Graphene is different from most optical materials in two important regards. First, it is usually a very thin material layer with a thickness as small as one atom. Second, it is usually characterized using a surface conductivity rather than a volumetric permittivity. The graphene material model in FDTD and MODE allows graphene to be accurately simulated as a 2D material without the need for an extremely small mesh, resulting in much faster simulations.

Occasionally, it may be useful to use a more conventional permittivity model, as described in the second half of this page.

## Graphene surface conductivity material model

As explained in [1], the surface conductivity for a single layer of graphene is given by

$$\sigma(\omega, \Gamma, \mu_c, T) = \sigma_{intra}(\omega, \Gamma, \mu_c, T) + \sigma_{inter}(\omega, \Gamma, \mu_c, T)$$

$$\sigma_{intra}(\omega, \Gamma, \mu_c, T) = \frac{-ie^2}{\pi \hbar^2 (\omega + i2\Gamma)} \int^\infty_0 \xi \left(\frac{\partial f_d(\xi)}{\partial \xi} - \frac{\partial f_d(-\xi)}{\partial \xi} \right) d\xi$$

$$\sigma_{inter}(\omega, \Gamma, \mu_c, T) = \frac{ie^2(\omega + i2\Gamma)}{\pi \hbar^2} \int^\infty_0\frac{f_d(-\xi) - f_d(\xi)}{(\omega + i2\Gamma)^2 - 4(\xi/\hbar)^2} d\xi$$

where

- \(\omega\) : angular frequency
- \(\Gamma\) : scattering rate
- \(\mu_c\) : chemical potential
- \(T\) : temperature
- \(e\) : electron charge
- \(\hbar\) : reduced Plank constant
- \(k_B\) : Boltzmann constant

and

$$f_d(\xi) \equiv \frac{1}{\mathrm{exp}((\xi - \mu_c)/(k_BT)) + 1}$$

is the Fermi-Dirac distribution. The two conductivity terms in (2) and (3) are referred to as the intraband and interband terms, respectively. The above surface conductivity material function is available in the material database in both FDTD and MODE as illustrated in Figure 1.

Fig. 1. Graphene material model based on a surface conductivity.

In strict terms, the surface conductivity material model specified by (1)-(4) is valid only for a single graphene layer. However, under some circumstances, this model can also be used to represent multiple layers by scaling the total conductivity by the number of layers. For this purpose, a conductivity scaling factor has been included in the material parameters of the graphene material type (see Figure 1); the scaling factor multiplies both (2) and (3). Once the desired scattering rate, chemical potential, temperature and scaling factor have been specified, the corresponding surface conductivity can be visualized in the material explorer. The conductivity plots shown in Fig. 2 were generated in the material explorer using the material parameters shown in Fig. 1.

Fig. 2. Real part (left) and imaginary part (right) of the surface conductivity of graphene (material parameters given in Fig. 1).

Once an instance of the graphene material type has been added to the material database, it can be assigned to any 2D rectangle. This provides a straightforward mechanism for introducing a graphene layer into a simulation. Figure 3 demonstrates how to add a 2D rectangle to a simulation project. Keep in mind that, because the graphene material type generates a surface conductivity that corresponds to a flat surface, it can only be used with geometric structures that represent flat surfaces.

Fig. 3. Adding a 2D rectangle to a simulation.

To generate the surface conductivity of graphene, the graphene material type evaluates the intraband conductivity term analytically and the interband conductivity term numerically. The intraband term is evaluated using the following formula:

$$\sigma_{intra}(\omega, \Gamma, \mu_c, T) = \frac{ie^2 k_B T}{\pi \hbar^2(\omega + i2\Gamma)} \left[\frac{\mu_c}{k_B T} + 2 \mathrm{ln} \left(\mathrm{exp} \left(-\frac{\mu_c}{k_B T} \right) +1 \right) \right]$$

## Creating an equivalent volumetric permittivity model

A graphene layer can be modeled using a uniaxial anisotropic permittivity if we assume that the graphene layer has a finite thickness as illustrated in Figure 4. By virtue of occupying a finite volume, the graphene layer must also have a reference permittivity value.

Fig. 4. Volumetric graphene layer with finite thickness.

To turn the surface conductivity of a graphene layer into a uniaxial anisotropic permittivity, two additional parameters must be therefore introduced:

- \(\Delta\) : sheet thickness
- \(\varepsilon_r\) : background relative permittivity

Using these two parameters, the two parallel (or in-plane) components and the perpendicular (or out-of-plane) component of the permittivity tensor are given by

$$\varepsilon_{||}(\omega, \Gamma, \mu_c, T) = \varepsilon_r + i\frac{\sigma(\omega, \Gamma, \mu_c, T)}{\varepsilon_0 \omega \Delta} \mathrm{\ and\ } \varepsilon_\bot = \varepsilon_r$$

The above uniaxial anisotropic material description can be introduced into a simulation in FDTD and MODE using the analytic material type. Because this material type requires explicit expressions for the entries of the permittivity tensor, it is necessary to use an approximation of the surface conductivity. As an example, let us employ a Drude-like approximation valid for mid-infrared wavelengths. This type of conductivity approximation is often employed in the graphene literature (see eq. (3) in reference [2]). In the mid-infrared, the intraband conductivity term usually dominates over the interband term. So, let us neglect the interband term and approximate the intraband term in (5) by a Drude-like expression. This leads to

$$\sigma(\omega, \Gamma, \mu_c, T) \approx \sigma_{intra}(\omega, \Gamma, \mu_c, T) \approx \frac{i e^2 \mu_c}{\pi \hbar^2(\omega + i\tau^{-1})} \mathrm{\ where \ } \tau = \frac{1}{2\Gamma}$$

Now, substituting (7) into (6) results in the following uniaxial anisotropic material description:

$$\varepsilon_{||} = \varepsilon_r - \left( \frac{ \mu_c e^2 \omega}{\pi \hbar^2(\omega^2 + \tau^{-2})}\right)\frac{1}{\varepsilon_0 \omega \Delta} + i\left( \frac{ \mu_c e^2}{\pi \hbar^2(\omega^2 + \tau^{-2})\tau}\right)\frac{1}{\varepsilon_0 \omega \Delta}\mathrm{\ and\ } \varepsilon_\bot = \varepsilon_r$$

The C (graphene) - Falkovsky (mid-IR) material found in the default material database is based on a similar approach, but uses a slightly different analytic model of graphene.

### Related Publications

[1] G. W. Hanson, "Dyadic Green’s functions and guided surface waves for a surface conductivity," J. Appl. Phys. Vol. 103, 064302 (2008).

[2] Optical properties of graphene, L.A. Falkovsky, Journal of Physics: Conference Series Vol. 129, 012004 (2008).