Transverse structure of the wave function of field emission electron beam determined by intrinsic transverse energy

The average transverse energy of field emission electrons at the cathode surface is one of the key factors that determines the virtual source size, hence the transverse spatial coherence of field emitters. In the past, the subject has been intensively studied by classical electron optics analysis but its wave optical studies are rare. In this work, we therefore aim to elucidate the influence of the transverse momentum in solid on the transverse structure of the wave function of field emission electrons. From the calculation extending the standard field emission theory within the WKB approximation for model planar free-electron metal, we obtained a Gaussian-beam-type wave function that exhibits a minimum transverse width at the cathode surface as determined by the average transverse energy and propagates the first few nanometers with a limited transverse spread. At far field, the wave function spreads as the electron propagates away from the cathode surface. Comparison with classical results indicated that, in the present planar field emitter model, the neglect of the three-dimensional potential around the tip apexes of actual field emitters underestimates the transverse spread up to a factor of 2. However, when the cathode size is finite and the electrons in the solid are phase-coherent within the source area, the transverse spread is much smaller than that of the point-source wave function. Our result indicates that the intrinsic transverse emittance of a finite size fully coherent field emitter is much smaller than the value predicted by classical analysis. Published by AIP Publishing.