Interface conditions on a graphene interface

[svletana]

Homework Statement

I'm trying to do all the calculations for the attached paper, and I'm having trouble with the boundary conditions for P polarization. My question is, how can I arrive to those conditions? The problem is 2D, an infinite dielectric cylinder coated with a layer of graphene, surrounded by another dielectric, and we analyze the cross section as a 2 dimensional problem. This would be for the theory section of the paper only.

Homework Equations

I understand that for this mode the conditions are $E_z = 0$ everywhere and $\frac{dH_z}{dr} \Big|_1 = \frac{dH_z}{dr}\Big|_2$ at the interface (this is condition 1). Also there's the Maxwell equation:
$$\nabla \times \textbf{H} = \frac{-i \omega}{c} \textbf{D}$$ (condition 2). Also the paper says the tangential component of E is continuous across the interface, and the tangential component of H has a "jump" across the interface that's proportional to the surface current of graphene.

The Attempt at a Solution

I started with the Maxwell equation, using cylindrical coordinates for the curl and also using that H is in the z direction only:

$$\frac{1}{r} \frac{dH_z}{d \phi} \hat{r} - \frac{dH_z}{dr} \hat{\phi} = \frac{-iw}{c} \epsilon \textbf{D}$$

now I calculate the difference between both media and evaluate at the interface, r=R. The dependence of the field on the variable $\phi$ is $e^{i n \phi}$.

$$\frac{i n}{R} (H_{z_1} - H_{z_2}) \hat{r} - \left ( \frac{dH_{z_1}}{dr} - \frac{dH_{z_2}}{dr} \right ) \hat{\phi} = \frac{-i \omega}{c} \left ( \epsilon_1 D_1 - \epsilon_2 D_2 \right )$$

Now, from condition 1 the $\hat{\phi}$ term on the left of the equation should be zero, which tells us the derivatives respect to r are the same for both media, but in the paper there's the epsilons dividing each derivative as well. Also since the tangential component of E is continuous then the right side of the equation is zero as well?

I don't seem to be arriving at the same conditions as they are stated on the paper, and I'm not sure what I'm doing wrong! Interface conditions are hard :(

 

[Nate810]

A:You have to consider the boundary conditions for the tangential $\mathbf E$ and $\mathbf H$ fields. The boundary conditions for the tangential $\mathbf E$ field are given by$$\mathbf E^{\parallel}_{1} = \mathbf E^{\parallel}_{2},$$where $\mathbf E^{\parallel}_{1}$ and $\mathbf E^{\parallel}_{2}$ denote the tangential components of the electric field in the two media at the interface, respectively. Similarly, the boundary conditions for the tangential $\mathbf H$ field are given by$$\mathbf H^{\parallel}_{1} = \mathbf H^{\parallel}_{2} + \frac{i \omega}{c} \sigma \mathbf E^{\parallel}_{\text{avg}},$$where $\mathbf H^{\parallel}_{1}$ and $\mathbf H^{\parallel}_{2}$ denote the tangential components of the magnetic field in the two media at the interface, respectively, and $\mathbf E^{\parallel}_{\text{avg}} = \frac{1}{2}\left(\mathbf E^{\parallel}_{1}+\mathbf E^{\parallel}_{2}\right)$. The last term on the RHS of the equation is due to the presence of a surface current on the graphene layer. Here, $\sigma$ denotes the surface conductivity of the graphene layer.