# Why electrons are relativistic in Graphene and non relativistic in vacuum?

If a free region in space has a potential difference of one volt, an electron in this region will acquire kinetic energy of 1 eV. Its speed will be much smaller than the speed of light hence it will be a non relativistic electron.

On the other hand conduction electrons in graphene are relativistic for the same potential difference.

Question is how come that when the electrons are in vacuum they are non relativistic, and when they are inside Graphene they are relativistic (for the same potential difference)?

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I think you assume from the first sentence $v\approx \sqrt{\frac{2 U_{eV}}{m_e}}<<c_0$ But your second sentence is a little bit misleading "conduction electrons in graphene are relativistic for the same potential difference". What do you mean? Electron mobility? Drift velocity? They are nonrelativistic. Or are you asking for the electron movement in a molecule? Well for Carbon the Schrödinger equation is a good approximation, you do not need the Dirac equation. You have to consider relativity for s electrons for heavy elements with high charge with e.g. ZORA - zeroth order relativistic approxi – Alex1167623 Feb 26 '12 at 0:10
Please do not answer questions if you are not familiar with the field. A quick google immediately brings up a wealth of information about the statement. In this case, there is no connection to the actual speed of light and the statement is a purely formal one regarding the equation of motion for quasiparticles. – genneth Feb 26 '12 at 2:32
@genneth You are right that this should not be an answer, but rather a comment on the given question to improve it. Pushed the wrong button :-(. But I am dissapointed too that a proper answer was not given by you, nor a reference. Hans de Vries finally clearified it, thnx. – Alex1167623 Feb 29 '12 at 9:33

As far as I understand, electrons in graphene are not relativistic, although quasiparticles in graphene are indeed described by the massless Dirac equation. However, for graphene, the speed velocity in this equation is replaced by the Fermi velocity, which is much smaller.

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@Revo: In my book, a particle is relativistic if its velocity is comparable to the velocity of light. If you use a different definition, please give me a reference to a reliable source (if you just alluded in your comment that the eigenvalues of velocity projections for a Dirac particle are always +-c due to Zitterbewegung, this does not seem to be relevant to your question). The velocity of the quasiparticles in graphene is always comparable to the "velocity of light" in the massless Dirac equation for graphene, but that "velocity of light" is not the genuine velocity of light. – akhmeteli Feb 9 '12 at 22:14
@Revo: No. I believe a particle is relativistic when its velocity is comparable to the velocity of light in vacuum. In most cases the velocity of speed in media is comparable to that in vacuum, so the clarification about vacuum is usually omitted. I agree that some exotic media may present exceptions. That does not mean that the particle that you describe must be described by a relativistic equation. It just so happens that quasiparticles in graphene can be described satisfactorily (to some extent) by an equation looking exactly like the massless Dirac equation with lesser "velocity of light" – akhmeteli Feb 10 '12 at 19:35
@Revo: I have two problems with your reasoning. While I agree that the standard Dirac equation is a relativistic equation and that it correctly describes a relativistic spin one half particle, that does not mean that if a particle is correctly described by the Dirac equation, it is necessarily relativistic, because the Dirac equation correctly describes slow particles as well. The above reasoning is correct, however, for the standard MASSLESS Dirac equation, as such an equation does not describe correctly slow particles. The other problem is described in another comment. – akhmeteli Feb 11 '12 at 1:19
@Revo: The other problem is as follows. The massless Dirac equation used for quasiparticles in graphene is not the standard massless Dirac particles for the following reasons. While it looks exactly like the standard massless Dirac equation, the speed constant in this equation is much less than the velocity of speed in vacuum, so it only describes particles that are slow compared to the velocity of light in vacuum. Furthermore, the equation is not relativistic in the sense that it is not invariant under Lorentz transforms, it is only correct in the frame of reference of the graphene lattice. – akhmeteli Feb 11 '12 at 1:30
@Revo: you are mistaken about the link between the Dirac equation and relativity. The Dirac equation correctly describes a single particle relativistically, but does not have to. One can use it to do other things. The statement "electrons in graphene are relativistic" is a purely formal statement about the lack of a rest mass for the quasiparticles. – genneth Feb 26 '12 at 2:30

The statement that in graphene the "conduction electrons are massless" is because the energy levels (bands) are proportional to their momenta.

So the $E = \sqrt{p^2+m^2}$ relation of a free electron becomes $E\propto p$ in graphene.

Massless particles travel all at the same speed because of the $E\propto p$ relation but this characteristic velocity in graphene is far below c though, only 0.3% of the speed of light.

The reason that the relation $E\propto p$ leads to a characteristic speed is due to the quantum mechanical wave character. $E$ is proportional to the phase changes in time, $p$ is proportional to the phase changes in space and therefor $p/E$ is proportional to the velocity. In the case that $E\propto p$ there is a characteristic velocity $v$ independent of the energy level.

The most striking aspect of graphene is that its electronic energy levels, or “bands,” produce conduction electrons whose energies are directly proportional to their momentum. This is the energy-momentum relationship exhibited by photons, which are massless particles of light. Electrons and other particles of matter normally have energies that depend on the square of their momentum.

When the bands are plotted in three dimensions, the photonlike energy-momentum relationship appears as an inverted cone, called a Dirac cone. This unusual relationship causes conduction electrons to behave as though they were massless, like photons, so that all of them travel at roughly the same speed (about 0.3 percent of the speed of light). This uniformity leads to a conductivity greater than copper.

Hans

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