Tags: Dispersion Relation / Permittivity

**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:- permittivity \(\varepsilon\left(\omega\right)\) and
- a three-dimensional conductivity \(\sigma\left(\omega\right)=\sigma_{2D}\left(\omega\right)/d\)
- 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

\[\begin{eqnarray*}\tanh\left\{ \sqrt{k_{p}^{2}-\varepsilon_{m}k_{0}^{2}}\frac{d}{2}\right\} & = & -\frac{\sqrt{k_{p}^{2}-\varepsilon_{m}k_{0}^{2}}\varepsilon_{d}}{\sqrt{k_{p}^{2}-\varepsilon_{d}k_{0}^{2}}\varepsilon_{m}}\ .\end{eqnarray*}\]

## Hints

In frequency space, **Ampère's law** reads

\[\nabla\times\mathbf{B}\left(\mathbf{r},\omega\right)=\mu_{0}\mathbf{j}\left(\mathbf{r},\omega\right)-\mathrm{i}\omega\mu_{0}\varepsilon\left(\mathbf{r},\omega\right)\mathbf{E}\left(\mathbf{r},\omega\right)\ .\]

Do you know a **relation** between \(\mathbf{j}\) and \(\mathbf{E}\) involving \(\sigma\)?

Can you interpret the conductivity term in Ampère's law as a **contribution to the permittivity**?

Let us now find out the permittivity of graphene!

We have to clearify two things to find the desired dispersion relation. First we have to find a relationship between **permittivity and conductivity**. Second we have to calculate the **vanishing thickness limit** of the SPP dispersion relation using the found relative “permittivity” of graphene with respect to this thickness.

## From Conductivity to Permittivity

In **Ampère's law**,

\[\begin{eqnarray*}\nabla\times\mathbf{B}\left(\mathbf{r},\omega\right) & = & \mu_{0}\mathbf{j}\left(\mathbf{r},\omega\right)-\mathrm{i}\omega\mu_{0}\varepsilon\left(\omega\right)\mathbf{E}\left(\mathbf{r},\omega\right)\end{eqnarray*}\]

we can use **Ohm's law** \(\mathbf{j}\left(\mathbf{r},\omega\right)=\sigma\left(\mathbf{r},\omega\right)\mathbf{E}\left(\mathbf{r},\omega\right)\)to find

\[\begin{eqnarray*}\nabla\times\mathbf{B}\left(\mathbf{r},\omega\right) & = & \mu_{0}\sigma\left(\mathbf{r},\omega\right)\mathbf{E}\left(\mathbf{r},\omega\right)-\mathrm{i}\omega\mu_{0}\varepsilon\left(\mathbf{r},\omega\right)\mathbf{E}\left(\mathbf{r},\omega\right)\\& = & \mu_{0}\left(\sigma\left(\mathbf{r},\omega\right)-\mathrm{i}\omega\varepsilon\left(\mathbf{r},\omega\right)\right)\mathbf{E}\left(\mathbf{r},\omega\right)\\& = & -\mathrm{i}\omega\mu_{0}\tilde{\varepsilon}\left(\mathbf{r},\omega\right)\mathbf{E}\left(\mathbf{r},\omega\right)\ .\end{eqnarray*}\]

The latter equation states that by using **Ohm's law** we can interprete the **conductivity** as a contribution to the **permittivity** \[\begin{eqnarray*}\tilde{\varepsilon}\left(\mathbf{r},\omega\right) & = & \varepsilon\left(\mathbf{r},\omega\right)+\mathrm{i}\frac{\sigma\left(\mathbf{r},\omega\right)}{\omega}\ .\end{eqnarray*}\]For graphene we have only given a **two-dimensional conductivity** and have to divide by the thickness to obtain a three-dimensional analog. We find the “permittivity” of graphene as\[\begin{eqnarray*}\tilde{\varepsilon}\left(\omega\right)=\varepsilon_{0}\varepsilon_{m}\left(\omega\right) & = & \varepsilon_{0}\left(1+\mathrm{i}\frac{\sigma_{2D}\left(\omega\right)}{\varepsilon_{0}\cdot\omega\cdot d}\right)\ .\end{eqnarray*}\]

## The Vanishing Thickness Limit

A **Taylor expansion** of the tangens hyperbolicus for small arguments is given by \[\begin{eqnarray*}\tanh\left(x\right) & = & x-\frac{x^{3}}{3}+\mathcal{O}\left(x^{5}\right)\ .\end{eqnarray*}\]We use this approximation in the dispersion relation and obtain\[\begin{eqnarray*}\tanh\left\{ \sqrt{k_{p}^{2}-\varepsilon_{m}k_{0}^{2}}\frac{d}{2}\right\} & \approx & \sqrt{k_{p}^{2}-\varepsilon_{m}k_{0}^{2}}\frac{d}{2}\\& = & -\frac{\sqrt{k_{p}^{2}-\varepsilon_{m}k_{0}^{2}}\varepsilon_{d}}{\sqrt{k_{p}^{2}-\varepsilon_{d}k_{0}^{2}}\varepsilon_{m}}\ .\end{eqnarray*}\]In the last step we have included the relative “permittivity” of graphene. However in the limit of vanishing thickness we can neglect the one in the denominator and find\[\begin{eqnarray*}k_{p}^{2} & \approx & \varepsilon_{d}k_{0}^{2}-\left(2\frac{\varepsilon_{0}\varepsilon_{d}\cdot\omega}{\sigma_{2D}\left(\omega\right)}\right)^{2}\\& = & \varepsilon_{d}k_{0}^{2}\left[1-\varepsilon_{d}\left(2\frac{\varepsilon_{0}c}{\sigma_{2D}\left(\omega\right)}\right)^{2}\right]\\ & = & \varepsilon_{d}k_{0}^{2}\left[1-\left(\frac{2}{\eta\cdot\sigma_{2D}\left(\omega\right)}\right)^{2}\right]\end{eqnarray*}\] where we have used the free space wave vector \(k_{0}=\omega/c\) and the impedance \(\eta=\sqrt{\mu_{0}/\varepsilon_{0}\varepsilon_{d}}\) of the surrounding medium in the last two steps. See Hanson's Dyadic Green's functions and guided surface waves for a surface conductivity model of graphene for a derivation of this relation using other methods.

The found **dispersion relation** allows some physical interpretation. We can see that if for a given frequency the conductivity approaches high values, \(k_{p}\rightarrow\varepsilon_{d}k_{0}^{2}\), hence graphene acts as a perfect metal as expected. On the other hand, if we look at the conductivity again and set \(\Gamma=0\), \(\sigma_{2D}\left(\omega\right)\) is a purely imaginary function. So for low conductivities \(k_{p}\gg k_{0}\).

This makes it possible to build **resonant graphene devices** that are **much smaller than the wavelength**. Antennas in THz frequencies for example can be in the order of just a few hundred nanometers for wavelengths of several tens of microns.

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**.

However, the outlined permittivity of graphene is widely used in scientific literature.