Blog entry

What the heck is a Dirac electron?

By Noel A. García ( INL)

We hear everywhere that electrons in graphene (and some other materials) are "Dirac electrons"... but what is a Dirac electron?

It is an electron that follows the Dirac equation. Easy, right?
This statement is true, but probably not very useful, like the cartoon...

To introduce the Dirac equation, we need to talk about a couple of things first, namely quantum mechanics and special relativity.
Quantum Mechanics is a set of laws that describes the behaviour of really tiny things, tiny like atoms, or a bunch of atoms. A complex molecule such a protein is already too big to see any quantum effect.
Special Relativity is a set of laws that describes the behaviour of things moving at velocities close to that of the light.

When you try to merge these two theories you find the problem (one of many) that Schroedinger equation, the one that describes quantum particles, treats time and space asymmetrically, like they were different things, but the main point of special relativity is precisely that there is not such a difference.

Dirac's equation was the solution to this problem. It describes the behaviour of electrons while respecting the symmetry between space and time that is observed in nature.
Or to put it in other way, Dirac's equation describes quantum particles in the relativistic regime.

According to this equation the energy of a particle has two terms, and one is dominant over the other depending on the limit you are working at.

The two limits I will refer here are the limit when the particle has no mass (and hence, it moves at the speed of light), and when the particle moves very very slowly (compared to c, the speed of light). In physics it is useful to define a quantity called "momentum", p that is just the mass times the velocity, so the less momentum a particle has, the slower it is moving.

These two limits are shown in Fig 1, it can be seen that for particles with small momentum the relation between Energy and momentum looks like a parabola, but for particles without mass it is a linear relation.

I can hear you complaining, particles without mass??!! Well, yes, photons for instance, the particles that make up light.

So we have just seen that there are two kind of particles those whose energy is quadratic (parabolic) with the momentum (these are particles with mass and small velocity), and those whose energy is linear with the momentum (massless particles that move at the speed of light).

Ok, linear, parabolic, special relativity, so what? what does this to do with condensed matter?
Well when you consider electrons in a material you expect them to have mass (as they do in vacuum) and hence you would expect somehow a parabolic relation between energy and momentum (when talking about electrons in a crystal this energy-momentum relation is called "bands"), of course the inside of a material will be very difficult to describe, with lots of interactions and lots of particles interacting with each other, but still the parabolic behaviour should appear somehow.

And for long time this was true. Semiconductors, insulators even metals, they all present parabolic-like bands, especially close to the Fermi energy (the energy  of the most energetic electron), for instance in insulators we would find maxima and minima close to this energy, and these are always parabolic.
And then graphene appeared (a.k.a was successfully synthesized with great effort, and restless years of hard work), and linear bands appeared at the Fermi energy (In the bands showed in Fig 2 this Fermi energy is at 0).

Wait, if electrons in graphene have a linear dispersion, does that mean that they do not have mass and/or that they are moving at the speed of light?

NO!! It means that they follow the exact same equation as the massless particles that travel that fast, so you would expect the same qualitative behaviour, but they are not moving at the speed of light, and they, for sure, are not massless.

Then... what is so impressive about them if they are just regular electrons only with a eccentric energy dispersion?
Well, even when they are regular electrons, they do not behave like that, they act like photons (they satisfy the same equations), so for instance they all move at the same velocity, and cannot stop, just like light...

After all this crazy stuff, what you need to keep in mind, is that Dirac's electrons, electrons in graphene, are just electrons... that behave like light!!! How awesome is that?!