Second-order Dirac equation of graphene electrons in an electromagnetic field and their novel spin

The second-order Dirac equation (DE) and its velocity operator of graphene electrons in an electromagnetic field are obtained according to tight-binding k.p method. With extra terms included, they demonstrate the motion of graphene electrons more completely through a more complete Ehrenfest theorem and present finer properties of graphene electrons. Eigen-energy given by the second-order DE for field-free graphene indicates that extra terms may affect the trembling motion of graphene electrons. For graphene in a magnetic field, eigen-energy given by the second-order DE suggests that graphene electrons have a new kind of spin of a boson other than true electronic spin and pseudo-spin of Dirac particles, which will modify graphene properties such as the optical spectra.