Generalized fractional Wentzel-Kramers-Brillouin approximation for electron tunnelling across rough metal interface

M. W. Ramzan, K. Riaz, M. Q. Mehmood*, M. Zubair*, Y. Massoud*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Article number20220600
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2270
StatePublished - Feb 22 2023

Bibliographical note

Funding Information:
This work is supported in part by the Innovative Technologies Laboratories—KAUST and in part by the ITU Pre-doctoral Fellowship.

Publisher Copyright:
© 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


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