Relation between permittivity,refractive index and conductivity in plasmonic elements

Here i want to discuss about relation between permittivity,refractive index and conductivity for plasmonic elements like gold,silver and graphene.

I quote definition permittivity from this link.

In electromagnetismpermittivity or absolute permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of resistance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium.

\epsilon=\epsilon_{r}\epsilon_{0}=\left(1+\chi\right)\epsilon_{0}

I quote definition conductivity from this link.

Electrical resistivity (also known as resistivityspecific electrical resistance, or volume resistivity) is an intrinsic property that quantifies how strongly a given material opposes the flow of electric current. A low resistivity indicates a material that readily allows the flow of electric current. Resistivity is commonly represented by the Greek letter ρ (rho). The SIunit of electrical resistivity is the ohmmetre (Ω⋅m).[1][2][3] As an example, if a 1 m × 1 m × 1 m solid cube of material has sheet contacts on two opposite faces, and the resistance between these contacts is 1 Ω, then the resistivity of the material is 1 Ω⋅m.

Electrical conductivity or specific conductance is the reciprocal of electrical resistivity, and measures a material’s ability to conduct an electric current. It is commonly represented by the Greek letter σ (sigma), but κ (kappa) (especially in electrical engineering) or γ (gamma) are also occasionally used. Its SI unit is siemens per metre (S/m) and CGSE unit is reciprocal second (s−1).

\sigma(conductivity)=\frac{1}{\rho(Resistace)}=\frac{J}{E}

I quote definition refractive index from this link.

In optics, the refractive index or index of refractionn of a material is a dimensionless number that describes how light propagates through that medium. It is defined as

n=\frac{c}{v}

I quote definition effective refractive index from this link.

For plane waves in homogeneous transparent media, the refractive indexn can be used to quantify the increase in the wavenumber (phase change per unit length) caused by the medium: the wavenumber is n times higher than it would be in vacuum. The effective refractive indexneff has the analogous meaning for light propagation in a waveguide with restricted transverse extension: the β value (phase constant) of the waveguide (for some wavelength) is the effective index times the vacuum wavenumber:

\beta={n_{eff}}k_{0}

Relation between permittivity and refractive index:

you can see proof from this link.

\epsilon_{r}=n^{2}

The relation between permittivity and conductivity

see this link.

\epsilon(w)=\epsilon^{'}(w)+\epsilon^{''}(w)=\epsilon_{r}(w)\epsilon_{0}+j\frac{\sigma(w)}{w}

 

Complex conductivity:

see this link.

\sigma_{c}=jw\epsilon_{c}=\sigma+jw\epsilon

 Graphene conductiviy  kubbo formula:

kubbo formula
Fig1:kubbo formula
fd
Fig2:fd(e)
sigmaintra
Fig3:sigmaintra
sigmainter
Fig4:sigmainter

 

 

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