Reconstruction of Fowler-Nordheim-type equations

Start date

01 July 2011

End date

20 July 2012


This project forms part of research in the theory of high electric field nanoscience.

Fowler-Nordheim (FN) tunnelling is electron tunnelling through an approximately triangular barrier. Deep FN tunnelling occurs at forwards energy levels well below the barrier top. Cold field electron emission (CFE) is electron emission in a regime where most electrons escape by deep FN tunnelling from states close to the Fermi level.

The family of Fowler-Nordheim-type equations describes CFE from the conduction band of large metallic emitters. They give the emission current i in terms of the notional emission area An, the emission current density J, the barrier field F and the local thermodynamic work-function , by expressions of the form [1]:

i  =  An J  = AnZa–1F2 PFexp[–Fb3/2/ F]. (1)

Here, a and b are the First and Second FN Constants, PF is a transmission pre-factor [2], and vF and Z are correction factors relating to the barrier and to electron supply. F is related to the applied voltage V by F=VV. The literature uses several variants of eq. (1).

Aims and objectives

This project aims to put FN-type equations and their use onto a secure, well-structured and transparent intellectual basis, and to assess the approximations involved in using customisations of eq. (1). Tunnelling theory is a separate project. Progress includes:

  1. Formulation of eq. (1) as a physically complete form, and provisional estimation of the uncertainties associated with the correction factors F, PF and Z [1];
  2. Restatement (with proof) of the principles involved in the summation over emitter states that is part of the derivation of FN-type equations [2];
  3. Modification of eq. (1) to apply to large-area field emitters (LAFEs), by introducing the concept of area efficiency of emission [3];
  4. For the SN barrier, detailed examination of the quantitative effects of a certain class approximations involved in determining values of PF and Z (with Mayer) [4];
  5. Development of better theory for analysing i-V characteristics using FN theory (in progress, with Mousa and Fischer).



J.H.B. Deane (Surrey); A. Mayer (Belgium); M.S. Mousa (Jordan).


References [2-5] are outputs in the academic period 2009-2011.

  1. R.G. Forbes, J. Vac. Sci. Technol. B 26, 788 (2008).
  2. R.G. Forbes, J. Vac. Sci. Technol. B 28, 1326 (2010).
  3. R. G. Forbes, J. Vac. Sci. Technol. B 27, 1200 (2009).
  4. A. Mayer, J.H.B. Deane & R.G. Forbes, Poster at 23rd IVNC, Palo Alto, July 2010.
  5. A. Fischer, M.S. Mousa & R.G. Forbes, Posters at 23rd IVNC, Palo Alto, July 2010.