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A general theoretical and experimental framework for nanoscale electromagnetism


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.

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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.

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.


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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).

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Authors and Affiliations



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.

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Correspondence to Yi Yang, Di Zhu or Thomas Christensen.

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

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Supplementary Information

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

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Yang, Y., Zhu, D., Yan, W. et al. A general theoretical and experimental framework for nanoscale electromagnetism. Nature 576, 248–252 (2019).

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