This example demonstrates the effectiveness of using FDTD to simulate graphene devices by using a simple test case where an analytic solution is available. The sample structure is a layer of graphene deposited on a glass substrate. Graphene is modeled as a true 2D sheet object in FDTD. This configuration allows for efficient simulation since it does require an extremely small mesh.
Overview
Understand the simulation workflow and key results
Step 1: MODE analysis
Get the mode profile and effective index of the P-polarized (TM) mode. It is then possible to estimate the optimal excitation angle for step 2.
Step 2: Get the reflection and transmission from FDTD
Measure the reflection and transmission versus incidence angle. Angles around 30 deg will excite the plasmon mode on the graphene surface. The simulation setup uses an Otto configuration, similar to how this measurement could be done experimentally.
Step 3: stackrt-analytical result
Compare with an analytical solution calculated with stackrt.
Run and results
Instructions for running the model and discussion of key results
Step 1: Calculation of the mode profile and effective index
- Open and run the MODE file.
Mode profile
Ex, Ey and Hz mode profile
Since the graphene layer is perpendicular to the x-axis, as expected, the Ex field component exhibits a discontinuity.
Estimate excitation angle to excite mode
The effective index of the surface plasmon mode is
$$ n_{eff} = 2.56 + i0.12 $$
The optimal coupling angle for a planewave can be estimated by
$$\theta_{SP} = \mathrm{arcsin}\left(\mathrm{Re}(n_{eff})/n_{prism} \right) \approx 30.8^{\circ}$$
where \(n_{prism}\) of 5 is the refractive index of the prism, see "
Important model settings" below.
Step 2: Simulate transmission and reflection
- Open the FDTD file and run the script to sweep the incident angle from 5 degree to 45 degree. The script will plot the reflection and transmission values.
Reflection and transmission
The following figure shows the simulated reflection (blue) and transmission (red) as a function of incident angle:
It is well known that for all source angles that satisfy
$$\theta \gt\theta_{TIR}$$
the fields in the air gap are evanescent, and therefore the fields should be reflected. However, observe that a dip in the reflected power occurs for the source angle given by θsp, as opposed to the total internal reflection. Such reflection drop is due to the fact that the evanescent field excites the surface plasmon of the graphene sheet.
Field profile
It is also interesting to plot the electric field intensity along the x direction as a function of source angle. The surface plasmon mode is visible around θsp.
Step 3: Get the analytical solution from stackrt
- This is automatically run as part of the previous script.
Analytic solution
The analytical reflection (red) and transmission (magenta) are also shown in the previous figure. The FDTD results show very good agreement with the analytic solution.
Important model settings
Description of important objects and settings used in this model
FDTD experimental setup
The FDTD simulation is setup in an Otto configuration, to mimic the way this measurement might be obtained experimentally. The source is in a high index material (the prism), separated from the graphene by a small air gap. At angles greater than ~23 deg, we expect total internal reflection. Coupling to the surface plasmon mode at around 32 deg results is visible as a drop in the reflected power.
Graphene material
The default Graphene surface conductivity material model is recommended for most graphene simulations. This 2D material model can only be used with a 2D structure object. This approach allows the use of much larger mesh sizes than the physical thickness of graphene.
Thickness for stackrt calculation
strackrt only supports bulk refractive index materials but not the 2D surface conductivity models used in the FDTD simulation. The method described in the second half of Graphene surface conductivity was used to calculate an equivalent bulk refractive index that could be used with stackrt.
Accuracy settings
In this specific example, a mesh accuracy of 4 is used to provide obtain better agreement with theory. When starting any new simulation, lower accuracy settings (e.g. coarse mesh) is strongly advised to keep the simulation time low. Only use high accuracy settings for convergent testing and final results. The number of Standard PML layers was increased to 24 to get better results. Similarly, we recommend using fewer points in your initial parameter sweeps to get your initial results more quickly.
Additional resources
Additional documentation, examples and training material
Related publications
- D.R. Mason, S.G. Menabde, and N. Park, "Unusual Otto excitation dynamics and enhanced coupling of light to TE plasmons in graphene," Opt. Express Vol. 22 (1), 847-858 (2014).
- F. Ramos-Mendieta, J.A. Hernández-López and M. Palomino-Ovando, "Transverse magnetic surface plasmons and complete absorption supported by doped graphene in Otto configuration," AIP Advances Vol. 4, 067125 (2014).
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