Plasmonics simulations with the MNPBEM toolbox: Consideration of substrates and layer structures

Jürgen Waxenegger; Andreas Trügler; Ulrich Hohenester

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Plasmonics simulations with the MNPBEM toolbox: Consideration of substrates and layer structures

Jürgen Waxenegger, Andreas Trügler, Ulrich Hohenester

arXiv:1412.5467v1Dec 17, 2014cond-mat.mes-hall

TL;DR

The paper addresses the need to simulate plasmonic nanoparticles in substrate and layer environments, which are common experimental settings but were not clearly supported by the potential-based BEM approach. It develops and implements a layer-aware MNPBEM methodology using tabulated reflected Green functions, and reports good agreement with DDA and quasistatic simulations. The resulting toolbox supports retarded and quasistatic substrate and layer simulations, while high-permittivity materials remain insufficiently tested.

Problem
The paper addresses the limited support for simulating plasmonic nanoparticles on substrates or embedded in layer structures with the potential-based BEM approach.
Method
The authors develop a potential-based BEM method for layer structures and accelerate simulations by tabulating reflected Green functions on suitable grids for later interpolation.
Results
The implementation shows good agreement with DDA simulations including substrates and with quasistatic simulations across the reported comparisons.
Takeaways & Limitations
MNPBEM provides substrate and layer-structure effects for retarded and quasistatic plasmonics simulations, with representative examples and reusable demo files.
Takeaways & Limitations
High-permittivity materials, including metals, have not been properly tested and may require refined particle discretizations.

Abstract

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Within the MNPBEM toolbox, developed for the simulation of plasmonic nanoparticles using a boundary element method approach, we show how to include substrate and layer structure effects. We develop the methodology for solving Maxwell's equations using scalar and vector potentials within the inhomogeneous dielectric environment of a layer structure. We show that the implementation of our approach allows for fast and efficient simulations of plasmonic nanoparticles situated on top of substrates or embedded in layer structures. The new toolbox provides a number of demo files which can also be used as templates for other simulations.

Program summary

MNPBEM is a Matlab-based, vectorized and parallelized toolbox for simulating plasmonic nanoparticles with the boundary element method.

Program summary: MNPBEM uses the boundary element method to simulate plasmonic nanoparticles on substrates or within layer structures.The toolbox supports Matlab 7.13.0 and requires at least 4 GByte RAM for typical data.

1. Introduction

The paper extends MNPBEM to plasmonic nanoparticles on substrates or embedded in layer structures, addressing the need to simulate experimentally common inhomogeneous environments. It develops the methodology and a fast implementation based on tabulated reflected Green functions.

1. Introduction: Figure 1 organizes the workflow into layer setup, Green-function grid generation, and construction of a tabulated Green-function object.The resulting setup supports electromagnetic-field calculations at additional grid points.
1. Introduction: MNPBEM uses boundary element methods for particles modeled as homogeneous, isotropic materials separated by abrupt interfaces.The toolbox already supported plane-wave and dipole excitations and several optical observables.
1. Introduction: The paper addresses how to simulate plasmonic nanoparticles located on substrates or embedded in layer structures.Such configurations are common in experiments, but their inclusion in the potential-based BEM approach had been unclear.
1. Introduction: Layer-structure simulations require defining dielectric materials, layer indices and interfaces, tabulating reflected Green functions, and then running BEM simulations.The tabulation is computed before simulation and reused through interpolation.
1. Introduction: The toolbox examples include planewave excitation of a gold nanosphere with and without a layer structure.The sphere has a 20 nm diameter, is 5 nm from the substrate, and uses 144 boundary vertices.

2. Getting started

Getting started involves installing the Matlab toolbox, running demonstration programs, and defining particle, excitation, layer, and tabulated Green-function objects. After these additions, the layer-structure simulation proceeds like the homogeneous-background workflow.

2.1. Installation of the toolbox: Installation adds the MNPBEM main directory and all subdirectories to the Matlab search path.The help pages are generated by running makemnpbemhelp from the main directory.
2.2.1. Scattering spectra of sphere without layer structure: A homogeneous-background sphere example defines dielectric functions, a 144-vertex 20 nm sphere, a retarded BEM solver, and plane-wave excitation.The scattering cross section is computed over wavelengths from 400 to 900 nm.
2.2.2. Scattering spectra of sphere with layer structure: The layer-structure example adds glass, defines an interface at z = 0, and includes the layer structure in the BEM options.The upper and lower materials are selected by the index array [1,3].
2.2.2. Scattering spectra of sphere with layer structure: Reflected Green functions are tabulated on an automatically generated grid, stored in a compgreentablayer object, computed for the wavelength array, and added to the options.The remaining simulation is identical to the workflow without a layer structure.
2.1. Installation of the toolbox: The toolbox provides demo files for layer-structure simulations, with stat and ret variants representing quasistatic and full-Maxwell calculations.The demos are recommended as templates for further simulations and are documented with typical runtimes.

3. Theory

The paper formulates boundary-element simulations for nanoparticles in layered dielectrics using scalar and vector potentials, reflected Green functions, and coupled surface-charge/current equations. The resulting method extends homogeneous-background BEM to substrate and layer structures while remaining numerically tractable.

