article  Google Scholar 

  69. Laturia, A., Van de Put, M. L. & Vandenberghe, W. G. Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk. NPJ 2D Mater. Appl. 2, 6 (2018).

  70. Pike, N. A., Dewandre, A., Van Troeye, B., Gonze, X. & Verstraete, M. J. Vibrational and dielectric properties of the bulk transition metal dichalcogenides. Phys. Rev. Mater. 2, 063608 (2018).

    Article  Google Scholar 

  71. Li, W. & Giustino, F. Many-body renormalization of the electron effective mass of InSe. Phys. Rev. B. 101, 035201 (2020).

    Article  ADS  Google Scholar 

  72. Garcia-Goiricelaya, P., Lafuente-Bartolome, J., Gurtubay, I. G. & Eiguren, A. Long-living carriers in a strong electron-phonon interacting two-dimensional doped semiconductor. Commun. Phys. 2, 81 (2019).

  73. Yan, C. et al. 2D group IVB transition metal dichalcogenides. Adv. Funct. Mater. 28, 1803305 (2018).

    Article  Google Scholar 

  74. Alexandrov, A. & Ranninger, J. Bipolaronic superconductivity. Phys. Rev. B. 24, 1164 (1981).

    Article  ADS  Google Scholar 

  75. Mounet, N. et al. Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nat. Nanotechnol. 13, 246 (2018).

    Article  ADS  Google Scholar 

  76. Rasmussen, F. A. & Thygesen, K. S. Computational 2D materials database: electronic structure of transition-metal dichalcogenides and oxides. J. Phys. Chem. C. 119, 13169 (2015).

    Article  Google Scholar 

  77. Sohier, T., Gibertini, M., Calandra, M., Mauri, F. & Marzari, N. Breakdown of optical phonons’ splitting in two-dimensional materials. Nano Lett. 17, 3758 (2017).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

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 (http://www.tacc.utexas.edu). 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.

Author information

Authors and Affiliations

  1. Institute of Applied Physics and Materials Engineering, University of Macau, Macao SAR, P. R. China

    Weng Hong Sio

  2. Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, USA

    Weng Hong Sio & Feliciano Giustino

  3. Department of Physics, The University of Texas at Austin, Austin, Texas, USA

    Feliciano Giustino

Authors
  1. Weng Hong Sio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feliciano Giustino
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

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

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sio, W.H., Giustino, F. Polarons in two-dimensional atomic crystals. Nat. Phys. 19, 629–636 (2023). https://doi.org/10.1038/s41567-023-01953-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-01953-4

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing