Schottky effect

Without the field, the surface barrier seen by an escaping Fermi-level electron has height W equal to the local work-function.

The electric field lowers the surface barrier by an amount ΔW, and increases the emission current.

This gives the equation[1][2] where J is the emission current density, T is the temperature of the metal, W is the work function of the metal, k is the Boltzmann constant, qe is the Elementary charge, ε0 is the vacuum permittivity, and AG is the product of a universal constant A0 multiplied by a material-specific correction factor λR which is typically of order 0.5.

For electric field strengths higher than 108 V m−1, so-called Fowler–Nordheim (FN) tunneling begins to contribute significant emission current.

[3] At even higher fields, FN tunneling becomes the dominant electron emission mechanism, and the emitter operates in the so-called "cold field electron emission (CFE)" regime.

Thermionic emission can also be enhanced by interaction with other forms of excitation such as light.

As an electron leaves the metal (left) towards vacuum (right), its potential energy is not the red static potential, but rather the blue curved potential due to its own image charge .
Schottky-emitter electron source of an Electron microscope