Resonance scattering across the superlattice barrier and the dimensional quantization

Abstract

Carbon nano-cluster cathodes exhibit a low threshold electron emission, which is 2-3 orders lower than on metals and semiconductors. We confirm the effect by direct experiments with graphene structures. We are suggesting a model based on the interference electrons wave function in 3D-space charge region of carbon structure interface with vacuum. The low-threshold emission is explained, in frames of the model, by the resonance properties of the barrier formed on the interface. Also in the following topics: interpretation of recent experimental findings for saturation of the field emission; local spectral analysis of multidimensional periodic lattices: dispersion via DN-map; examples of iso-energetic surfaces associated with solvable models of periodic lattice; Lagrangian version of the operator extension algorithm; solvable models of selected one-body spectral problems; quantum dot attached to the node of a quantum graph; a solvable model of a discrete lattice and spectral structure of a 1D superlattice via analytic perturbation procedure.Keywords: periodic interface, carbon cover, Weyl-Titchmarsh function, resonance, field emission, DN-map. Received: 23 August 2016Revised: 1 September 2016 An example: resonance concepts of the low-threshold field emissionIn numerous recent experiments, see for instance [1][2][3][4][5][6][7] extremely low-threshold field emission from metallic cathodes under carbon deposit was observed for electric fields (10 4 − 10 5 V /cm). This is a surprisingly strong effect, because the field initiating a noticeable emission (10 −10 − 10 −9 A) from these materials is by 2-3 orders of magnitude less than the field required for the field-emission from the traditional metals and semiconductors. Despite an obviously unusual nature of the effect, numerous authors, see for instance [5,6] attempted to explain the low-threshold phenomenon trivially with use of the classical Fowler-Nordheim techniques, based on enhancing of the field at the micro-protrusions. They assume that the local field F s near the emitting center is calculated as F 0 = γF 0 , where γ is the field enhancement coefficient, defined by the micro-geometry, and F 0 is the field of the equivalent flat capacitor. This completely classical explanation of the low-threshold emission phenomenon is not universal, and certainly non-valid for deposit, considered in our recent papers [3,7] because the surface of the carbon flakes, obtained by the detonation synthesis, are perfectly smooth, see the flakes (see Fig. 1) under maximal magnifications.with rare and relatively small protrusions. These protrusions are able to lower the threshold 5-fold, while 10 2 times lowering is observed in our experiments. We suggested in in [3,7] an alternative explanation of the threshold lowering (field enhancement) based on the dimensional (size-) quantization in the under-surface spacecharge region. The classical Fowler-Nordheim techniques for calculating the transmission coefficient T for simple rectangular potential barrier, see [8], gives an exponentially...

Model of tunnelling through double quantum layer in a magnetic field

“…Here, e is the electron charge,h is the Planck's constant. The geometry of a nanostructure can influence transmission coefficient significantly [17].…”