Low-dimensional approach to pair production in an oscillating electric   field: Application to bandgap graphene layers
  The production of particle-antiparticle pairs from the quantum field theoretic ground state in the presence of an external electric field is studied. Starting with the quantum kinetic Boltzmann-Vlasov equation in four-dimensional spacetime, we obtain the corresponding equations in lower dimensionalities by way of spatial compactification. Our outcomes in $2+1$-dimensions are applied to bandgap graphene layers, where the charge carriers have the particular property of behaving like light massive Dirac fermions. We calculate the single-particle distribution function for the case of an electric field oscillating in time and show that the creation of particle-hole pairs in this condensed matter system closely resembles electron-positron pair production by the Schwinger effect. 
Ibrahim Akal, Reinhold Egger, Carsten M\"uller, Selym   Villalba-Ch\'avez
hep-ph cond-mat.mes-hall