- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Time-domain study of surface plasmon polariton propagation in silver nanowires

Wenhua Zhao1,2,*, Álvaro Rodríguez Echarri1, Alberto Eljarrat3,4, Hannah C. Nerl3, Thomas Kiel2, Benedikt Haas3,4, Henry Halim5, Yan Lu5,6, Christoph T. Koch3,4,† et al.

Kurt Busch2,1,‡

  • *Contact author: wzhao@physik.hu-berlin.de
  • †Contact author: christoph.koch@hu-berlin.de
  • ‡Contact author: kurt.busch@physik.hu-berlin.de
PDF

Phys. Rev. B 113, 085425 – Published 18 February, 2026

DOI: https://doi.org/10.1103/gpqj-4v29

Abstract

Electron microscopy techniques such as electron energy-loss spectroscopy (EELS) facilitate the spatiospectral characterization of plasmonic nanostructures. In this work, a time-dependent perspective is presented that significantly enhances the utility of EELS. In particular, this approach facilitates the analysis of the dynamics of plasmonic excitations that repeatedly interact with swift electrons in a STEM-EELS configuration. This includes the bulk plasmon mode, which can only be excited by penetrating electron beams, and the fundamental surface plasmon polariton modes propagating along the wire, which can be excited by both penetrating and aloof trajectories. In addition, the role of higher-order azimuthal surface plasmon polariton modes, often overlooked for very thin wires, is observed and analyzed in both the energy-loss spectrum and from the dynamical perspective. Such a complete understanding of the interaction of electrons and plasmonic excitations is key for the design of efficient plasmonic sensors, the study of hot electron dynamics in metals, and applications in the context of electron quantum optics, where full control of the spatial and temporal characteristics of the fields at the nanometer and femtosecond scales is highly desirable.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (64)

