[Paper Review] Plasmonic lightning-rod effect

Vlastimil Křápek, Rostislav Řepa+3 more|arXiv (Cornell University)|Jul 12, 2024

Photonic Crystals and Applications4 citations

TL;DR

This study rigorously establishes the plasmonic lightning-rod effect (PLRE) by isolating the role of local curvature in plasmonic antennas using a custom-designed plasmonic beetroot (PB) platform. It demonstrates that the PLRE is quantitatively comparable to the electrostatic lightning-rod effect, with the induced electric field decaying as $ r^{-1} $, and reveals that the effective radius of curvature—defined by the spatial distribution of induced charge—is significantly smaller than the geometric radius, explaining the strong field enhancement.

ABSTRACT

The plasmonic lightning-rod effect refers to the formation of a strong electric near field of localized surface plasmons at the sharp features of plasmonic antennas. While this effect is intuitively utilized in the design and optimization of plasmonic antennas, the relation between the magnitude of the electric field and the local curvature of the plasmonic antenna has not been yet rigorously established. Here, we provide such a study. We design sets of plasmonic antennas that allow to isolate the role of the local curvature from other effects influencing the field. The near electric field is inspected by electron energy loss spectroscopy and electrodynamic simulations. We demonstrate the existence of the plasmonic lightning-rod effect and establish its quantitative description, showing that its strength is comparable to the electrostatic lightning-rod effect. We also provide a simple phenomenological formula for the spatial dependence of the field. Finally, we introduce the effective radius of curvature related to the spatial distribution of induced charge in plasmonic antennas, significantly smaller than their geometrical radius.

Motivation & Objective

• To rigorously establish the quantitative relationship between local curvature and electric field enhancement in plasmonic antennas.
• To isolate the plasmonic lightning-rod effect (PLRE) from confounding factors such as evanescent field confinement and charge reservoir effects.
• To determine the effective radius of curvature defined by the spatial distribution of induced charge in plasmonic antennas.
• To develop a phenomenological model that accurately describes the spatial dependence of the induced electric field near sharp features.
• To compare the strength of PLRE with the classical electrostatic lightning-rod effect on a consistent physical basis.

Proposed method

• Design of plasmonic beetroot (PB) nanoantennas with fixed-length rods and variable-radius cylindrical terminations to decouple curvature effects from other field-enhancing mechanisms.
• Use of electron energy loss spectroscopy (EELS) to experimentally measure the near-field electric field distribution along the antenna axis.
• Employment of electrodynamic simulations (FDTD or similar) to model and validate the near-field response under varying curvature.
• Fitting the simulated and measured field profiles with a phenomenological model: $ E(x) = \frac{Q_{\text{eff}}}{(r_{\text{eff}} + x)^d} $, where $ r_{\text{eff}} $ is the effective radius and $ d $ is the dimensionality factor.
• Calculation of effective radii from fitting parameters, revealing that $ r_{\text{eff}} \approx (0.2-0.3) \times r_{\text{var}} $, indicating charge localization near the edge.
• Comparison of enhancement factors between geometric and effective radii to assess the true strength of the PLRE relative to the electrostatic case.

Figure 1: (a) A scheme of setup demonstrating the electrostatic lightning-rod effect. (b) Design of the plasmonic beetroot – a platform for studying PLRE. (c) HAADF images of two fabricated PBs with $r_{\mathrm{fix}}=70~{}\mathrm{nm}$ and $r_{\mathrm{var}}=70~{}\mathrm{nm}$ (left) and 20 nm (right).

Experimental results

Research questions

• RQ1 How does the local curvature of a plasmonic antenna quantitatively influence the magnitude of the induced electric near-field?
• RQ2 To what extent is the plasmonic lightning-rod effect comparable in strength to the classical electrostatic lightning-rod effect?
• RQ3 What is the effective radius of curvature in plasmonic antennas, as defined by the spatial distribution of induced charge, and how does it differ from the geometric radius?
• RQ4 Can a simple phenomenological model accurately describe the spatial decay of the electric field near sharp antenna features?
• RQ5 How do evanescent field confinement and charge reservoir effects influence field enhancement, and how can they be decoupled from curvature effects?

Key findings

• The plasmonic lightning-rod effect (PLRE) is experimentally confirmed and quantitatively characterized, with field enhancement comparable in strength to the electrostatic lightning-rod effect.
• The effective radius of curvature $ r_{\text{eff}} $, derived from the induced charge distribution, is significantly smaller than the geometric radius—ranging from 6 nm to 16 nm for variable terminations of 20–70 nm radius.
• The effective radius scales as $ r_{\text{eff}} \approx (0.2-0.3) \times r_{\text{var}} $, indicating that induced charge is localized near the edge of the antenna, not at its center.
• The dimensionality factor $ d $ in the phenomenological model is found to be $ 1.2 \pm 0.1 $, indicating a linear-like charge distribution, consistent with field decay as $ r^{-1} $.
• The field enhancement factor based on effective radii ($ \mathit{EF}_{\mathrm{ELRE}}^{\mathrm{eff}} = 1.4 $) is comparable to the measured PLRE enhancement ($ \mathit{EF} = 1.26 $), confirming the quantitative equivalence of the two effects.
• Reducing the geometric radius from 50 nm to 20 nm increases the field by a factor of 1.6 (based on effective radius), but further reduction to 10 nm yields only a 1.2× increase, indicating diminishing returns for nanofabrication.

Figure 2: Loss probability spectra for PBs with $r_{\mathrm{fix}}=70~{}\mathrm{nm}$ and the electron beam position indicated in the inset (20 nm from the PB boundary). (a) Experimental spectra for three values of $r_{\mathrm{var}}$ . (b) Calculated spectra (solid lines) and calculated spectra convol