Invariant expansion for the trigonal band structure of graphene

We present a symmetry analysis of the trigonal band structure in graphene, elucidating the transformational properties of the underlying basis functions and the crucial role of time-reversal invariance. Group theory is used to derive an invariant expansion of the Hamiltonian for electron states near the $\mathbf{K}$ points of the graphene Brillouin zone. Besides yielding the characteristic $k$-linear dispersion and higher oder corrections to it, this approach enables the systematic incorporation of all terms arising from external electric and magnetic fields, strain, and spin-orbit coupling up to any desired order. Several new contributions are found, in addition to reproducing results obtained previously within tight-binding calculations. Physical ramifications of these new terms are discussed.

Figure 1

(Color online) (a) Honeycomb structure of graphene. Atoms in sublattice $A$ $\left(B\right)$ are marked with open (closed) circles. (b) Brillouin zone and its two inequivalent corner points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$. The remaining corners are related with $\mathbf{K}$ or ${\mathbf{K}}^{\prime }$ by reciprocal lattice vectors. (c) Dispersion $E\left(k\right)$ near the $\mathbf{K}$ point. We have $\mathbf{k}\equiv \mathbf{\kappa }-\mathbf{K}$.