Massless Dirac equation and graphene

[Theo1]

I am reading about the electron flow in graphene and the article said this

"This behavior is not described by the traditional mathematics (Schrödinger equation) but by the mass-less Dirac equation"

What does this mean and what is the massless Dirac equation...

the whole paragraph is this...if it helps:
"Electrons flowing through the special structure of graphene (hexagons in a one atom thick layer) behave like electrons traveling in a vacuum close to the speed of light. This behavior is not described by the traditional mathematics (Schrödinger equation) but by the mass-less Dirac equation"

 

[Born2bwire]

My recollection is that what they mean by electrons behaving like mass-less is that they have a linear dispersion curve. The band diagram of graphene is a bit complicated but there are a set of six points where the bands are continuous. These points lie at the Fermi level for graphene and thus the propagation of electrons occurs at these points. It turns out that the dispersion curve for the electrons at these points is linear, not quadratic.

If we have a quadratic dispersion curve, we can treat the movement of the particle as if it was moving through free space but with an effective mass different from the normal mass. You may have heard of the electron having an effective mass when talking about semiconductors, same thing. But since the dispersion curve is not quadratic, the effective mass is zero. That doesn't mean that it moves about instantaneously but that its dispersion is like that of a photon.

However, all the treatments of graphene that I have read about (and I think the most recent was when I reviewed Supriyo Datta's text, "Quantum Transport: Atom to Transistor," (a good introductory book)) use the Schroedinger equation and you get this result. I don't think you need to use the Dirac equation and I can't recall a treatment that did. Perhaps they are referring specifically to the behavior of the electron as a massless particle (in the Schroedinger treatment the electron still has mass and we take into account tight bonding models and so forth).

 

[K^2]

There is a lengthy discussion in this thread. Just try to ignore Mr Vibrating.

 

[Born2bwire]

There is a lengthy discussion in this thread. Just try to ignore Mr Vibrating.

Oh boy... I missed that discussion.

EDIT: But yeah... If you (the OP) want a good introductory treatment then take a look at Datta's Quantum Transport text. He actually goes over the graphene and carbon nanotube examples with a very light treatment of quantum mechanics. But I do warn you that you need to go through maybe the first 5-7 chapters to understand the full treatment though the analysis of the graphene is a fairly short subject.

 

[Theo1]

Oh boy... I missed that discussion.

EDIT: But yeah... If you (the OP) want a good introductory treatment then take a look at Datta's Quantum Transport text. He actually goes over the graphene and carbon nanotube examples with a very light treatment of quantum mechanics. But I do warn you that you need to go through maybe the first 5-7 chapters to understand the full treatment though the analysis of the graphene is a fairly short subject.

do u know where i could get that?
thx

 

[Born2bwire]

do u know where i could get that?
thx

Outside of borrowing it from a university library or purchasing it from, say, Amazon, no. It's only a few years old so I do not think that Datta has released an e-copy.

 

[ZapperZ]

This colloquium gives an intro to transport properties of graphene.

http://arxiv.org/abs/1007.2849

Zz.