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Quantitative sampling of atomic-scale electromagnetic waveforms

Nature Photonics volume 15, pages 143–147 (2021)Cite this article

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Abstract

Tailored nanostructures can confine electromagnetic waveforms in extremely sub-wavelength volumes, opening new avenues in lightwave sensing and control down to sub-molecular resolution. Atomic light–matter interaction depends critically on the absolute strength and the precise time evolution of the near field, which may be strongly influenced by quantum-mechanical effects. However, measuring atom-scale field transients has remained out of reach. Here we introduce quantitative atomic-scale waveform sampling in lightwave scanning tunnelling microscopy to resolve a tip-confined near-field transient. Our parameter-free calibration employs a single-molecule switch as an atomic-scale voltage standard. Although salient features of the far-to-near-field transfer follow classical electrodynamics, we develop a comprehensive understanding of the atomic-scale waveforms with time-dependent density functional theory. The simulations validate our calibration and confirm that single-electron tunnelling ensures minimal back-action of the measurement process on the electromagnetic fields. Our observations access an uncharted domain of nano-opto-electronics where local quantum dynamics determine femtosecond atomic near fields.

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Fig. 1: Near-field waveform sampling by superposition.
Fig. 2: Quantitative single-molecule peak-voltage sensor.
Fig. 3: Calibrated atomic-scale near-field waveform.
Fig. 4: Quantum-mechanical simulation of atomic-scale light–matter interaction.

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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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Acknowledgements

We thank C. Meineke, A. Pöllmann, C. Rohrer and M. Furthmeier for assistance and F. Evers, H. Appel and S. Ohlman for discussions. We acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID 314695032—SFB 1277 (Subproject B02), Research Grants HU1598/3 and HU1598/8, the Cluster of Excellence ‘Advanced Imaging of Matter’ (AIM, EXC 2056, ID 390715994) and from Grupos Consolidados (IT1249-19), the European Research Council (ERC-2015-AdG694097), the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 895747 and the Flatiron Institute, a division of the Simons Foundation.

Author information

Author notes

  1. These authors contributed equally: D. Peller, F. Bonafé.

Authors and Affiliations

  1. Department of Physics, University of Regensburg, Regensburg, Germany

    D. Peller, C. Roelcke, L. Z. Kastner, T. Buchner, A. Neef, J. Hayes, R. Huber & J. Repp

  2. Max Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, Hamburg, Germany

    F. Bonafé, D. Sidler, M. Ruggenthaler & A. Rubio

  3. Center for Computational Quantum Physics, Simons Foundation Flatiron Institute, New York, NY, USA

    A. Rubio

  4. Universidad del País Vasco, UPV/EHU, San Sebastián, Spain

    A. Rubio

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Contributions

D.P., C.R., L.Z.K., T.B., A.N., J.H., J.R. and R.H. conceived, set up and carried out the experiments. D.P. and A.N. implemented and carried out the classical finite-element simulations. F.B., D.S., M.R. and A.R. conceived, implemented and carried out the TDDFT simulations. All authors analysed the data and wrote the manuscript.

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Correspondence to A. Rubio or R. Huber.

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The authors declare no competing interests.

Additional information

Peer review information Nature Photonics thanks Kazuhiko Hirakawa, Christoph Lienau and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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

Extended data

Extended Data Fig. 1 Calibrated near-field waveform for different gap sizes.

