  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.
