**Graphene** is a monolayer of carbon atoms and offers extremely interesting physics. Its **unique properties** may very well make it the silicon of the 21st century. For practical applications it is of great importance to **model** this material containing the main electromagnetic features. Find out in this problem how graphene can be represented as a **thin layer** with a **certain permittivity** and how this can be used for next-generation electromagnetic devices.

## Problem Statement

Find the **dispersion relation** of transverse magnetic surface plasmon polaritons (SPPs) supported by graphene embedded in some medium with relative permittivity \(\varepsilon_{d}\)! You may want to follow these steps:

- Start from
**Ampère's law**including the displacement current to figure out a relation between \(\varepsilon\left(\omega\right)\) and a three-dimensional \(\sigma\left(\omega\right)=\sigma_{2D}\left(\omega\right)/d\) assuming a certain thickness \(d\) of graphene. - Regard graphene now as a
**metallic layer**with thickness \(d\) and relative permittivity \(\varepsilon_{m}\left(\omega\right)\) - Use the
**dispersion relation**of transverse magnetic SPPs to find the wave-vector in the limit of**vanishing thickness**\(d\).

The dispersion relation for the wave vector \(k_{p}\left(\omega\right)\) of such an SPP supported by the given layer reads as\[\begin{eqnarray*}\tanh\left\{ \sqrt{k_{p}^{2}-\varepsilon_{m}k_{0}^{2}}\frac{d}{2}\right\} & = & -\frac{\sqrt{k_{p}^{2}-\varepsilon_{m}k_{0}^{2}}\varepsilon_{d}}{\sqrt{k_{p}^{2}-\varepsilon_{d}k_{0}^{2}}\varepsilon_{m}}\ .\end{eqnarray*}\]

Background: Graphene - a Material with Tunable Conductivity

The main results for the physical quantities of **graphene** follow from calculations which are a little too complex to be presented here. In short, the properties of the material are related to its **high conductivity** \(\sigma_{2D}\). Furthermore, \(\sigma_{2D}\) can be **tuned** using the electric field effect (EFE) applying a certain voltage difference to a surrounding material. These two properties can be seen as the reason why enormous effort is put into research relating graphene at the moment.

The EFE changes the so-called **chemical potential** \(\mu_{c}\) which enters the **conductivity** \(\sigma_{2D}\) of graphene:\[\begin{eqnarray*}\sigma_{2D}\left(\omega\right) & = & \mathrm{i}\frac{1}{\pi\hbar^{2}}\frac{e^{2}k_{B}T}{\omega+\mathrm{i}2\Gamma}\left\{ \frac{{\color{red}\mu_{c}}}{k_{B}T}+2\ln\left[\exp\left(-\frac{{\color{red}\mu_{c}}}{k_{B}T}\right)+1\right]\right\} \\& & +\mathrm{i}\frac{e^{2}}{4\pi\hbar}\ln\left[\frac{2\left|{\color{red}\mu_{c}}\right|-\hbar\left(\omega+\mathrm{i}2\Gamma\right)}{2\left|{\color{red}\mu_{c}}\right|+\hbar\left(\omega+\mathrm{i}2\Gamma\right)}\right]\ .\end{eqnarray*}\]Here, \(T\) is the temperature and \(\Gamma\) a rate corresponding to relaxation processes. We are furthermore in the frequency (Fourier) domain with an \(\exp\left(-\mathrm{i}\omega t\right)\) time dependence. In the literature you can find \(\mu_{c}\approx50\dots1000\) meV and \(\Gamma\lesssim1\) meV as **characteristic values**.

Now we want to model this two dimensional conductivity to be able to do electrodynamic **calculations**, e.g. on a computer. A very decent way to do so is to find a representation of graphene as a **thin three-dimensional layer** with a certain permittivity supporting collective electron oscillation called **surface plasmon polaritons** (SPPs), see right figure. With this trick, sophisticated computations are possible and applications for graphene can be designed, see also Graphene Plasmonics: Antennas for THz Spectroscopy.

Note that the permittivity we have derived is just the dispersion relation of a kind of supported modes, transverse magnetic SPPs, not of anything that can happen! The quantity has to be used with care which I want to emphasize using the **quotation marks**.