Understanding DFT & Graphene: Schrodinger vs Dirac

[knghrts17]

Hello all,


I have read a few papers lately that have used DFT based techniques to investigate metallic adatom adsorption on top of graphene (see for instance PRB 77, 235430 2008). I was under the impression that electrons in graphene are described by the Dirac equation and not the Schrödinger equation. How then can the electronic properties be accurately described using DFT techniques based in the Shcrodinger equation.


Thanks in advance

-E

 

[alxm]

I have read a few papers lately that have used DFT based techniques to investigate metallic adatom adsorption on top of graphene (see for instance PRB 77, 235430 2008). I was under the impression that electrons in graphene are described by the Dirac equation and not the Schrödinger equation. How then can the electronic properties be accurately described using DFT techniques based in the Shcrodinger equation.

Well, there are relativistic DFT models. But in general, no, it's not necessary to use the Dirac equation. The relativistic effects are quite small in a light element such as carbon. Smaller than the error of any DFT method.

The main technical issue with DFT (which I mentioned in a post just yesterday), as pertains to adsorption to graphene, is that most DFT functionals do not accurately reproduce van der Waals forces.There's a lot of http://arxiv.org/abs/cond-mat/0306033" in this area, often using graphite/graphene/fullerene structures as testbeds.

 

[f95toli]

The relativistic effects are quite small in a light element such as carbon. Smaller than the error of any DFT method.

I don't think knghrts17 was referring to "real" relativistic effects. Graphene is very strange material and one reason for that is that the effective mass is (ideally) zero in some directions (the gap is essentially "V" shaped, not parabolic); graphene is very different from other forms of carbon.
Some people have suggested that one should therefore use the Dirac equation instead of the Schroedinger equation in order to calculate some of the properties of graphene.
There are even suggestions that it might eventually be possible to repeat some of the experiments normally done in particle accelerators in graphene.

There is still a lot of hype surrounding graphene so it is not surprising that there are all sorts of more or less realistic ideas floating around.

 

[alxm]

Some people have suggested that one should therefore use the Dirac equation instead of the Schroedinger equation in order to calculate some of the properties of graphene.

Well that's reasonable of course. And I'm a bit unsure of the value of DFT methods overall when it comes to electronic structure (what with the 'physicality' of Kohn-Sham orbitals being a subject of constant debate).

So, to be more specific, calculating 'molecular' (as opposed to Solid State/electronic structure) properties, such as bond strengths, forces, energies and the like, the relativistic effects on those properties will be small. I'd wager ~1 kJ/mol. In the case of adsorption, which is essentially all London forces, the largest error by far will come from how accurately the correlation functional reproduces vdW effects.

 

[knghrts17]

Thank you all for all your input. I have thought about this some more and would like to throw out a question concerning my previous post. For a long time it was know through tight-binding calculations that a monolayer hexagonal lattice possessed a linear dispersion relation with a velocity that was independent of of both energy and momentum. To the best of my knowledge, there is nothing relativistic about the traditional tight-binding approach though I'm sure it can be reformulated in such a manner. Could it therefore be that the Schrödinger equation, while able to predict the band structure and behavior about the Dirac points cannot explain the other ancillary phenomena that arise due to the linear dispersion relation (i.e anomalous QHE, Klein paradox, etc)


Thanks in advance

-E

 

[genneth]

The so-called "relativistic" aspects of graphene are not a manifestation of anything deeper than a linear dispersion relation around the points of the Fermi surface. Graphene is well-described by simply solving the Schrödinger equation (or the equivalent 2nd quantised form).

 

[rodrigooeiras]

The relativistic effects are important to explain a QHE. A magnetic field is applied in graphene structure and using tight-biding approach is possible to calculate the resistance rho_{xy} which related with the landau levels. A good reference to understand the methodology is a paper of Castro Neto. Search for this paper in web of science site.