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.

  • Article
  • Published:

A general theoretical and experimental framework for nanoscale electromagnetism

Abstract

The macroscopic electromagnetic boundary conditions, which have been established for over a century1, are essential for the understanding of photonics at macroscopic length scales. Even state-of-the-art nanoplasmonic studies2,3,4, exemplars of extremely interface-localized fields, rely on their validity. This classical description, however, neglects the intrinsic electronic length scales (of the order of ångström) associated with interfaces, leading to considerable discrepancies between classical predictions and experimental observations in systems with deeply nanoscale feature sizes, which are typically evident below about 10 to 20 nanometres5,6,7,8,9,10. The onset of these discrepancies has a mesoscopic character: it lies between the granular microscopic (electronic-scale) and continuous macroscopic (wavelength-scale) domains. Existing top-down phenomenological approaches deal only with individual aspects of these omissions, such as nonlocality11,12,13 and local-response spill-out14,15. Alternatively, bottom-up first-principles approaches—for example, time-dependent density functional theory16,17—are severely constrained by computational demands and thus become impractical for multiscale problems. Consequently, a general and unified framework for nanoscale electromagnetism remains absent. Here we introduce and experimentally demonstrate such a framework—amenable to both analytics and numerics, and applicable to multiscale problems—that reintroduces the electronic length scale via surface-response functions known as Feibelman d parameters18,19. We establish an experimental procedure to measure these complex dispersive surface-response functions, using quasi-normal-mode perturbation theory and observations of pronounced nonclassical effects. We observe nonclassical spectral shifts in excess of 30 per cent and the breakdown of Kreibig-like broadening in a quintessential multiscale architecture: film-coupled nanoresonators, with feature sizes comparable to both the wavelength and the electronic length scale. Our results provide a general framework for modelling and understanding nanoscale (that is, all relevant length scales above about 1 nanometre) electromagnetic phenomena.

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Fig. 1: Framework, experimental structure and measured nonclassical shifts.
Fig. 2: Measurement setup and sample micrographs.
Fig. 3: Systematic measurement of the complex surface-response function d(ω) of the Au–AlOx interface.
Fig. 4: Robustness to nonclassical corrections.

Similar content being viewed by others

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.

References

  1. Maxwell, J. C. A dynamical theory of the electromagnetic field. Philos. Trans. R. Soc. Lond. 155, 459–512 (1865).

    Article  ADS  Google Scholar 

  2. Nielsen, M., Shi, X., Dichtl, P., Maier, S. A. & Oulton, R. F. Giant nonlinear response at a plasmonic nanofocus drives efficient four-wave mixing. Science 358, 1179–1181 (2017).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  3. Chikkaraddy, R. et al. Single-molecule strong coupling at room temperature in plasmonic nanocavities. Nature 535, 127–130 (2016).

    Article  ADS  CAS  Google Scholar 

  4. Akselrod, G. M. et al. Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas. Nat. Photon. 8, 835–840 (2014).

    Article  ADS  CAS  Google Scholar 

  5. Kreibig, U. & Genzel, L. Optical absorption of small metallic particles. Surf. Sci. 156, 678–700 (1985).

    Article  ADS  CAS  Google Scholar 

  6. Tiggesbäumker, J., Köller, L., Meiwes-Broer, K.-H. & Liebsch, A. Blue shift of the Mie plasma frequency in Ag clusters and particles. Phys. Rev. A 48, R1749–R1752 (1993).

    Article  ADS  Google Scholar 

  7. Cottancin, E. et al. Optical properties of noble metal clusters as a function of the size: comparison between experiments and a semi-quantal theory. Theor. Chem. Acc. 116, 514–523 (2006).

    Article  CAS  Google Scholar 

  8. Ciracì, C. et al. Probing the ultimate limits of plasmonic enhancement. Science 337, 1072–1074 (2012).

    Article  ADS  Google Scholar 

  9. Scholl, J. A., Koh, A. L. & Dionne, J. A. Quantum plasmon resonances of individual metallic nanoparticles. Nature 483, 421–427 (2012).

