Verifications of Schottky's Conjecture

Abstract

Schottky's Conjecture posits that the geometric field enhancement produced by a hybrid shape formed from a small perturbation on a larger base is the product of the individual field enhancement factors of the base and perturbation in isolation. This is a powerful concept with practical applications to understanding field emitter design and operation, as actual field emitters have complicated surface shapes with structure and, therefore, contributions to field enhancement, occurring simultaneously on many length scales. Recent studies of the Conjecture imply that the degree to which it accurately predicts a hybrid structure's total field enhancement depends on the degree of self-similarity between the base and perturbation shapes. To explore these aspects of the Conjecture, we have used the zero-potential surface produced by simple charge distributions to produce compound shapes with small perturbations on larger base structures. In the limit of small perturbation strength, these simple models adequately approximate idealized compound shapes, such a hemisphere-on-hemisphere geometry. Changing the base shape and the location of the perturbation allows us to selectively degrade the self-similarity present in the problem geometry. Proofs of the Schottky Conjecture in the limit of small perturbation strength will be provided, using these techniques, for hemisphere-on-hemisphere and half-cylinder-on-half-cylinder systems—the latter being the geometry considered by Schottky and for which the Conjecture was first proposed, but not proven.

Field emitter electrostatics: a review with special emphasis on modern high-precision finite-element modelling

“…Various other methods of investigating two-stage fieldenhancement effects and/or the limitations of equation ( 89) have been explored in the literature, for example [113,146,148,[151][152][153][154][155][156][157][158][159][160].…”

Field emitter electrostatics: a review with special emphasis on modern high-precision finite-element modelling

“…Various other methods of investigating two-stage fieldenhancement effects and/or the limitations of eq. ( 90) have been explored in the literature, for example [109,142,144,[147][148][149][150][151][152][153][154][155][156].…”

Analytical proof of Schottky Conjecture for multi-stage field emitters

“…Despite the simplicity of this argument, there is no general proof of SC, except for the very particular cases presented in [20,28]. Moreover, there is no analytical demonstration of SC for multi-stage field emitters even in this situation, although there are interesting evidences obtained from techniques using charge-models [32][33][34]. Indeed, there is no analytical proof of SC even in the case of the superposition of two cylindrical protrusions, the case originally studied by Schottky when he proposed his conjecture [27].…”

“…Indeed, there is no analytical proof of SC even in the case of the superposition of two cylindrical protrusions, the case originally studied by Schottky when he proposed his conjecture [27]. Nevertheless, some point-or line-charge models are able to provide shapes of emitters very similar to the case of ideal geometries, such as superimposed spherical, elliptical or cylindrical protrusions, and it is proved that these models may provide the FEFs predicted by SC for the idealized structures under certain limits [33,34].…”