# 3.3.2 Thermionic emission current

Thermionic emission is the process where electrons are emitted across a barrier. The driving force of this process is the thermal energy which provides a non-zero density of carriers at energies larger than the confining barrier. The current density associated with this process is obtained from:

(te1)

where q is the electronic charge, n(E) is the density of electrons per unit energy and per unit volume and vx(E) is the velocity of the electrons with which they approach the barrier. The integral is to be taken over all electron energies large enough energy to surmount the barrier and must include only electrons moving towards the barrier. The electron density is obtained by multiplying the density of states function with the Fermi function yielding:

(te2)

The energy can be written as a function of the electron velocity using:

(te3)

which yields:

(te4)

where the Fermi function is approximated by the Maxwell-Boltzmann distribution function. This is based on the assumption that the Fermi energy is at least 3kT below the top of the barrier as is typically the case. The energy can be further expressed as a function of the velocity components in the x, y and z direction:

(te5)

so that the integral can be written as a product of three integrals, one for each velocity component. The intergral over vy and vz extends from minus to plus infinity, while the integral over vx start from the minimum velocity in the positive x direction needed to overcome the barrier, vx,min, to infinity, yielding:

(te6)

The integrals over vy and vz can be solved using the following definite integral:

(te7)

while the minimum velocity in the x direction required to overcome the barrier is obtained by setting the kinetic energy equal to the barrier height:

(te8)

which yields:

(te9)

This can be rewritten as:

(te10)

where AR is refered to as being the Richardson constant and is given by:

(te11)

The expression for the current due to thermionic emission can also be written as a function of the average velocity with which the electrons at the interface approach the barrier. This velocity is refered to as the Richardson velocity given by:

So that the current density becomes:

3.3.1 ¬ ­ ® 3.3.3