Berry phase and the role of trigonal warping in graphene systems

In general Dirac fermions exhibit a Berry phase of $\pi$ whereas non-relativistic free-like particles have a Berry phase of $2\pi $. The two cases have been reported, both theoretically and experimentally, in single-layer and bi-layer graphene respectively. On the other hand, if one considers, for example, bi-layer graphene in more detail, its band structure shows both the Dirac-type, coming from the trigonal warping, and the parabolic, non-relativistic aspect in different energy regimes. This gives an unprecedented opportunity to investigate the \textbf{crossover between non- relativistic and Dirac fermions in a Berry phase formulation}. Here, we propose a scattering type-experiment (reflecting the Berry phase) to demonstrate this crossover. We present our theoretical results on scattering cross-sections taking into account the trigonal warping term, which confirm the jump in the Berry phase from $\pi$ to $2\pi$ as the energy of the incoming electrons is increased. This jump in the Berry phase can be understood from a general theorem relating the Berry phase and the band structure of a material.