Nanoscale electromagnetism with the boundary element method

In Yang et al. [Nature 576, 248 (2019)], the authors introduced a general theoretical framework for nanoscale electromagnetism based on Feibelman parameters. Here quantum effects of the optically excited electrons at the interface between two materials are lumped into two complex-valued and frequency-dependent parameters which can be incorporated into modified boundary conditions for Maxwell's equations, the so-called mesoscopic boundary conditions. These modifications can, in principle, be implemented in any Maxwell solver, although the technicalities can be subtle and depend on the chosen computational approach. In this paper, we show how to implement mesoscopic boundary conditions in a boundary element method approach based on a Galerkin scheme with Raviart-Thomas shape elements for the representation of the tangential electromagnetic fields at the boundary. We demonstrate that the results of our simulations are in perfect agreement with Mie theory, including Feibelman parameters, and that for typical simulation scenarios the computational overhead is usually small.

Figure 1

Schematics of (a) boundary integral method and (b) boundary element method. (a) We consider a nanoparticle with a homogeneous and local permittivity function ${\varepsilon }_1\left(\omega \right)$ that is embedded in a background medium with permittivity ${\varepsilon }_2\left(\omega \right)$. The sharp nanoparticle boundary is denoted with $\partial \mathrm{\Omega }$ and the outer surface normal with $\widehat{\mathit{n}}$. Once the tangential electromagnetic fields ${\mathit{u}}^{E,H}$ are known at the boundary, they can be computed everywhere else using the representation formula of Eq. (4). (b) In the boundary element method, the nanoparticle boundary is discretized using boundary elements of triangular shape and (c) the tangential electromagnetic fields are approximated using Raviart-Thomas shape elements ${\mathit{f}}_{\nu }^e$. The working equations can be expressed in the form of matrix-vector multiplications, and the solutions ${\mathit{u}}^{E,H}$ are obtained through matrix inversion.

Figure 2

Extinction cross sections for gold nanosphere (diameter 20 nm, dielectric function taken from Ref. [39]) embedded in water (refractive index $n_b=1.33$). We use large and frequency-independent Feibelman parameters of (1) $d_{\perp }=0.5$ nm, $d_{\parallel }=0$, (2) $d_{\perp }=-0.5$ nm, $d_{\parallel }=0$, (3) $d_{\perp }=0, d_{\parallel }=0.5$ nm, (4) $d_{\perp }=0, d_{\parallel }=-0.5$ nm. The circle symbols report results of our BEM approach using the sphere discretization with 400 vertices shown in the inset of (a), the cross symbols report results of Mie theory including Feibelman parameters. BEM and Mie results are in perfect agreement throughout. The dashed line shows for comparison results of standard Mie theory without Feibelman parameters.

Figure 3

Extinction cross section for coupled gold nanospheres with a diameter of 20 nm and for gap distances of 2, 4, and 8 nm, as indicated on the right-hand side (we use $n_b=1.33$). The spectra are offset for clarity, and the polarization of the incoming light is along the symmetry axis of the coupled spheres. The two peaks are associated with bonding and anti-bonding dimer modes, and the splitting increases with decreasing gap distance owing to the increased coupling strength.

Figure 4

Computation of resonance modes and optical spectra for nanoellipsoids using a dielectric function representative for gold [18]. (a) Extinction cross sections for prolate nanoellipsoids with a short axis of 20 nm and a long axis of 40 nm (circles) and 60 nm (squares); the polarization of the incoming light is parallel to the long axis. The open symbols show results from the full BEM simulations, the open symbols from the resonance mode approximation, and the dashed lines from simulations without Feibelman parameters. (b) Complex frequency plane. The solid line shows the contour used in the computation of the resonance modes and the symbols the location of the resonance mode energies.

Figure 5

Error between tangential electric fields computed within Mie theory and BEM simulations, evaluated at the centroids of the boundary elements, see Eq. (25). We consider a gold nanosphere with 20 nm diameter and the same material and simulation parameters as listed in the caption of Fig. 2, and compare simulation results for boundary discretizations with a varying number of vertices $N$ and for a wavelength of 520 nm. With decreasing mesh size, the error becomes smaller monotonously.