    Article  ADS  CAS  Google Scholar 

  10. Raza, S. et al. Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS. Nanophotonics 2, 131–138 (2013).

    Article  ADS  CAS  Google Scholar 

  11. Boardman, A. D. Electromagnetic Surface Modes (John Wiley & Sons, 1982).

  12. Fernández-Domínguez, A. I., Wiener, A., García-Vidal, F. J., Maier, S. A. & Pendry, J. B. Transformation-optics description of nonlocal effects in plasmonic nanostructures. Phys. Rev. Lett. 108, 106802 (2012).

    Article  ADS  Google Scholar 

  13. Raza, S., Bozhevolnyi, S. I., Wubs, M. & Mortensen, N. A. Nonlocal optical response in metallic nanostructures. J. Phys. Condens. Matter 27, 183204 (2015).

    Article  ADS  Google Scholar 

  14. Zhu, W. et al. Quantum mechanical effects in plasmonic structures with subnanometre gaps. Nat. Commun. 7, 11495 (2016).

    Article  ADS  CAS  Google Scholar 

  15. Skjølstrup, E. J. H., Søndergaard, T. & Pedersen, T. G. Quantum spill-out in few-nanometer metal gaps: effect on gap plasmons and reflectance from ultrasharp groove arrays. Phys. Rev. B 97, 115429 (2018).

    Article  ADS  Google Scholar 

  16. Zuloaga, J., Prodan, E. & Nordlander, P. Quantum description of the plasmon resonances of a nanoparticle dimer. Nano Lett. 9, 887–891 (2009).

    Article  ADS  CAS  Google Scholar 

  17. Teperik, T. V., Nordlander, P., Aizpurua, J. & Borisov, A. G. Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response. Phys. Rev. Lett. 110, 263901 (2013).

    Article  ADS  CAS  Google Scholar 

  18. Feibelman, P. J. Surface electromagnetic fields. Prog. Surf. Sci. 12, 287–407 (1982).

    Article  ADS  CAS  Google Scholar 

  19. Liebsch, A. Electronic Excitations at Metal Surfaces (Springer, 1997).

  20. Halperin, W. P. Quantum size effects in metal particles. Rev. Mod. Phys. 58, 533–606 (1986).

    Article  ADS  CAS  Google Scholar 

  21. Yan, W., Wubs, M. & Mortensen, N. A. Projected dipole model for quantum plasmonics. Phys. Rev. Lett. 115, 137403 (2015).

    Article  ADS  Google Scholar 

  22. Christensen, T., Yan, W., Jauho, A.-P., Soljačić, M. & Mortensen, N. A. Quantum corrections in nanoplasmonics: shape, scale, and material. Phys. Rev. Lett. 118, 157402 (2017).

    Article  ADS  Google Scholar 

  23. Apell, P. & Ljungbert, A. A general non-local theory for the electromagnetic response of a small metal particle. Phys. Scr. 26, 113–118 (1982).

    Article  ADS  CAS  Google Scholar 

  24. Liebsch, A. Dynamical screening at simple-metal surfaces. Phys. Rev. B 36, 7378–7388 (1987).

    Article  ADS  CAS  Google Scholar 

  25. Liebsch, A. Surface-plasmon dispersion and size dependence of Mie resonance: silver versus simple metals. Phys. Rev. B 48, 11317–11328 (1993).

    Article  ADS  CAS  Google Scholar 

  26. Feibelman, P. J. Comment on “Surface plasmon dispersion of Ag”. Phys. Rev. Lett. 72, 788 (1994).

    Article  ADS  CAS  Google Scholar 

  27. Lalanne, P., Yan, W., Vynck, K., Sauvan, C. & Hugonin, J.-P. Light interaction with photonic and plasmonic resonances. Laser Photonics Rev. 12, 1700113 (2018).

    Article  ADS  Google Scholar 

  28. Yan, W., Faggiani, R. & Lalanne, P. Rigorous modal analysis of plasmonic nanoresonators. Phys. Rev. B 97, 205422 (2018). 

