The C (graphene) - Falkovsky (mid-IR) material model provides an *alternate* to the default graphene surface conductivity model. The Falkovsky model is implemented using the analytic permittivity material model, and it is defined by

$$ \tau^{-1} = \frac{e \nu_f^2}{\mu \mu_c}$$

where

- \(\nu_f\) : Fermi velocity
- \(\mu\) : carrier mobility

$$ \varepsilon_{xx} = \varepsilon_{yy} = x_6 - \frac{x_1^2 x_3}{\pi x_2^2}\frac{w}{w^2 + \left( \frac{x_1 x_4^2}{x_5 x_3}\right)^2} \frac{1}{w x_8 x_7} + i\frac{x_1^2 x_3}{\pi x_2^2}\frac{w}{w^2 + \left( \frac{x_1 x_4^2}{x_5 x_3}\right)^2} \frac{x_1 x_4^2}{w x_5 x_3 x_8 x_7}$$

The analytic material type does not support arbitrary variable names, so the real and imaginary parts of the permittivity tensor entries were entered using variable names x_{1} through x_{8}:

where

- \(x_1 = e\) (elementary charge [coulomb])
- \(x_2 = \hbar\) (reduced Plank constant [joule\(\cdot\)second])
- \(x_3 = \mu_c\) (chemical potential or Fermi energy [joule])
- \(x_4 = \nu_f\) (Fermi velocity [meter/second])
- \(x_5 = \mu\) (carrier mobility [meter^2/volt/second])
- \(x_6 = \varepsilon_r\) (background relative permittivity)
- \(x_7 = \Delta\) (graphene thickness [meter])
- \(x_8 = \varepsilon_0\) (vacuum permittivity [farad/meter])

When using the C (graphene) - Falkovsky (mid-IR) material model, it is important to keep two important constraints in mind. First, the layer thickness that is entered into the material model (x_{7}) must match the thickness of the rectangle structure used in your simulation. Second, the material model assumes that the graphene layer is parallel to the XY plane. If you want to orient the graphene in some other plane, three permittivity entries in (10) must be swapped accordingly.

### Related publications

- G. W. Hanson, "Dyadic Green’s functions and guided surface waves for a surface conductivity," J. Appl. Phys. Vol. 103, 064302 (2008).
- Optical properties of graphene, L.A. Falkovsky, Journal of Physics: Conference Series Vol. 129, 012004 (2008).