a, Three waveforms obtained in a way analogous to Fig. 3b are shown. The transients were recorded with the same molecular sensor and identical atomic shape of the tip, but only the relative tip height was varied. The exponential dependence of the tunnelling rate with tip height limits the possible variations to approximately 0.5 Å, for which the current changes already by more than a factor of three. The absolute tip-molecule distance is unknown, but can be estimated to be on the order of a few ångströms. All three datasets exhibit very similar waveforms, only the one at closest distance (yellow) is slightly reduced in amplitude. In a classical regime of electrodynamics, we expect that the gap voltage does not depend on the width of the junction, in consistency with the data. These statistical measurements at relative tip heights of 0.5 Å, 0.3 Å and 0.0 Å exhibit standard deviations of 10 mV, 5 mV and 5 mV, respectively. b, The same waveform as in panel a (0.3 Å relative tip height, apex #1) is shown together with a transient acquired with a slightly different tip termination (apex #2, standard deviation 4 mV) obtained through mechanical modification by gently indenting the tip into the sample. The latter waveform has been induced by a stronger far field. It is rescaled by a factor 0.76 in the panel for better visual comparison. The scaled profiles of both transients agree extremely well, indicating that the atomistic details of the tip apex do not affect the shape of the near-field waveform.

Extended Data Fig. 2 Parameter-free classical simulation of atomic-scale near-field coupling.

a,b, Electron microscope images of the etched tip employed in the experiments allow accurate modelling of the macroscopic geometry, opening angle and apex radius of curvature in a numerical simulation. Maxwell’s equations are solved on a graded three-dimensional mesh for complex electromagnetic fields of a given frequency. c, A cross section of the obtained field distribution for a frequency of 1 THz visualizes the real part of the electric field component perpendicular to the substrate, Ez, (normalized to this component’s excitation strength) in a colour-coded map. The pattern reveals interference of incident wavefronts with reflections off the substrate and, in the vicinity of the tip apex, field enhancement and phase retardation effects. d, Across the 1 nm gap between tip and sample, the electric field lines are vertical with field strengths enhanced by ~2 × 105. e, Extracting the field enhancement (black solid line) that the geometry induces within the tunnelling gap for different frequencies, we find a f−1-like behaviour as predicted by antenna theory (black dashed line). The simulated phase retardation is approximately constant at -π/3 rad (grey line). Both amplitude and phase of this complex transfer function exhibit the same minor periodic modulation as observed in the waveform measurement. Analysis of the spatial field distributions reveals that a weak standing wave across the etched region of the tip causes this subtle structure.

Extended Data Fig. 3 Standing wave across etched region of the tip.

a, The finite-element simulation confirms that the coupling efficiency of external lightwaves to the tunnelling junction exhibits a slight frequency-periodic modulation. The panel shows the simulated field enhancement and phase retardation at the tip apex from Extended Data Fig. 2e with arrows highlighting two local maxima and a local minimum of the modulation. b–d, The spatial field distribution at the front-most segment of the tip illustrates that the tapered region (~200 μm long) of the etched tip serves as a resonator for surface plasmons. We observe a standing wave pattern (antinodes indicated by grey arrows), giving rise to a sequence of frequencies with slightly increased or attenuated coupling efficiency.

Supplementary information

Supplementary Information

Supplementary Note 1 and Figs. 1 and 2.

Supplementary Video 1

Simulated temporal evolution of the Hartree potential comparing the set-up with molecule in the junction and the free junction. Video showing the vertical cross-section of the Hartree potential, similar to Fig. 4a,b, as it evolves over time when driven by an external waveform. Locally, the calculated Hartree potential without the molecule (left panel) and including the molecule in the junction (right panel) vary strongly. Every frame corresponds to a time step of 215 as. The colour scale indicates the Hartree potential in electronvolts.

Supplementary Video 2

Simulated temporal evolution of the Hartree potential comparing two different tip orientations. Video showing the vertical cross-section of the Hartree potential as in Supplementary Video 1 (same time step per frame). Comparing a tilted tip configuration (left panel) with a symmetric geometry (right panel), we obtain very similar spatial distributions of the potential in the vicinity of the molecule, causing similar near-field profiles. Hence the near-field is not strongly dependent on the tip symmetry or orientation. The colour scale indicates the Hartree potential in electronvolts.

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Peller, D., Roelcke, C., Kastner, L.Z. et al. Quantitative sampling of atomic-scale electromagnetic waveforms. Nat. Photonics 15, 143–147 (2021). https://doi.org/10.1038/s41566-020-00720-8

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