Scaling in large area field emitters and the emission dimension

Electrostatic shielding is an important consideration for large area field emitters (LAFE) and results in a distribution of field enhancement factors even when the constituent emitters are identical. Ideally, the mean and variance together with the nature of the distribution should characterize a LAFE. In practice however, it is generally characterized by an effective field enhancement factor obtained from a linear fit to a Fowler-Nordheim plot of the $\text{I V}$ data. An alternate characterization is proposed here based on the observation that for a dense packing of emitters, shielding is large and LAFE emission occurs largely from the periphery, while well separated emitter tips show a more uniform or 2-dimensional emission. This observation naturally leads to the question of the existence of an emission-dimension, $D_e$ for characterizing LAFEs. We show here that the number of patches of size $L_P$ in the ON-state (above average emission) scales as $N(L_P) \sim L_P^{-D_e}$ in a given LAFE. The exponent $D_e$ is found to depend on the applied field (or voltage) and approaches $D_e = 2$ asymptotically.