  1. M. Kauranen and A. V. Zayats, Nonlinear plasmonics, Nature Photon 6, 737 (2012).
  2. X. Xu, H. Li, D. Hasan, R. S. Ruoff, A. X. Wang, and D. Fan, Near-field enhanced plasmonic-magnetic bifunctional nanotubes for single cell bioanalysis, Adv. Funct. Mater. 23, 4332 (2013).
  3. B. Sharma, R. R. Frontiera, A.-I. Henry, E. Ringe, and R. P. Van Duyne, Sers: Materials, applications, and the future, Mater. Today 15, 16 (2012).
  4. S. I. Bozhevolnyi, Plasmonic nano-guides and circuits, in Plasmonics and Metamaterials (Optica Publishing Group, Rochester, New York, 2008), p. MWD3.
  5. H.-K. Yu, B.-D. Liu, W.-L. Wu, and Z.-Y. Li, Surface plasmaons enhanced light-matter interactions, Acta Phys. Sin. 68, 149101 (2019).
  6. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, Biosensing with plasmonic nanosensors, Nat. Mater. 7, 442 (2008).
  7. F. Arcadio, L. Noel, D. Prete, D. Maniglio, M. Seggio, O. Soppera, N. Cennamo, A. Bossi, and L. Zeni, Soft molecularly imprinted nanoparticles with simultaneous lossy mode and surface plasmon multi-resonances for femtomolar sensing of serum transferrin protein, Sci. Rep. 13, 11210 (2023).
  8. N. Jo and Y.-B. Shin, Enhancing biosensing sensitivity of metal nanostructures through site-selective binding, Sci. Rep. 10, 1024 (2020).
  9. R. Liu, Z.-K. Zhou, Y.-C. Yu, T. Zhang, H. Wang, G. Liu, Y. Wei, H. Chen, and X.-H. Wang, Strong light-matter interactions in single open plasmonic nanocavities at the quantum optics limit, Phys. Rev. Lett. 118, 237401 (2017).
  10. A. Gonzalez-Tudela, A. Reiserer, J. García-Ripoll, and F. García-Vidal, Light–matter interactions in quantum nanophotonic devices, Nat. Rev. Phys. 6, 166 (2024).
  11. S. Slussarenko and G. Pryde, Photonic quantum information processing: A concise review, Appl. Phys. Rev. 6, 041303 (2019).
  12. D. Smirnova and Y. Kivshar, Multipolar nonlinear nanophotonics, Optica 3, 1241 (2016).
  13. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Surface plasmon subwavelength optics, Nature (London) 424, 824 (2003).
  14. B. Wang, P. Yu, W. Wang, X. Zhang, H.-C. Kuo, H. Xu, and Z. Wang, High‐Q plasmonic resonances: Fundamentals and applications, Adv. Opt. Mater. 9, 2001520 (2021).
  15. M. Mayer, L. Scarabelli, K. March, T. Altantzis, M. Tebbe, M. Kociak, S. Bals, F. J. García de Abajo, A. Fery, and L. M. Liz-Marzán, Controlled living nanowire growth: Precise control over the morphology and optical properties of AgAuAg bimetallic nanowires, Nano Lett. 15, 5427 (2015).
  16. M. Rothe, Y. Zhao, G. Kewes, Z. Kochovski, W. Sigle, P. A. van Aken, C. Koch, M. Ballauff, Y. Lu, and O. Benson, Silver nanowires with optimized silica coating as versatile plasmonic resonators, Sci. Rep. 9, 3859 (2019).
  17. A. Baburin, A. Merzlikin, A. Baryshev, I. Ryzhikov, Y. Panfilov, and I. Rodionov, Silver-based plasmonics: Golden material platform and application challenges, Opt. Mater. Express 9, 611 (2019).
  18. R. F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope (Springer New York, NY, 2011), pp. XII, 491.
  19. A. Liu, T. Davis, T. Coenen, S. Hari, L. Voortman, Z. Xu, G. Yuan, P. Ballard, A. Funston, and J. Etheridge, Modulation of cathodoluminescence by surface plasmons in silver nanowires, Small 19, 2207747 (2023).
  20. J. Nelayah, M. Kociak, O. Stéphan, F. J. García de Abajo, M. Tencé, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzán, and C. Colliex, Mapping surface plasmons on a single metallic nanoparticle, Nat. Phys. 3, 348 (2007).
  21. M. Bosman, V. Keast, M. Watanabe, A. Maaroof, and M. Cortie, Mapping surface plasmons at the nanometre scale with an electron beam, Nanotechnology 18, 165505 (2007).
  22. C. Colliex, M. Kociak, and O. Stéphan, Electron energy loss spectroscopy imaging of surface plasmons at the nanometer scale, Ultramicroscopy 162, A1 (2016).
