Friday, June 1, 2012

Dirac points

During my presentation, the Dirac point was mentioned.
This also popped up in Andy's paper.


From an article:  Pairs of bands are degenerate at a point of high symmetry of the Brillouin zone.
Electromagnetic one-way edge modes were first predicted in 2008, 2006, to occur in systems possessing Dirac points.
Here, they take a Dirac point to be points s.t. modes near each degeneracy point can be described by Dirac Hamiltonian. (We may recall Dirac Hamiltonian from Quantum, similar to Schrodinger, but with spin with

We may interpret the Dirac points to be points of high symmetry of the Brillouin zone, which is where electrons can be described by the Dirac equation. This is consistent with description given in another article of where two energy bands intersect linearly and the electrons behave as relativistic Dirac fermions.
The points of high symmetry would be the centre and corners of the Brilouin zone (wikipedia), verified in graphene since this is where the kx and ky must be equal (in magnitude), with the periodic nature of Bloch model.


This was related to my presentation with the plot Dave pointed out on Page 9, this figure, with the band structure of of the Bi2Se3.
 
For Figure a, we see our Dirac point is at the centre of the Brillouin zone. We see our surface band meets our bulk band at the one point at the Dirac point, which is a condition for being a topological insulator.
We have the yellow for a high electron density of states. Then there is a higher density of states at the bulk band, which we can infer as most electrons are in the bulk state, i.e. insulating. Thus there are fewer states in the surface state, i.e. conducting
This the conduction is small compared to the insulation, as we would expect since conduction is only at the edges.
Sidenote: Figure b would be the dispersion relation for the material. Resembles the cone of graphene.
So our Fermi surface would be circles, of diameter proportional to the Fermi energy/wavenumber.

4 comments:

  1. Hmmm, interesting...

    I think I understand why the dirac point makes it a topographical insulator, but I don't understand your last point. Why is the fermi surface made up of circles?

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  2. Wen we construct a Fermi surface, don't we first assume our Fermi surface is a circle (for 2D), from free electron?
    Then deform the shape with Bragg plane terms etc?

    If there's nothing to deform it, then it remains a circle (sphere in 3D)?

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  3. Yes, that's true. Thanks Ann. Also, I just realised that I wasn't seeing the whole figure :P

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  4. Wait, so was it first predicted in 2006 or 2008? ;)

    Ahh, Dirac fermions, that makes more sense. I was wondering where the name came from, and was thinking it had something to do with the dirac delta function, as it looked like it might have been a singularity or something.

    Actually, it was interesting in the 'another article' article, where it mentioned that two dirac points when merged annihilate each other.... which goes against what i would have expected, and what this article seems to state http://prb.aps.org/pdf/PRB/v80/i15/e153412

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