D.29 The thermionic emission equation

This note derives the thermionic emission equation for a typical metal following [42, p. 364ff]. The derivation is semi-classical.

To simplify the analysis, it will be assumed that the relevant electrons in the interior of the metal can be modeled as a free-electron gas. In other words, it will be assumed that in the interior of the metal the forces from surrounding particles come from all directions and so tend to average out.

(The free-electron gas assumption is typically qualitatively
reasonable for the valence electrons of interest if you define the
zero of the kinetic energy of the gas to be at the bottom of the
conduction band. You can also reduce errors by replacing the true
mass of the electron by some suitable effective mass.

But the zero of the energy drops out in the final expression, and the
effective mass of typical simple metals is not greatly different from
the true mass. See chapter 6.22.3 for more on these issues.)

Assume that the surface through which the electrons escape is normal
to the

where

An electron can only escape if its energy

where

That is because the initial exponential is a rewritten Maxwell-Boltzmann distribution (6.21) that gives the number of electrons per state, while the remainder is the number of states in the energy range according to the density of states (6.6).

Normally, the typical thermal energy

Further, even if an electron has in principle sufficient energy to
escape, it can only do so if enough of its momentum is in the

where the final equality applies since the kinetic energy is proportional to the square momentum.

Since the velocity for the escaping electrons is mostly in the

Putting it all together, the current density becomes

Rewriting in terms of a new integration variable

If an external electric field

The bottom line is that it seems to the escaping electron that it is
pulled back not by surface charges, but by a positron mirror image of
itself. Therefore, including now an additional external electrical
field, the total potential in the later stages of escape is:

where

workon the image. If that is confusing, just write down the force on the electron and integrate it to find its potential energy.

If there is no external field, the maximum potential energy that the
electron must achieve occurs at infinite distance