Graphene is a two-dimensional material formed by carbon atoms in a honeycomb lattice. Because of the symmetry of the honeycomb lattice, the electrons in graphene obey a linear dispersion relation $E\propto|\vec{p}|$ (as in Dirac case) rather than a quadratic one $E\propto|\vec{p}|^2$ (as in Schroedinger case). Nevertheless, the speed of light c appearing in linear dispersion relation (found for Dirac case) is replaced by the Fermi velocity $\vec{u_F}$ of electrons. In explicit terms, dispersion relation of electrons on graphene is given by $E =\vec{u_F} |\vec{p}| \times20$ where $ \vec{p} = p\hat{x}x + p\hat{y }y$. (A proportionality constant is absorbed in the Fermi speed υF .)

I was trying to construct this Hamiltonian in xy-plane and i don know how i can do for honeycomb lattice .


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