On the transverse wave function and coherence of field emission electrons

Field emitter is an important cathode to probe atomic structures of large unit-cell crystal samples by electron imaging and diffraction experiments because of the capability to produce high brightness electron beam with a large transverse coherence length. Despite the small energy spread, normally only the near-axis part of the beam is utilized because of the large geometrical angular spread at the expense of total electron flux in typical applications such as the high-resolution electron microscopy instruments [1]. Still, achieving high current is important for recently emerging applications such as the coherent electron beam shaping experiments [2] as well as the ultra-fast time-resolved electron imaging and diffraction [3]. Despite the established use of field emitters in high-resolution transmission electron microscopy, theoretical analysis of intrinsic transverse coherence length is rare [4,5]. In this work, we extend the standard field emission theory for the calculation of electron current density [6] to the calculations of the statistically weighted wave function and compare with experiments: In recent experiments, we were able to observe the low-energy electron diffraction from suspended graphene using the double-gate field emitter array cathodes, when the field emission beamlet produced from multiple metal nanotips were maximally collimated by the on-chip beam collimator [7]. From the present theory, we found that this mathematically justified the wave function in the analysis of the transverse coherence, that was assumed by physical consideration.