# Impact of Graphene on the Polarizability of a Neighbour Nanoparticle: A Dyadic Green’s Function Study

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## Abstract

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## 1. Introduction

## 2. Dyadic Green’s Function for the Electric Field

#### 2.1. Free-Space Dyadic Green’s Function

#### 2.2. Weyl’s or Angular Spectrum Representation of the Dyadic Green’s Function: A Useful Formulation for Interfaces

#### 2.3. Source and Scattered Green’s Functions: Scattering at a Planar Interface

## 3. Renormalization of the Polarizability of a Quantum Emitter Near a Graphene Sheet and a Graphene-Based Grating

#### 3.1. Polarizability of a Quantum Emitter in a Homogeneous Medium

#### 3.2. Polarizability of a Quantum Emitter in Proximity to a Planar Interface

#### 3.3. Renormalized Polarizability of an Isotropic Quantum Emitter Near a Continuous Graphene Sheet

#### 3.4. Renormalized Polarizability of an Isotropic Quantum Emitter Near a Plasmonic Graphene Grating

#### 3.4.1. Optical Properties of a Plasmonic Graphene Grating

#### 3.4.2. Renormalization of the Polarizability of a Quantum Emitter

## 4. Extension of the Formalism When the Quantum Emitter Has Both an Electric and a Magnetic Dipole

#### 4.1. Free-Space Electric, Magnetic and Mixed Green’s Functions

#### 4.2. Weyl’s or Angular Spectrum Representation of Magnetic and Mixed Green’s Functions

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

SNOM | Scanning near-fieldoptical microscope |

RHS | Right-hand-side |

NP | Nanoparticle |

THz | Terahertz |

## Appendix A. Derivation of the Wave Equation

## Appendix B. Green’s Function for the Helmholtz Equation

## References

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**Figure 1.**The two systems considered in this paper: a graphene sheet (

**a**) and a graphene grid of ribbons (

**b**) located in between two dielectrics. A nanoparticle is located at position ${\mathbf{r}}_{0}=(0,0,{z}_{0})$ and is characterized by a polarizability tensor ${\alpha}_{0}$ in vacuum. In addition, a plane wave impinges on the nanoparticle and on graphene coming from $z=+\infty $.

**Figure 2.**Representation of the primary field (${\mathbf{E}}_{0}^{\pm}$), emitted by a point dipole (represented by the golden ball with an arrow), and the reflected (${\mathbf{E}}_{r}$) and the transmitted (${\mathbf{E}}_{t}$) field due to the presence of the interface at $z=0$. The ± sign in ${\mathbf{E}}_{0}^{\pm}$ indicates whether the field is emitted along the positive/negative z direction.

**Figure 3.**Top panels: Real (

**a**) and imaginary (

**b**) parts of the renormalized polarizability of a gold nanoparticle with radius $R=50\phantom{\rule{0.166667em}{0ex}}$nm located at a distance of ${z}_{0}=151\phantom{\rule{0.166667em}{0ex}}$nm from a graphene sheet with a Fermi energy of 1 eV and damping parameter of $\hslash \gamma =4.1$ meV supported by a dielectric of permittivity ${\u03f5}_{2}=2$. The solid red line represents the $xx$ component, and the black dotted line represents the $zz$ component of the polarizability in the presence of graphene. For comparison, the $xx$ component of the polarizability of the nanoparticle is also represented in the absence of graphene (but in the presence of the dielectric interface), as ${\alpha}_{xx}^{NG}$. One can appreciate the increase in the imaginary part of the polarizability by about two orders of magnitude when the particle is near doped graphene. (

**c**) The imaginary part of the nanoparticle polarizability in a vacuum. In all the panels, the parameters used in the Drude model for dielectric function of gold are: $\hslash {\omega}_{p}=7.9$ eV and ${\mathsf{\Gamma}}_{0}=0.053$ eV.

**Figure 4.**Real (

**a**) and imaginary (

**b**) parts of the renormalized polarizability of a CdSe nanoparticle with $R=50\phantom{\rule{0.166667em}{0ex}}$nm located at a distance of ${z}_{0}=151\phantom{\rule{0.166667em}{0ex}}$nm from a graphene sheet with a Fermi energy of 1 eV and damping parameter of $\hslash \gamma =4.1$ meV, supported by a dielectric of permittivity ${\u03f5}_{2}=2$. The solid red line represents the $xx$ component of the polarizability, and the black dotted line represents the $zz$ component. The dashed blue line is the $xx$ component of the polarizability in the absence of graphene. The parameters used in both panels for the Lorentz model for the dielectric function of CdSe are: ${\u03f5}_{\infty}=6.2$, ${\omega}_{\mathrm{LO}}=211$ cm${}^{-1}$, ${\omega}_{\mathrm{TO}}=169$ cm${}^{-1}$ and ${\mathsf{\Gamma}}_{\mathrm{TO}}=5$ cm${}^{-1}$.

**Figure 5.**Real (blue dashed line) and imaginary (orange line) of the function ${\mu}_{0}\chi \left(\omega \right)$. The parameters of the grating are $L=0.5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m and $w=L/2$. The Fermi energy of graphene is ${E}_{F}=1$ eV. The real part has a pronounced resonance due to the excitation of a surface plasmon polariton of that frequency (∼87 THz).

**Figure 6.**Real (

**a**) and imaginary (

**b**) renormalized polarizability of a gold nanoparticle in close proximity to a plasmonic graphene-based grating. The solid red line represents the $xx$ component of the polarizability in the presence of graphene, the dashed brown line represents the $yy$ component, and the black dotted line represents the $zz$ component. The dashed blue line is the $xx$ component of the polarizability in the absence of graphene. Note that ${\alpha}_{xx}\ne {\alpha}_{yy}$, due to lack of rotational symmetry in the $xy-$plane introduced by the ribbon structure. The parameters of the grating are $L=0.5\phantom{\rule{0.166667em}{0ex}}\mu $m and $w=L/2$. The parameters for the graphene conductivity and for the Drude dielectric function of gold are the same as in Figure 3.

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## Share and Cite

**MDPI and ACS Style**

Amorim, B.; Gonçalves, P.A.D.; Vasilevskiy, M.I.; Peres, N.M.R.
Impact of Graphene on the Polarizability of a Neighbour Nanoparticle: A Dyadic Green’s Function Study. *Appl. Sci.* **2017**, *7*, 1158.
https://doi.org/10.3390/app7111158

**AMA Style**

Amorim B, Gonçalves PAD, Vasilevskiy MI, Peres NMR.
Impact of Graphene on the Polarizability of a Neighbour Nanoparticle: A Dyadic Green’s Function Study. *Applied Sciences*. 2017; 7(11):1158.
https://doi.org/10.3390/app7111158

**Chicago/Turabian Style**

Amorim, B., P. A. D. Gonçalves, M. I. Vasilevskiy, and N. M. R. Peres.
2017. "Impact of Graphene on the Polarizability of a Neighbour Nanoparticle: A Dyadic Green’s Function Study" *Applied Sciences* 7, no. 11: 1158.
https://doi.org/10.3390/app7111158