  23. G. Ruthemann, Diskrete energieverluste mittelschneller elektronen beim durchgang durch dünne folien, Ann. Phys. 437, 113 (1948).
  24. R. Vincent and J. Silcox, Dispersion of radiative surface plasmons in aluminum films by electron scattering, Phys. Rev. Lett. 31, 1487 (1973).
  25. H. Watanabe, Experimental evidence for the collective nature of the characteristic energy loss of electrons in solids–studies on the dispersion relation of plasma frequency–, J. Phys. Soc. Jpn. 11, 112 (1956).
  26. D. Rossouw, M. Couillard, J. Vickery, E. Kumacheva, and G. Botton, Multipolar plasmonic resonances in silver nanowire antennas imaged with a subnanometer electron probe, Nano Lett. 11, 1499 (2011).
  27. D. Rossouw and G. Botton, Plasmonic response of bent silver nanowires for nanophotonic subwavelength waveguiding, Phys. Rev. Lett. 110, 066801 (2013).
  28. X. Zhou, A. Hörl, A. Trügler, U. Hohenester, T. Norris, and A. Herzing, Effect of multipole excitations in electron energy-loss spectroscopy of surface plasmon modes in silver nanowires, J. Appl. Phys. 116, 223101 (2014).
  29. D. Nabben, J. Kuttruff, L. Stolz, A. Ryabov, and P. Baum, Attosecond electron microscopy of sub-cycle optical dynamics, Nature (London) 619, 63 (2023).
  30. R. H. Ritchie, Plasma losses by fast electrons in thin films, Phys. Rev. 106, 874 (1957).
  31. F. J. García de Abajo, Optical excitations in electron microscopy, Rev. Mod. Phys. 82, 209 (2010).
  32. C. H. Chen and J. Silcox, Detection of optical surface guided modes in thin graphite films by high-energy electron scattering, Phys. Rev. Lett. 35, 390 (1975).
  33. K. Busch, M. König, and J. Niegemann, Discontinuous Galerkin methods in nanophotonics, Laser Photonics Rev. 5, 773 (2011).
  34. P. E. Stamatopoulou, W. Zhao, A. Rodríguez Echarri, N. A. Mortensen, K. Busch, C. Tserkezis, and C. Wolff, Electron beams traversing spherical nanoparticles: Analytic and numerical treatment, Phys. Rev. Res. 6, 013239 (2024).
  35. B. Schröder, T. Weber, S. V. Yalunin, T. Kiel, C. Matyssek, M. Sivis, S. Schäfer, F. von Cube, S. Irsen, K. Busch, C. Ropers, and S. Linden, Real-space imaging of nanotip plasmons using electron energy loss spectroscopy, Phys. Rev. B 92, 085411 (2015).
  36. P. B. Johnson and R. W. Christy, Optical constants of the noble metals, Phys. Rev. B 6, 4370 (1972).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/gpqj-4v29 for the technical details and additional information regarding our study. Specifically, we give a derivation of the temporal analysis of EELS, address the surface plasmon polariton dispersion relation of infinite nanowires, expand the technical formalism to solve the electromagnetic problem numerically via the DGTD Maxwell solver, and provide further details about the characteristics of the surface plasmon polariton modes in silver long nanowires. Finally, the influence of the substrate is addressed.
  38. A. Rodríguez Echarri, W. Zhao, K. Busch, and F. J. García de Abajo, Relativistic electron energy-loss spectroscopy in cylindrical waveguides and holes, Phys. Rev. B 111, 205436 (2025).
  39. V. Lucarini, J. Saarinen, K. Peiponen, and E. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, Berlin, Heidelberg, 2005).
  40. J. L. Walsh, The Cauchy-Goursat theorem for rectifiable Jordan curves, Proc. Natl. Acad. Sci. USA 19, 540 (1933).
  41. V. G. Kravets, A. V. Kabashin, W. L. Barnes, and A. N. Grigorenko, Plasmonic surface lattice resonances: A review of properties and applications, Chem. Rev. 118, 5912 (2018).
  42. R. H. Ritchie and A. Howie, Inelastic scattering probabilities in scanning transmission electron microscopy, Philos. Mag. A 58, 753 (1988).
  43. F. J. García de Abajo and M. Kociak, Probing the photonic local density of states with electron energy loss spectroscopy, Phys. Rev. Lett. 100, 106804 (2008).
  44. U. Hohenester, H. Ditlbacher, and J. R. Krenn, Electron-energy-loss spectra of plasmonic nanoparticles, Phys. Rev. Lett. 103, 106801 (2009).
