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### Maximal Anderson localization and suppression of surface plasmons in two-dimensional random Au networks

##### J. Schultz, K. Hiekel, P. Potapov, R. A. Römer, P. Khavlyuk, A. Eychmüller, and A. Lubk

##### Phys. Rev. Research **6**, 033221 – Published 26 August 2024

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#### Abstract

Two-dimensional random metal networks possess unique electrical and optical properties, such as almost total optical transparency and low sheet resistance, which are closely related to their disordered structure. Here we present a detailed experimental and theoretical investigation of their plasmonic properties, revealing Anderson (disorder-driven) localized surface plasmon resonances of very large quality factors and spatial localization close to the theoretical maximum, which couple to electromagnetic waves. Moreover, they disappear above a geometry-dependent threshold of approximately 1.7eV in the investigated Au networks, explaining their large transparencies in the optical spectrum.

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- Received 16 July 2021
- Accepted 18 July 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.033221

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

#### Physics Subject Headings (PhySH)

- Research Areas

Anderson localizationSurface plasmons

- Physical Systems

2-dimensional systemsGels

- Techniques

Dipole approximationElectron energy loss spectroscopyScanning transmission electron microscopy

Condensed Matter, Materials & Applied Physics

#### Authors & Affiliations

J. Schultz^{1,*}, K. Hiekel^{2}, P. Potapov^{1}, R. A. Römer^{3}, P. Khavlyuk^{2}, A. Eychmüller^{2}, and A. Lubk^{1,4,†}

^{1}Leibniz Institute for Solid State and Materials Research Dresden, Helmholtzstraße 20, 01069 Dresden, Germany^{2}Physical Chemistry, TU Dresden, Zellescher Weg 19, 01069 Dresden, Germany^{3}Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom^{4}Institute of Solid State and Materials Physics, TU Dresden, Haeckelstraße 3, 01069 Dresden, Germany

^{*}Contact author: j.schultz@ifw-dresden.de^{†}Contact author: a.lubk@ifw-dresden.de

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#### Images

###### Figure 2

(a)Experimental setup. (b)2D slice of the 3D dataset [$\mathrm{\Gamma}(x,y,\omega )$] at $\hslash \omega =0.6\phantom{\rule{0ex}{0ex}}\mathrm{eV}$. The color scale corresponds to the spatially resolved loss probability; the gray arrows illustrate propagators of the surface plasmons eventually interfering constructively at random hot spots. (c)Spectrally resolved loss probability at a specific scan positioyn.

###### Figure 3

(a) As-recorded and absorption-corrected loss probability map at 1.62 eV. (b) EELS spectra from different scanning regions indicated by the solid (overall image) and dashed blue rectangle (upper right corner) in the bottom image of (a). (c) High-angle annular dark-field (HAADF) image of EELS scanning region (a), and corresponding absorption-corrected loss probability maps at different energies (the contours of the metal web are highlighted for better visibility). The intensity scaling of the absorption-corrected loss probability maps is individual to compensate for the effect of decreasing loss probability with increasing energy for better visibility of the hotspots.

###### Figure 4

(a)Optical transmission of a macroscopic web (mm in size, coverage $\approx 0.4$) compared to the spatially averaged loss probability ${\overline{\mathrm{\Gamma}}}_{\text{n}}^{\text{exp}}\left(\omega \right)$ as well as the simulated loss probability ${\overline{\mathrm{\Gamma}}}^{\text{sim}}\left(\omega \right)={\mathcal{N}}_{\text{res}}\left(\omega \right)\left|\overline{\mathit{P}}\left(\omega \right)\right|$. (b)Azimuthally averaged autocorrelation $R(r,\omega )$ of resonant LSP modes at 0.8eV energy loss. The width of the blue shaded area indicates the FWHM $\xi $ of the central peak (correlation length). (c)Spectral dependence of the inverse participation number $1/p\left(\omega \right)$ and correlation length $1/\xi \left(\omega \right)$. (d)Spectral dependence of the number of resonant eigenmodes, ${\mathcal{N}}_{\text{res}}\left(\omega \right)$, of the simulated system of coupled electric dipoles. The simulated data were averaged over an ensemble of 10 disorder configurations in all cases.

###### Figure 5

Spatial distribution of the induced dipole moments of selected resonant eigenmodes of the simulated system of coupled dipoles at different energies. The number of dipoles participating to a resonant mode (participation number) decreases with increasing energy, revealing stronger localization with higher energy.

###### Figure 6

Comparison of the mean dipole moment, in-plane (${\overline{P}}_{\perp}$) and perpendicular to the nano-oblates (${\overline{P}}_{z}$).

###### Figure 7

Simulated inverse participation number for different coverages (red and blue curves) and material (orange curve). The green and black graphs correspond to simulations of the quasistatic case (neglecting retardation effects) and loss-free (no dielectric damping) material, both for gold and coverage of 0.4.

###### Figure 8

Effect of diagonal (NP geometry variation only) and off-diagonal (position randomization only) disorder on the inverse participation ratio. Only randomizing the positions did not yield resonant modes according to the resonance criterion, which is indicated by a dashed line.