Quantum corrected model for plasmonic nanoparticles: A boundary element method implementation

Ulrich Hohenester

Phys. Rev. B 91, 205436 – Published 26 May 2015

Abstract

We present a variant of the recently developed quantum corrected model (QCM) for plasmonic nanoparticles [Nat. Commun. 3, 825 (2012)] using nonlocal boundary conditions. The QCM accounts for electron tunneling in narrow gap regions of coupled metallic nanoparticles, leading to the appearance of new charge-transfer plasmons. Our approach has the advantages that it emphasizes the nonlocal nature of tunneling and introduces only contact resistance, but not ohmic losses through tunneling. Additionally, it can be implemented much more easily in boundary element method (BEM) approaches. We develop the methodology for the QCM using nonlocal boundary conditions and present simulation results of our BEM implementation, which are in good agreement with those of the original QCM.

Received 19 March 2015
Revised 11 May 2015

DOI:https://doi.org/10.1103/PhysRevB.91.205436

Authors & Affiliations

Ulrich Hohenester^*

Institute of Physics, University of Graz, Universitätsplatz 5, 8010 Graz, Austria

^*ulrich.hohenester@uni-graz.at

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Issue

Vol. 91, Iss. 20 — 15 May 2015

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Images

Figure 1
Schematics of the quantum corrected model (QCM). (a) Volume-based implementation of Esteban et al. [11, 13] where artificial dielectric materials are placed inside the gap. The conductivities of these materials are set to the gap-size-dependent tunnel conductivities. (b) Boundary-element-based implementation of this work, with nonlocal artificial boundary conditions which are chosen in order to obtain the proper tunnel current between boundary positions $s_{a}$ and $s_{b}$ . Inset: The pillbox (with outer surface normal ${\hat{n}}_{a})$ over which Gauss' law is integrated to obtain the artificial boundary conditions. For details, see text.
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Figure 2
Comparison of the volume quantum corrected model (QCM) of Esteban et al. [11, 13] with the boundary QCM of this work. We use two spheres with diameters of 50 nm and a Drude-type dielectric function representative of gold, and a single layer of artificial tunnel material. The light polarization is along the nanoparticle connection. The material covers a distance range between the gap size $d_{gap}$ and $d_{gap} + 0.2$ nm, and the artificial dielectric function is $ɛ_{2 t} (d_{gap} + 0.1 nm)$ . The figure shows the gap-size-dependent extinction cross section (offset for clarity, with gap distance given on left axis) for the volume QCM and compares them with results of the boundary QCM. In the latter approach, we consider quantum tunneling in the same distance window as in the volume QCM, and set the tunneling dielectric function to the same value as in the volume QCM.
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Figure 3
Volume and boundary QCM for the same spheres as in Fig. 2 and for $d_{gap} = 0.075$ nm. In the volume QCM, we consider an onionlike sequence of five materials $ɛ (ℓ)$ , with $ℓ$ covering the region from $d_{gap}$ to $d_{gap} + 0.4$ nm. In the boundary QCM, we use the $ɛ (ℓ)$ values for the respective boundary element distances. Volume QCM1 refers to the model of Ref. [11] and volume QCM2 refers to a simulation where the light excitation and the scattered far fields are computed without the artificial materials. Boundary QCM1 refers to simulations where opposite boundary elements of the flipped spheres are connected (with a refined mesh at the poles) and boundary QCM2 refers to a simulation where the respective closest boundary elements of the neighboring spheres are connected.
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Figure 4
Extinction cross section for a dimer, using a classical electrodynamic (gray dashed line) and a QCM simulation (blue line) with polarization along the nanoparticle connection, as well as a QCM simulation for a trimer (red line). The sphere diameters are 50 nm and the gap distances are 0.1 nm. For the trimer, the optical spectra do not depend on the polarization direction of the incoming light (light propagation direction perpendicular to trimer plane).
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