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

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