  29. Yang, Y., Miller, O. D., Christensen, T., Joannopoulos, J. D. & Soljačić, M. Low-loss plasmonic dielectric nanoresonators. Nano Lett. 17, 3238–3245 (2017).

    Article  ADS  CAS  Google Scholar 

  30. Yang, J., Giessen, H. & Lalanne, P. Simple analytical expression for the peak-frequency shifts of plasmonic resonances for sensing. Nano Lett. 15, 3439–3444 (2015).

    Article  ADS  CAS  Google Scholar 

  31. Jin, D. et al. Quantum-spillover-enhanced surface-plasmonic absorption at the interface of silver and high-index dielectrics. Phys. Rev. Lett. 115, 193901 (2015).

    Article  ADS  Google Scholar 

  32. Groner, M. D., Elam, J. W., Fabreguette, F. H. & George, S. M. Electrical characterization of thin Al2O3 films grown by atomic layer deposition on silicon and various metal substrates. Thin Solid Films 413, 186–197 (2002).

    Article  ADS  CAS  Google Scholar 

  33. Banerjee, A. et al. Optical properties of refractory metal based thin films. Opt. Mater. Express 8, 2072–2088 (2018).

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

We thank J. Daley, S. E. Kooi and M. Mondol for assistance in sample fabrication and measurement. We thank F. Niroui and T. Zhu for lending us equipment. We acknowledge discussions with V. Bulović, O. D. Miller and N. A. Mortensen. We thank P. Rebusco for reading and editing the manuscript. K.K.B. thanks A. Chu and J. Wanapun for support. This work was partly supported by the US Army Research Office through the Institute for Soldier Nanotechnologies under contract number W911NF-18-2-0048 and W911NF-13-D-0001, and by Air Force Office of Scientific Research (AFOSR) under grant contract number FA9550-18-1-0436. Y.Y. was partly supported by the MRSEC Program of the National Science Foundation under grant number DMR-1419807. D.Z. was supported by a National Science Scholarship from A*STAR, Singapore. W.Y. was supported by Programme IdEx Bordeaux-LAPHIA (grant number ANR-10- IDEX-03-02) and project ‘Resonance’ (grant number ANR-16-CE24-0013) of the French National Agency for Research (ANR). M.Z. was supported by the National Natural Science Foundation of China (grant number 11574078) and the China Scholarship Council. T.C. was supported by the Danish Council for Independent Research (grant number DFFC6108-00667).

Author information

Authors and Affiliations

Authors

Contributions

Y.Y. and T.C. conceived the idea. D.Z. fabricated the samples. Y.Y. and D.Z. designed the experiment, built the setup, conducted the scattering measurements and performed the ellipsometry. T.C. derived the mesoscopic boundary conditions. Y.Y., W.Y. and T.C. developed the numerical methods and Y.Y. performed the numerical calculations. W.Y. proposed the auxiliary-potential method, performed density functional theory calculations and implemented the QNM-based perturbation analysis. D.Z. performed the atomic-force microscopy measurement. A.A. and D.Z. performed the transmission electron microscopy measurement. D.Z. and M.Z. characterized nanoparticle size statistics. Y.Y., D.Z., W.Y. and T.C. analysed the data. Y.Y., D.Z. and T.C. drafted the manuscript with extensive input from all authors. J.D.J., P.L., T.C., K.K.B. and M.S. supervised the project.

Corresponding authors

Correspondence to Yi Yang, Di Zhu or Thomas Christensen.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Peer review information Nature thanks Michal Lipson and Javier Aizpurua for their contribution to the peer review of this work.

Supplementary information

Supplementary Information

This file contains supplementary text sections S1–S16, which includes supplementary figures and tables.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, Y., Zhu, D., Yan, W. et al. A general theoretical and experimental framework for nanoscale electromagnetism. Nature 576, 248–252 (2019). https://doi.org/10.1038/s41586-019-1803-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-019-1803-1

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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