Coherent field emission image of graphene predicted with a microscopic theory

Abstract

Electrons in the mono-layer atomic sheet of graphene have a long coherence length of the order of micrometers. We will show that this coherence is transmitted into the vacuum via electric field assisted electron emission from the graphene edge. The emission current density is given analytically. The parity of the carbon pi-electrons leads to an image whose center is dark as a result of interference. A dragonfly pattern with a dark body perpendicular to the edge is predicted for the armchair edge whose emission current density is vanishing with the mixing angle of the pseudo-spin. The interference pattern may be observed up to temperatures of thousand Kelvin as evidence of coherent field emission. Moreover, this phenomenon leads to a novel coherent electron line source that can produce interference patterns of extended objects with linear sizes comparable to the length of the graphene edge. Nano-emitters of various shapes and materials have been used for cold field electron emission (CFE) with the aim of constructing a practical microelectronic vacuum electron source that may be used in flat-panel displays, electron microscopes and parallel e-beam lithography systems. The feature most important for such applications is the high aspect ratio of nano-tips enhancing the apex field to the point where field electron emission can be driven by macroscopic electric fields of about ten volts per micrometer or even less. Two successful examples of this approach are Spindt-type cathodes[1] and single atom field emission tips. [2,3] In the context of the present paper we should note that in the latter the emission volume is the terminating atom so that emitted electrons are spatially coherent as proven succinctly in in-line electron holography.[4] The coherent field emission of carbon-based nano-emitters has been confirmed with the perfect interference fringes produced by the electron beam from carbon nanotube CFE. [5,6] Several groups have demonstrated that graphene does show promising CFE properties such as a low emission threshold field and large emission current density.[7-13] What we will show in this paper is that the unique electronic properties of graphene and its monolayer atomic structure result in a novel electron line source with coherent field emission that can produce interference patterns of extended objects with linear sizes comparable to the length of the graphene edge. What makes graphene unique is that its two-dimensional electron system is well described by a superposition of 2p carbon orbitals and that its low energy excitations behave like Dirac particles. [14][15][16] Due to the perfect atomic structure, * corresponding author: stslzb@mail.sysu.edu.cn electrons in graphene have long coherent lengths (a few micrometers at room temperatures [17]). We will show that the emission electrons will carry this coherence of the quantum states of graphene into the vacuum.The challenge is to explain/predict the coherent aspect of the CFE image. Because of its wave nature coherent emission electrons must have been ...

Field emission patterns showing symmetry of electronic states in graphene edges

“…Araidai and Watanabe carried out theoretical studies and predicted the contributions of dangling bonds to FE. Li et al . predicted a dragonfly‐shaped FEM image of graphene that was obtained by calculation of coherent interference of electrons emitted from π orbitals of the armchair edge.…”

Manifesting pseudo-spin polarization of graphene with field emission image

“…A uniform macroscopic electric field F 0 is applied on the graphene in the negative x direction by an anode that is far away from the graphene presumably. In a proper approximation [18], the potential barrier for the p-th eigenstate of graphene has the form…”

“…The z-directional distribution is given by the square-normalized Let Δ be the edge energy shift with respect to the intrinsic Fermi level. A finite Δ may be originated from the band-bending as a result of the field penetration [16,18].…”

Edge field emission of large-area single layer graphene

“…Theoretical considerations which take into account a specific band structure of graphene and the field penetration effect also predict deviations from FN law [14] and reveal additional peaks in energy distribution of emitted electrons [15]. Moreover, relativistic [16] and electron interference [17,18] effects can in principle be identified in the study of FE patterns and currentvoltage characteristics of graphene emitters. Therefore, the experimental study of FE can be a powerful probing technique able to reveal various features of graphene's electronic properties.…”