Analytic Solutions for Space-Charge-Limited Current Density From a Sharp Tip

While analytic equations for space-charge limited current density (SCLCD) have been derived for planar and nonplanar geometries, the SCLCD for emission from a sharp tip has not been derived. In this article, we use variational calculus (VC) to derive an exact analytic equation for SCLCD for a 1-D ti...

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Bibliographic Details
Published in:IEEE transactions on electron devices 2021-12, Vol.68 (12), p.6525-6531
Main Authors: Harsha, N. R. Sree, Garner, Allen L.
Format: Article
Language:English
Subjects:
Anodes
Axes of rotation
Calculus of variations
Cathodes
Charge density
Circles (geometry)
Conformal mapping
Current density
diodes
Electric potential
electron sources
Engineering
Exact solutions
Geometry
high power microwave generation
Mathematical analysis
Measurement
Nanoscale devices
Physics
space charge
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Summary:While analytic equations for space-charge limited current density (SCLCD) have been derived for planar and nonplanar geometries, the SCLCD for emission from a sharp tip has not been derived. In this article, we use variational calculus (VC) to derive an exact analytic equation for SCLCD for a 1-D tip-to-tip geometry, represented by hyperboloids in the prolate spheroidal coordinates, and recover the SCLCD for a tip-to-plate geometry as a special case. We then consider circles in the extended Poincaré disk, which is the stereographic projection of hyperboloids onto a plane, as conformal transformations to derive the SCLCD for a misaligned tip-to-tip geometry, where the axes of rotation of the hyperboloids are displaced by a small distance. This mapping technique is also applied to study the effect of a small angle tilt in a tilted tip-to-tip geometry, where the axes of rotation of the hyperboloids meet at an angle.
ISSN:0018-9383
1557-9646
DOI:10.1109/TED.2021.3122393