  45. A. Losquin and M. Kociak, Link between cathodoluminescence and electron energy loss spectroscopy and the radiative and full electromagnetic local density of states, ACS Photonics 2, 1619 (2015).
  46. N. Talebi, W. Sigle, R. Vogelgesang, M. Esmann, S. F. Becker, C. Lienau, and P. A. van Aken, Excitation of mesoscopic plasmonic tapers by relativistic electrons: Phase matching versus eigenmode resonances, ACS Nano 9, 7641 (2015).
  47. J. C. Ashley and L. C. Emerson, Dispersion relations for nonradiative surface plasmons on cylinders, Surf. Sci. 41, 615 (1974).
  48. A. Polman, M. Kociak, and F. J. García de Abajo, Electron-beam spectroscopy for nanophotonics, Nat. Mater. 18, 1158 (2019).
  49. D. B. Williams and C. B. Carter, Transmission Electron Microscopy (Springer, New York, NY, 2009).
  50. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, Theory of surface plasmons and surface-plasmon polaritons, Rep. Prog. Phys. 70, 1 (2007).
  51. S. Zhang, H. Wei, K. Bao, U. Håkanson, N. J. Halas, P. Nordlander, and H. Xu, Chiral surface plasmon polaritons on metallic nanowires, Phys. Rev. Lett. 107, 096801 (2011).
  52. U. Hohenester and A. Trügler, MNPBEM – A Matlab toolbox for the simulation of plasmonic nanoparticles, Comput. Phys. Commun. 183, 370 (2012).
  53. Note that we use the same experimental setup both in Fig. 3 and Fig. S2 but with different energy ranges and consequently different energy sampling δE. Specifically, in Fig. 3 we have δE=0.003eV while in Fig. S2 we have δE=0.006eV.
  54. H. Wei, D. Pan, S. Zhang, Z. Li, Q. Li, N. Liu, W. Wang, and H. Xu, Plasmon waveguiding in nanowires, Chem. Rev. 118, 2882 (2018).
  55. K. Luke, Y. Okawachi, M. R. E. Lamont, A. L. Gaeta, and M. Lipson, Broadband mid-infrared frequency comb generation in a Si3N4 microresonator, Opt. Lett. 40, 4823 (2015).
  56. V. Mkhitaryan, K. March, E. Tseng, X. Li, L. Scarabelli, L. M. Liz-Marzán, S.-Y. Chen, L. H. G. Tizei, O. Stéphan, J.-M. Song, M. Kociak, F. J. García de Abajo, and A. Gloter, Can copper nanostructures sustain high-quality plasmons? Nano Lett. 21, 2444 (2021).
  57. B. E. A. Saleh and M. C. Teich, Guided-wave optics, in Fundamentals of Photonics, edited by J. W. Goodman, B. E. A. Saleh, and M. C. Teich (John Wiley & Sons, Ltd, 1991), Chap. 7, p. 238.
  58. M. Rothe, Y. Zhao, J. Müller, G. Kewes, C. T. Koch, Y. Lu, and O. Benson, Self-assembly of plasmonic nanoantenna–waveguide structures for subdiffractional chiral sensing, ACS Nano 15, 351 (2021).
  59. M. Hu, J. feng Gao, Y. Dong, S. Yang, and R. K. Li, Rapid controllable high-concentration synthesis and mutual attachment of silver nanowires, RSC Adv. 2, 2055 (2012).
  60. O. L. Krivanek, T. C. Lovejoy, M. F. Murfitt, G. Skone, P. E. Batson, and N. Dellby, Towards sub-10 meV energy resolution STEM-EELS, J. Phys.: Conf. Ser. 522, 012023 (2014).
  61. F. de la Peña, E. Prestat, J. Lähnemann, V. T. Fauske, P. Burdet, P. Jokubauskas, T. Furnival, C. Francis, M. Nord, T. Ostasevicius, K. E. MacArthur, D. N. Johnstone, M. Sarahan, J. Taillon, T. Aarholt, pquinn dls, V. Migunov, A. Eljarrat, J. Caron, T. Nemoto, et al., hyperspy/hyperspy: V2.3.0 (v2.3.0), Zenodo (2025), https://doi.org/10.5281/zenodo.14956374.
  62. C. Geuzaine and J.-F. Remacle, Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng. 79, 1309 (2009).
  63. M. R. Gonçalves, H. Minassian, and A. Melikyan, Plasmonic resonators: Fundamental properties and applications, J. Phys. D: Appl. Phys. 53, 443002 (2020).
  64. N. Talebi, Interaction of electron beams with optical nanostructures and metamaterials: From coherent photon sources towards shaping the wave function, J. Opt. 19, 103001 (2017).

Outline

Information

Phys. Rev. B 113, 085425– Published 18 February, 2026
  • Received 6 November 2025
  • Accepted 15 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation