^{1}, S. Lasquellec

^{1}and C. Brosseau

^{1,a)}

### Abstract

We present a numerical model we have created and verified to characterize the frequency dependence of the effective magnetic permeability and permittivity of a core-shell (CS) nanostructure composed of a magnetic core and a plasmonic shell with well-controlled dimensions for different geometries and polarizations. Two principal ingredients in our model are as follows: (i) we consider two-dimensional (or cross sections of infinite three-dimensional parallel, infinitely long, identical, cylinders, where the properties and characteristics are invariant along the perpendicular cross sectional plane) three-phase heterostructure, and (ii) while strictly valid only in a dc situation, our analysis can be extended to treat electric fields that oscillate with time provided that the wavelengths associated with the fields are much larger than the microstructure dimension in order that the homogeneous (effective medium) representation of the composite structure makes sense. Such nanostructures simultaneously possess both magnetic gyromagnetic resonance and plasmonicresonance (PLR) resonances. To illustrate the effects of shape anisotropy of the CS structure, we analyze several possible shell shapes involving sharp edges and tips. Geometric parameters of the CS nanostructures and excitation polarized parallel and perpendicular to the antenna axis permit to finely tune the PLR. Changing the internal geometry of the nanostructure not only shifts its resonance frequencies but can also strongly modify the relative magnitudes of the electric field enhancement, independently of nanoparticle shape. The model sets the foundation of quantitatively determining the spatial confinement of the electric field in regions ≈20 nm in linear dimension. Because of its resonant nature, we found nanolocalized terahertz fields corresponding to large electric field enhancement two orders of magnitude higher in amplitude than the excitation optical field. The simulations in this paper are important because magnetoplasmonic CS nanostructures are currently being explored as candidates for resonant optical nanoantennas for biosensing applications.

The Lab-STICC is Unité Mixte de Recherche CNRS 3192. We are grateful to the Ph.D. funding program (grant program 211-B2-9/ARED) of the Conseil Régional de Bretagne for funding this work.

I. INTRODUCTION

II. Preliminaries and the model

A. Model of the CS nanostructures and details of the calculations

B. Numerical method

III. RESULTS and DISCUSSION

IV. Concluding remarks and future directions

### Key Topics

- Nanostructures
- 40.0
- Nanomagnetism
- 30.0
- Electric fields
- 21.0
- Plasmons
- 20.0
- Permittivity
- 18.0

## Figures

Illustrative examples of CS nanostructures considered: (a) isotropic case; (b), (c), and (d) refer to representative anisotropic CS nanostructures situations where the shell of the CS nanostructures involves sharp edges and tips.

Illustrative examples of CS nanostructures considered: (a) isotropic case; (b), (c), and (d) refer to representative anisotropic CS nanostructures situations where the shell of the CS nanostructures involves sharp edges and tips.

Cross-sectional view for the infinite CS nanostructure (i.e., for type a) investigated. The square cell has size in the direction and is infinitely extended in the direction. We assume that the cylinders are centered about the origin of the coordinates. Since typical values of the radius and shell thickness are and , respectively, the shell thickness is greatly exaggerated in this figure.

Cross-sectional view for the infinite CS nanostructure (i.e., for type a) investigated. The square cell has size in the direction and is infinitely extended in the direction. We assume that the cylinders are centered about the origin of the coordinates. Since typical values of the radius and shell thickness are and , respectively, the shell thickness is greatly exaggerated in this figure.

(a) Calculated values of the region of the complex plane inside which the allowed values of the effective complex permittivity of the CS structure a of Fig. 1 exist from the Wiener (black solid and dashed lines) and HS (gray solid and dashed lines) bounds. . The FE calculated value of the effective magnetic permeability gives . (b) The real, , and imaginary, , parts of the effective complex permittivity plotted as a function of near the resonance of the CS structure a of Fig. 1. The solid line represents the values of deduced from using the KK relationship.

(a) Calculated values of the region of the complex plane inside which the allowed values of the effective complex permittivity of the CS structure a of Fig. 1 exist from the Wiener (black solid and dashed lines) and HS (gray solid and dashed lines) bounds. . The FE calculated value of the effective magnetic permeability gives . (b) The real, , and imaginary, , parts of the effective complex permittivity plotted as a function of near the resonance of the CS structure a of Fig. 1. The solid line represents the values of deduced from using the KK relationship.

(a) The real part of the complex effective magnetic permeability is plotted versus frequency of the magnetic field, for the CS structures considered in Fig. 1 and . The letters and corresponding lines (solid: a, dashed: b, dotted: c, and dash-dotted: d) refer to the structures displayed in Fig. 1. The magnetic field is defined to be along the -axis. (b) Same as in (a) for the imaginary part of the complex effective magnetic permeability. (c) Same as in (a) for the modulus of the impedance. (d) Same as in (c) for the phase of the impedance.

(a) The real part of the complex effective magnetic permeability is plotted versus frequency of the magnetic field, for the CS structures considered in Fig. 1 and . The letters and corresponding lines (solid: a, dashed: b, dotted: c, and dash-dotted: d) refer to the structures displayed in Fig. 1. The magnetic field is defined to be along the -axis. (b) Same as in (a) for the imaginary part of the complex effective magnetic permeability. (c) Same as in (a) for the modulus of the impedance. (d) Same as in (c) for the phase of the impedance.

Same as in Fig. 3 for .

Same as in Fig. 3 for .

Spatial distribution of the MFE for the gyresonance mode (4.53 GHz) for the 2D CS structure considered in the inset of Fig. 1. , , and . The arrows indicate the orientation of the magnetic field.

Spatial distribution of the MFE for the gyresonance mode (4.53 GHz) for the 2D CS structure considered in the inset of Fig. 1. , , and . The arrows indicate the orientation of the magnetic field.

(a) Evolution of the spectral behavior of the real part of the effective permittivity for the CS structures considered in Fig. 1 and . The letters and corresponding lines refer to the structures displayed in Fig. 1 and polarization of the electric field. The electric field is defined to be either along the -axis or the -axis . (b) Same as in (a) for the imaginary part of the effective permittivity of the complex effective permittivity. (c) Same as in (a) for the modulus of the impedance. (d) Same as in (c) for the phase of the impedance.

(a) Evolution of the spectral behavior of the real part of the effective permittivity for the CS structures considered in Fig. 1 and . The letters and corresponding lines refer to the structures displayed in Fig. 1 and polarization of the electric field. The electric field is defined to be either along the -axis or the -axis . (b) Same as in (a) for the imaginary part of the effective permittivity of the complex effective permittivity. (c) Same as in (a) for the modulus of the impedance. (d) Same as in (c) for the phase of the impedance.

Same as in Fig. 6 for .

Same as in Fig. 6 for .

Top: visualization of the EFE for CS structure a and corresponding to . The electric field is oriented along the -axis. Middle: same as in top for CS structure d and . The electric field is oriented along the -axis. Bottom: same as in middle for . The electric field is oriented along the -axis.

Top: visualization of the EFE for CS structure a and corresponding to . The electric field is oriented along the -axis. Middle: same as in top for CS structure d and . The electric field is oriented along the -axis. Bottom: same as in middle for . The electric field is oriented along the -axis.

Evolution of the spectral behavior of the modulus of the impedance for several variants of the d CS structure displayed in Fig. 1 and . The letters and corresponding lines, refer to the d structure with values of and listed in Table I. The electric field is defined to be either along the -axis or the -axis . (b) Same as in (a) for the phase of the impedance.

Evolution of the spectral behavior of the modulus of the impedance for several variants of the d CS structure displayed in Fig. 1 and . The letters and corresponding lines, refer to the d structure with values of and listed in Table I. The electric field is defined to be either along the -axis or the -axis . (b) Same as in (a) for the phase of the impedance.

(a) Evolution of the spectral behavior of the real part of the effective permittivity for several variants of the d CS structure displayed in Fig. 1 and . The letters and corresponding lines refer to the d structure with values of and listed in Table I. The electric field is directed along the -axis . (b) Same as in (a) for the imaginary part of the effective permittivity of the complex effective permittivity. (c) Same as in (a) for the modulus of the impedance. (d) Same as in (c) for the phase of the impedance.

(a) Evolution of the spectral behavior of the real part of the effective permittivity for several variants of the d CS structure displayed in Fig. 1 and . The letters and corresponding lines refer to the d structure with values of and listed in Table I. The electric field is directed along the -axis . (b) Same as in (a) for the imaginary part of the effective permittivity of the complex effective permittivity. (c) Same as in (a) for the modulus of the impedance. (d) Same as in (c) for the phase of the impedance.

Same as in Fig. 10 when the electric field is oriented along the -axis .

Same as in Fig. 10 when the electric field is oriented along the -axis .

Top: visualization of the EFE for CS structure and corresponding to . Same as in top for CS structure and . Same as in top for structure and . Same as in top for structure and 275.2 THz. The electric field is oriented along the -axis.

Top: visualization of the EFE for CS structure and corresponding to . Same as in top for CS structure and . Same as in top for structure and . Same as in top for structure and 275.2 THz. The electric field is oriented along the -axis.

Same as in Fig. 12 when the electric field is oriented along the -axis. From top to bottom, the frequencies are for : 279.1 THz, : 232 THz, : 207.4 THz, and : 252.1 THz, respectively.

Same as in Fig. 12 when the electric field is oriented along the -axis. From top to bottom, the frequencies are for : 279.1 THz, : 232 THz, : 207.4 THz, and : 252.1 THz, respectively.

## Tables

Values of and considered for the calculations leading to Figs. 10–14 (see Fig. 1 for their respective definition).

Values of and considered for the calculations leading to Figs. 10–14 (see Fig. 1 for their respective definition).

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