Quantum corrections to drift-diffusion equations in graphene with smoothed energy-band

Quantum corrections to the semiclassical drift-diffusion equation are obtained for electrons in graphene with a regularized energy-band. The derivation starts from the single-particle, single-band Wigner equation and exploits the quantum maximum entropy principle together with the classical Chapman-Enskog method. The functional calculus in phase-phase space is then used to expand the model to second order in the scaled Planck’s constant. The model is shown to be singular in the limit where the regularization parameter goes to zero.