For the Fowler-Nordheim tunneling current density :
- J = K1 × E2 × e-K2/E
"The point is that the current increases with the voltage squared multiplied by an exponential increase with inverse voltage. While the second factor, E2, obviously increases rapidly with voltage, the third factor, the exponential, deserves another sentence"
Compare Fowler-Nordheim with the Richardson-Dushman Equation for Thermionic Emission:
http://www.virginia.edu/ep/SurfaceScience/thermion.html
http://www.virginia.edu/ep/SurfaceScience/electron.html
"Jellium model. The charge of the ion cores is spread over the solid (jellium) and the electrons then move in the potential produced by this jellium. Density functional theory is used where the properties of the electron "gas" depends only on the electron density. This is sometimes refined by adding non-local corrections to the properties. We note that a uniform electron gas is not a good approximation at the surface"
Surface dipole
"In the jellium model, the positive background terminates abruptly at the surface (jellium edge). The electrons are allowed to readjust. The finite wavelength of the electrons causes Friedel oscillations in the electron density near the surface (this is analogous to what happens when one tries to express a step function as a sum of sinusoidal functions up to a maximum frequency). The sharpness of the jellium and the spread of the electron density (which decays exponentially outside the solid) produces a deficit of electrons just inside the jellium edge and an excess outside. This produces a dipole layer. This dipole attracts electrons to the surface and produces a step in the surface potential"
"The total potential seen by the electrons (inner potential) is the electrostatic potential caused by the distribution of charge density (Poisson equation), plus the exchange-correlation potential produced by electron-electron correlations. The exchange-correlation potential evolves into the image potential outside the solid. The electrostatic potential includes the surface dipole whose value depends on the roughness of the surface, both at the atomic scale and that produced by steps. Thus, the work function, which is the inner potential minus the Fermi energy, depends on the crystallographic orientation of the face of the crystal. For instance, the work function of Cu (fcc) is 4.94 eV, 4.59 eV and 4.48 eV for the (111), (100) and (110) surfaces, respectively. The work function will be changed when permanent or induced dipoles are added during adsorption of gases on the surface. These additional dipoles can increase or decrease the w! ork function."
