Graphene wavefunction expressed in tight binding form

[PRB147]

TL;DR Summary

Does the wavefunction for A or B atom have the spinorial components (2 components)?

In the framework of tight binding approximation, does the wavefunction for atom A (or B) has two spinorial components(2 components) in "real space"? If so how does this spinorial component propagate in the graphene?

 

[TeethWhitener]

I’m not quite sure what you mean by the wavefunction for an individual atom in graphene. The wavefunction describes the behavior of the whole system, and in a condensed matter system, the real space wavefunction for a given wave vector $\mathbf{k}$ is delocalized over lots of atoms.

Do you maybe mean Wannier functions? These are usually more localized than the eigenfunctions of the Hamiltonian for a condensed matter system.

 

[PRB147]

I’m not quite sure what you mean by the wavefunction for an individual atom in graphene. The wavefunction describes the behavior of the whole system, and in a condensed matter system, the real space wavefunction for a given wave vector $\mathbf{k}$ is delocalized over lots of atoms.

Do you maybe mean Wannier functions? These are usually more localized than the eigenfunctions of the Hamiltonian for a condensed matter system.

Thank you very much for your help, I mean is the tight binding wavefunction (lattice coefficient) at atom A (or B) two-component spinor? If so, how the spinorial lower component propogate? If not, how tight binding wavefunction match the continuum model? In continuum model, the wavefunction evolves between atom A and atom B in two-component spinor form.

 

[Dr Transport]

https://en.wikipedia.org/wiki/User:Jatosado

 

[TeethWhitener]

Ok just to clarify, what the literature calls the continuum model of graphene is the tight binding model, but where you expand the Hamiltonian around the Dirac cones. So the tight binding Hamiltonian looks like
$$H = \begin{pmatrix}
\epsilon & -tf(\mathbf{k}) \\
-tf^*(\mathbf{k}) & \epsilon \\
\end{pmatrix}$$
where
$$f(\mathbf{k}) = \sum_{\mathbf{\delta}}\exp{i\mathbf{k}\cdot\mathbf{\delta}}$$
and the $\mathbf{\delta}$ are the nearest neighbor vectors in real space.
Expanding to first order about the Dirac cone at $\mathbf{k}=\mathbf{K}$ gives:
$$H_{\xi} = \hbar v_F \begin{pmatrix}
0 & \xi k_x-ik_y \\
\xi k_x+ik_y & 0 \\
\end{pmatrix}$$
where we've picked up the valley index $\xi$ which is $+1$ near $\mathbf{K}$ and $-1$ near $\mathbf{K'}$.
The continuum model allows us to write down the wavefunction explicitly:
$$\psi_{\pm,\xi}=\frac{1}{\sqrt{2}}\begin{pmatrix}
1 \\
\pm\xi e^{i\xi\theta_{\mathbf{k}}}\\
\end{pmatrix} e^{i\mathbf{k}\cdot\mathbf{r}}$$
The spinor nature of this wavefunction doesn't change when you switch to the full tight binding model. (When you say the A/B atoms, I think you're referring to the sublattices of graphene with which each component of the spinor is associated.) But the components of the spinor become functions of $f(\mathbf{k})$, and I'm not sure if they can be written in closed form.

Did that get a little closer to what you were talking about?

 

[TeethWhitener]

This paper (possibly paywalled, but couldn't find the arxiv version) has some info which might help you:
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.195450

 

[PRB147]

This paper (possibly paywalled, but couldn't find the arxiv version) has some info which might help you:
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.195450

Thank you very much , with my best wishes.

 

[PRB147]

https://en.wikipedia.org/wiki/User:Jatosado
[/QUOTEThank you! Best regards