Transition from Nottingham-heating to Henderson-cooling during field emission

When (on average) an electron is field emitted from above the Fermi level there is a cooling effect, called Henderson-cooling; when (on average) it is emitted from below the Fermi level then Nottingham-heating occurs. This work compares our earlier used simple estimation to predict this emitted electron energy difference (average energy of emitted electron minus the Fermi level in the cathode) during field emission (FE) from a carbon nanotube [1,2] with a more numerically intensive formula that better captures the effects of Henderson-cooling and Nottingham-heating and the transition between the two regimes [3-5].