The conductive rough surfaces act as an integral part of several electron devices and systems. Electron tunnelling through the potential barrier imposed by the rough metal-vacuum interface is an important mechanism of charge transport in vacuum electron devices. Here, we analytically derive a generalized current-voltage relationship with a fractional image potential barrier that considers the reduced space-dimensionality encountered by the tunnelling electrons at a rough interface, in an effective manner. The traditional Schottky-Nordhiem equation based on the Schottky image potential barrier is shown to be a limiting case of our model for a perfectly flat surface. The fractional-dimension parameter used in this model accounts for the barrier reduction due to the geometrical roughness and it can be determined by fitting our model to a given current-voltage measurement. It is shown that the application of this model could reduce the error between measured current-voltage response and theoretical estimates based on the conventional model. This work provides an analytical framework for efficient design and engineering of quantum tunnelling in practical electron devices.
|Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
|Published - Feb 22 2023
Bibliographical noteFunding Information:
This work is supported in part by the Innovative Technologies Laboratories—KAUST and in part by the ITU Pre-doctoral Fellowship.
© 2023 The Author(s).
- fractal media
- fractional calculus
- quantum tunnelling
- rough metal interface
- rough surface
- Wentzel-Kramers-Brillouin approximation
ASJC Scopus subject areas
- General Mathematics
- General Engineering
- General Physics and Astronomy