Analytical treatment of cold field electron emission from a nanowall emitter, including quantum confinement effects

Abstract

An elementary approximate analytical treatment of cold field electron emission (CFE) from a classical nanowall (i.e. a blade-like conducting structure on a flat surface) is presented. This paper first discusses basic CFE theory for situations where quantum confinement occurs transverse to the emitting direction. It develops an abstract CFE equation more general than Fowler–Nordheim type (FN-type) equations, and then applies this to classical nanowalls. With sharp emitters, the field in the tunnelling barrier may diminish rapidly with distance; an expression for the on-axis transmission coefficient for nanowalls is derived by conformal transformation. These two effects interact to generate complex emission physics, and lead to regime-dependent equations different from FN-type equations. Thus: (i) the zero-field barrier height H R for the highest occupied state at 0 K is not equal to the local thermodynamic work-function ϕ , and H R rather than ϕ appears in equations; (ii) in the exponent, the power dependence on macroscopic field F M can be F −2 M rather than F −1 M ; (iii) in the pre-exponential, explicit power dependences on F M and H R differ from FN-type equations. Departures of this general kind are expected when nanoscale quantum confinement occurs. FN-type equations are the equations that apply when no quantum confinement occurs.