Consider a graphene MOSFET, and calculate its current (Id vs Vgs and Vds) in a manner that follows our ECE212B prescription. You could calculate the ballisitic current for a FET given graphene channel and a gate oxide of 2 nm thickness, with SiO2.
There are some expressions that you could use (p. 38 in lecture notes 9). You should get results that are similar to the curves on p. 46, 47, 48. It will take some work to get the integrals correct, and to take into account the fermi levels at the source and drain, and then in addition to the electron current consider the hole current (for this, the drain and source will be interchanged). It will be an interesting piece of work. If you complete this, you could examine the results for different oxide thicknesses, as an example, and compare with experimental results. You could speculate about what happens if there is scattering in the channel, but this is beyond what would be needed.
Lecture Notes -
https://www.dropbox.com/s/ef2z2waxyf3yb9y/Lecture%20Notes.rar?dl=0