Why Use the Dirac Equation for Graphene Instead of the Schrödinger Equation?

[jamie.j1989]

Hi, I'm doing a project on graphene and don't really understand why we use the Dirac equation instead of the Schrödinger equation. The fermi velocity of electrons in graphene is not relativistic, I know the particles are considered as quasiparticles but don't see how this changes things. My only reasoning is that the effective mass is zero and the 1/m dependence in the Schrödinger equation is incompatible with this? Thanks.

Scrodinger equation

$$i\hbar\frac{\partial}{\partial{t}}\psi({\textbf{r},t})=\left[-\frac{\hbar^2}{2m}\nabla^2V(\textbf{r})+E\right]\psi({\textbf{r},t})$$

Dirac equation

$$\left[\gamma^{\mu}\partial_{\mu}+\frac{c}{\hbar}m\right]\psi(\textbf{r},t)=0$$

 

[Brage]

Hi jamie.j1989, I am not sure what aspects of graphene you are investigating, but for example the conductance of graphene is very good, which translates to very fast moving electrons (definitely within the range where we see relativistic effects). Also the fermi velocity is only relevant at 0K, so for temperatures above that the velocities of the fastest electrons will exeed the fermi velocity.

 

[atyy]

Hi, I'm doing a project on graphene and don't really understand why we use the Dirac equation instead of the Schrödinger equation. The fermi velocity of electrons in graphene is not relativistic, I know the particles are considered as quasiparticles but don't see how this changes things.

The fundamental equation for graphene is the Schroedinger equation. However, an equation with the form of the Dirac equation "emerges" from the Schroedinger equation as an excellent approximation at low energy. The "speed of light" in the equation with the form of the Dirac equation is not the speed of light. It is ~10^6 m/s.
http://arxiv.org/abs/cond-mat/0509330
http://www.physics.upenn.edu/~kane/pedagogical/295lec3.pdf