Angle-Dependent van Hove Singularities in a Slightly Twisted Graphene Bilayer

Abstract

Recent studies show that two low-energy van Hove singularities (VHSs) seen as two pronounced peaks in the density of states could be induced in a twisted graphene bilayer. Here, we report angle-dependent VHSs of a slightly twisted graphene bilayer studied by scanning tunneling microscopy and spectroscopy. We show that energy difference of the two VHSs follows $\Delta E_{\mathrm{vhs}}\sim \hslash {\nu }_F\Delta K$ between 1.0° and 3.0° [here ${\nu }_F\sim 1.1\times {10}^6\mathrm{m}/\mathrm{s}$ is the Fermi velocity of monolayer graphene, and $\Delta K=2K\sin (\theta /2)$ is the shift between the corresponding Dirac points of the twisted graphene bilayer]. This result indicates that the rotation angle between graphene sheets does not result in a significant reduction of the Fermi velocity, which quite differs from that predicted by band structure calculations. However, around a twisted angle $\theta \sim 1.3°$, the observed $\Delta E_{\mathrm{vhs}}\sim 0.11\mathrm{eV}$ is much smaller than the expected value $\hslash {\nu }_F\Delta K\sim 0.28\mathrm{eV}$ at 1.3°. The origin of the reduction of $\Delta E_{\mathrm{vhs}}$ at 1.3° is discussed.