Modeling of nanoscale devices with carriers obeying a three-dimensional density of states

While aggressively nanoscale field-effect transistors commonly used in CMOS technology exhibit strong quantum confinement of charge carriers in one or two dimensions, few devices have been recently proposed whose operation reminds that of vacuum tube triodes and bipolar transistors, since charge carriers are ballistically injected into a three-dimensional k-space. In this work we derive, under the parabolic band approximation, the analytical expressions of the first three directed ballistic moments of the Boltzmann transport equation (current density, carrier density, and average kinetic energy), suitable to describe ballistic and quasi-ballistic transport in such devices. The proposed equations are applied, as an example, to describe the ballistic transport in graphene-based variable-barrier transistors.