  Experimental Fowler-Nordheim plots taken from orthodoxly behaving carbon
nanotube (CNT) field electron emitters are known to be linear. This shows that,
for such emitters, there exists a characteristic field enhancement factor (FEF)
that is constant for a range of applied voltages and applied macroscopic fields
$F_\text{M}$. A constant FEF of this kind can be evaluated for classical CNT
emitter models by finite-element and other methods, but (apparently contrary to
experiment) several past quantum-mechanical (QM) CNT calculations find
FEF-values that vary with $F_\text{M}$. A common feature of most such
calculations is that they focus only on deriving the CNT real-charge
distributions. Here we report on calculations that use density functional
theory (DFT) to derive real-charge distributions, and then use these to
generate the related induced-charge distributions and related fields and FEFs.
We have analysed three carbon nanostructures involving CNT-like nanoprotrusions
of various lengths, and have also simulated geometrically equivalent classical
emitter models, using finite-element methods. We find that when the
DFT-generated local induced FEFs (LIFEFs) are used, the resulting values are
effectively independent of macroscopic field, and behave in the same
qualitative manner as the classical FEF-values. Further, there is fair to good
quantitative agreement between a characteristic FEF determined classically and
the equivalent characteristic LIFEF generated via DFT approaches. Although many
issues of detail remain to be explored, this appears to be a significant step
forwards in linking classical and QM theories of CNT electrostatics. It also
shows clearly that, for ideal CNTs, the known experimental constancy of the FEF
value for a range of macroscopic fields can also be found in appropriately
developed QM theory.
