# Graphene 2D Interband term

Hello,

I am currently working with graphene 2D approach, and I am wondering how is computed the interband term. From the kubo formula, it is possible to solve the intraband term analytically, that is fine. However, I don’t find any details on the interband term. It comes from the second part of the kubo formula, where the integral is non solvable analytically. Could you give me clue on the way lumerical compute this interband conductivity term ?

Thank you.

Regards,

David

Hi David,

You are right. The interband term cannot be approximated analytically except in some special cases, so we just calculate the integral numerically. The integral is shown here. There is nothing special about this integral; you can calculate it using quadrature, for example with the quad function in Matlab.

Thank you very much.
I did my own calculation, and realize that 3D graphene permittivity, based on intraband conductivity, is not suitable for Fermi energy lower than 0.3eV.
Just a last question, the 2D graphene material, based on the full optical conductivity of graphene (intra+inter bands terms) has no particular condition of use (Fermi energy, spectral range) ? I mean, apart that it has to be flat surface.

Cheers,

David

You are welcome. The surface conductivity model implemented in Lumerical’s optical solvers is based on Kubo’s formula. There are some assumptions:

• The photon energy is small enough (usually less than 3eV) so that carriers excited optically remain close to the K point. In that regime only nearest pi bands are sufficient for the band structure calculation and the bands are described by the Dirac cone.
• The scattering rate is assumed to be the same for inter- and intraband terms. Furthermore, it is independent of energy.
• The sheet is flat and in principle extends to infinity (this can be relaxed if the size of the sheet is large compared to the period of the underlying atomic lattice).

For more details please take a look at these references (and those cited therein):

Hanson, G. W. (2008): https://arxiv.org/abs/cond-mat/0701205
Falkovsky, L. A., et al. (2006): https://arxiv.org/abs/cond-mat/0606800

Great, thank you very much. Really helpful.
I have another question about the 2D material algorithm. I couldn’t find any reference talking about it on the website.
Could you please update me ?
I did some bibliography and I found at least two different approach :
But also this one :
http://ieeexplore.ieee.org/document/7093590/?tp=&arnumber=7093590
The different papers seem to use different algorithm to compute the 2D graphene.

Best regards,

David

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The actual algorithm for the 2D material implementation is something we cannot discuss here because it is proprietary of Lumerical. As you can see in the application examples here, we have validated the algorithm by comparing it with analytical and published results. Are you concerned about the performance of the model in some specific scenario?

Hi,

I got good results, according to the theory. I just want to justify and cite the key algorithm used for computing 2D graphene in FDTD (maybe link the optical conductivity with one of the electro-magnetic field). Are there any possibility to get that ?

Sorry I didn’t have a chance to reply before. Our implementation of graphene is basically a FDTD implementation of the theory described in Hanson’s paper: G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity,” J. Appl. Phys. Vol. 103, 064302 (2008).