Current density and conductivity through modified gravity in the   graphene with defects
  We propose a model describing the evolution of the free electron current density in graphene.   Based on the concept of Mp-branes, we perform the analysis using the difference between curvatures of parallel and antiparallel spins. In such a framework an effective graviton emerges in the form of gauge field exchange between electrons. In a plain graphene system, the curvatures produced by both kinds of spins neutralize each other. However, in the presence of defects, the inequality between curvatures leads to the emergence of current density, modified gravity and conductivity. Depending on the type of the defects, the resulting current density can be negative or positive. 
Alireza Sepehri, Richard Pincak, Kazuharu Bamba, Salvatore   Capozziello, Emmanuel N. Saridakis
gr-qc cond-mat.mes-hall hep-th