The tunneling potential for field emission from nanotips

Abstract

In the quasi-planar approximation of field emission, the potential energy due to an external electrostatic field E 0 is expressed as −eγE 0 ∆s where ∆s is the perpendicular distance from the emission site and γ is the local field enhancement factor on the surface of the emitter. We show that for curved emitter tips, the current density can be accurately computed if terms involving (∆s/R 2 ) 2 and (∆s/R 2 ) 3 are incorporated in the potential where R 2 is the second (smaller) principle radius of curvature. The result is established analytically for the hemiellipsoid and hyperboloid emitters and it is found that for sharply curved emitters, the expansion coefficients are equal and coincide with that of a sphere. The expansion seems to be applicable to generic emitters as demonstrated numerically for an emitter with a conical base and quadratic tip. The correction terms in the potential are adequate for R a 2 nm for local field strengths of 5 V/nm or higher. The result can also be used for nano-tipped emitter arrays or even a randomly placed bunch of sharp emitters.

Shielding effects in random large area field emitters, the field enhancement factor distribution, and current calculation

“…The strides achieved in the past decades in our ability to pattern arrays of pointed emitters [3][4][5] and the discovery of carbon nanotubes (CNT) as a suitable material 6,7 for stable operation, have led to vigorous research activity in this direction. These efforts are supported by theoretical studies on large area field emitters (LAFE) [8][9][10][11][12][13][14] and corrections for nano-tipped emitters [15][16][17] to the planar Fowler-Nordheim (FN) formula for current density [18][19][20][21] . Despite this progress, there are several challenges in our ability to predict theoretically, the current emitted by a single emitter or a cluster of emitters placed randomly or in an array.…”

Approximate universality in the tunneling potential for curved field emitters -- a line charge model approach

“…Eq. ( 4) was also found to hold for other shapes such as a rounded cone and cylinder, both of which were modelled numerically using a suitable nonlinear line charge distribution 22 . This paper aims to put Eq.…”

“…The question of correction to the potential (V ext ) due to an applied external electric field near a sharp tip has been considered earlier 21,22 . The first correction along the axis of an axially symmetric emitter was shown to be 21…”

“…In Ref. [22], it was established that for a hemiellipsoid, a hyperboloid and a hemisphere diode, the first correction (Eq. 3) holds for other points near the apex as well.…”

Higher order curvature corrections to the field emission current density

“…Using analytically solvable models such as the hemiellipsoid and hyperboloid diodes, it was subsequently 18 shown in 2018 that at points close to the apex, the potential has the form…”