A quantum mechanical model of electron field emission from two dimensional materials. Application to graphene

We implement a new time-independent perturbative quantum method to study quantitatively electron field emission from two dimensional materials and, in particular, from graphene. The Bardeen transfer Hamiltonian formalism is coupled to a detailed description of the electronic structure of the material. This calculation method is first validated on the standard Fowler-Nordheim (FN) model of a three-dimensional (3D) free-electron gas. Then, it is used to study emission from a two-dimensional (2D) free-electron gas and from graphene represented by a tight-binding model. In the case of graphene, we show that a full electronic band model of the material is necessary to obtain reasonable results because emission is not restricted to the vicinity of the Fermi level near the Dirac points. The graphene emitted current density follows a modified FN law with respect to the applied field, with a prefactor exponent for the field n approximate to 1.5 intermediate between the one for the cases of 2D ( n = 0) and 3D ( n = 2) free-electron gases. However, the emitted current level is low because the kinetic energy of the electrons corresponds to a motion parallel to the emitting surface, which is not efficient in promoting emission. Our study gives a firm ground to the idea that emission from graphene results almost exclusively from defects.