The BEM represents electromagnetic fields with scalar and vector potentials, while surface charges and currents enforce Maxwell boundary conditions at dielectric interfaces.
Layer interactions couple the perpendicular vector-potential component with surface charge, requiring coupled equations for h⊥ and σ rather than the decoupled homogeneous-background treatment.
For layer structures, the homogeneous-medium Green function is replaced by a layered Green function, and reflected Green functions are computed from cylinder-wave expansions and interface matching.
The layered formulation solves for surface charges and currents through matrix inversion, using matrices of order 2N instead of order N for the original BEM approach.
For arbitrary numbers of media, ε must be treated as a diagonal matrix, and G2 and ε2 must not be exchanged because reflected Green functions can connect different layers and media.
Reflected Green-function calculations remain fast because the required matrix inversions have low order, determined by the number of layer media multiplied by two or four.

4. Implementation

The toolbox represents layer structures through dielectric media, interface positions, and tabulated reflected Green functions used to accelerate BEM simulations. It also addresses interpolation grids, interface singularities, and validation against established simulations.

Layer structure: A layer structure is defined by dielectric functions, interface positions, and an index array ordering the media from top to bottom.The uppermost medium is epstab{ind(1)}, followed by the media associated with successive indices.
Layer structure: Layer-structure options specify tolerances, minimum radial and vertical distances, integration parameters, and ODE solver settings for Sommerfeld integrals.Boundary elements closer than ztol to an interface are assigned to the layer.
Tabulated Green functions: Reflected Green functions are precomputed on grids of r, z1, and z2 values, then interpolated to reduce the computational bottleneck of repeated evaluations.The toolbox supports automatically generated logarithmic grids or manually supplied grids, with separate tables required for points in different media.
Tabulated Green functions: The interpolation represents G, Fr, and Fz using a functional form based on radial distance and the sum of distances to nearby interfaces, with g, fr, and fz tabulated.This form matches the small-distance shape of quasistatic image-charge Green functions and is expected to vary weakly across layer structures.
Tabulated Green functions: Precomputed Green-function tables can be reused across simulations sharing a layer structure, although their initial wavelength-dependent construction may be time-consuming.The toolbox provides compatibility checking for previously computed tables.
Interface treatment: Interface boundary elements require special treatment because reflected Green functions produce singular surface-derivative contributions analogous to the standard BEM diagonal singularity.Elements within ztol of an interface are treated as interface elements, while the remaining contribution is smoothly integrated.
Validation: For nanospheres, nanodisks, and nanotriangles a few nanometers in size above glass, retarded and quasistatic scattering spectra show excellent agreement for all shapes.The quasistatic substrate is modeled with image charges, while the full calculations use retarded simulations.

5. Testing the toolbox

The toolbox agrees well with independent DDA and quasistatic calculations and remains stable across substrate-contact and discretization tests. Substrate proximity produces a significant plasmon redshift, while converged spectra are robust to several implementation choices.

5.1. Comparison with DDA: The BEM absorption cross sections agree very well with DDA simulations for a 50 nm gold nanosphere 1 nm above glass across the wavelength regime.
5.2. Comparison with quasistatic simulations: Full Maxwell and quasistatic simulations show excellent agreement for nanospheres, nanodisks, and nanotriangles near a glass substrate.
5.3. Disk above substrate: Decreasing the gap between a 60 nm diameter, 10 nm high gold nanodisk and glass causes a significant redshift of the plasmon scattering peak.
5.3. Disk above substrate: The disk 0.1 nm above glass and directly on the substrate produce very similar scattering results, despite requiring different integration refinement.The 0.1 nm case uses refine=3 and npol=40, whereas the on-substrate singular contribution can use refine=1 and npol=10.
5.3. Disk above substrate: Changing disk discretization, placing the lower plate slightly below the interface, or using different mesh resolutions gives practically indistinguishable or excellent-agreement spectra.For the membrane-supported nanotriangle, nrad=5,7,15 correspond to 823, 1256, and 2142 boundary elements, with only a small redshift at finer discretization.
5.3. Disk above substrate: Convergence requires sufficiently fine discretization near lower-plate edges, although coarse meshes remain similar for substrates and layers with not too high permittivities.

6. Summary

The authors implement substrate and layer effects in MNPBEM for retarded and quasistatic plasmonic simulations, using a potential-based BEM methodology with reflected-Green-function interpolation. Comparisons show good agreement and typical elementary simulations run in a few minutes, but high-permittivity materials remain insufficiently tested.

6. Summary: The implemented potential-based BEM methodology incorporates substrate and layer effects into the Matlab MNPBEM toolbox.
6. Summary: Computing reflected Green functions on a suitable grid and interpolating them later provides a significant speedup, with elementary nanoparticle simulations typically taking a few minutes.
6. Summary: The implementation agrees well with DDA simulations including substrates and with quasistatic simulations, supporting the viability of the approach.
6. Summary: The toolbox supports substrate and layer structures in both retarded and quasistatic simulations, including plane-wave, scattering, extinction, and dipole-related calculations.
6. Summary: The toolbox has primarily been tested with low-permittivity substrates and layers; high-permittivity materials, including metals, have not been properly tested.Such materials might require refined particle discretizations, so the authors advise caution with those results.

Appendix A. Deriving the BEM working equations

The appendix derives the layer-structure BEM equations by eliminating the parallel field variables and expressing them through perpendicular fields and surface charge. Substitution and rearrangement then produce the two coupled working equations.

Appendix A. Deriving the BEM working equations: The derivation first expresses h∥ in terms of h⊥ and σ using Eq. (8a).
Appendix A. Deriving the BEM working equations: With Σ = HG−1, the derivation rewrites terms from Eq. (9) before introducing abbreviations for the resulting expressions.
Appendix A. Deriving the BEM working equations: Substituting Eq. (A.1) into Eq. (9) yields the first working equation, Eq. (10a).
Appendix A. Deriving the BEM working equations: Rewriting Eq. (8b) and rearranging produces the second working equation, Eq. (10b).